Periodic Monopoles and Difference Modules (Lecture Notes in Mathematics) 3030944999, 9783030944995

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Lecture Notes in Mathematics  2300

Takuro Mochizuki

Periodic Monopoles and Difference Modules

Lecture Notes in Mathematics Volume 2300

Editors-in-Chief Jean-Michel Morel, CMLA, ENS, Cachan, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Alessio Figalli, ETH Zurich, Zurich, Switzerland Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Sylvia Serfaty, NYU Courant, New York, NY, USA Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

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Takuro Mochizuki

Periodic Monopoles and Difference Modules

Takuro Mochizuki Research Institute for Mathematical Sciences Kyoto University Kyoto, Japan

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-030-94499-5 ISBN 978-3-030-94500-8 (eBook) https://doi.org/10.1007/978-3-030-94500-8 Mathematics Subject Classification: 53C07, 58E15, 14D21, 81T13 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to Professor Carlos Simpson on the occasion of his 60th birthday.

Preface

In this monograph, we shall study the relationship between polystable parabolic difference modules and singular monopoles on R3 with one periodicity. It is a new equivalence between objects in algebraic geometry and in differential geometry. On one hand, a monopole on R3 is a vector bundle E with a Hermitian metric h, a unitary connection ∇ and an anti-Hermitian endomorphism φ satisfying the Bogomolny equation F (∇) = ∗∇φ, where F (∇) denotes the curvature of ∇, and ∗ is the Hodge star operation. Recall that the Bogomolny equation is a dimensional reduction of the anti-self duality equation in the gauge theory, and that it is a nonabelian generalization of Maxwell’s equation under some constraints. A monopole (E, h, ∇, φ) is called periodic if it is invariant under the translation by a vector. Similarly, it is called doubly periodic (resp. triply periodic) if it is invariant under the translation by two (resp. three) linearly independent vectors. More generally, we shall consider a monopole (E, h, ∇, φ) which can be singular at a discrete subset S, i.e., (E, h, ∇, φ) is defined on R3 \S. We should impose some asymptotic conditions to the monopoles, called of GCK-type. On the other hand, for  ∈ C, a -difference module means, in this monograph, a finite dimensional C(y)-vector space V equipped with a C-linear automorphism Φ ∗ such that Φ ∗ (f s) = f (y + )Φ ∗ (s) for any f ∈ C(y) and s ∈ V . If  = 0, it is just a C(y)-vector space with a C(y)-linear automorphism. We shall introduce the notion of parabolic structure for -difference modules in this monograph, as we consider a parabolic structure for a vector bundle on a punctured Riemann surface. A -difference module V equipped with a parabolic structure is called a parabolic -difference module, and denoted by V ∗ . We obtain the number deg(V ∗ ) ∈ R called the degree, which is an analogue of the degree of a vector bundle on a compact Riemann surface. The number μ(V ∗ ) = deg(V ∗ )/ dimC(y) V is called the slope. Any -difference submodule V  ⊂ V is naturally enhanced to a parabolic -difference submodule V ∗ ⊂ V ∗ . In this situation, it is standard in algebraic geometry to define that V ∗ is stable if the slope of V ∗ is strictly smaller than the slope of V ∗ for any -difference proper submodule V  of V , and that V ∗ is polystable if it is a direct sum of stable ones V i∗ with the same slope.

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It is our purpose in this monograph to show that there exist equivalences between periodic singular monopoles and polystable parabolic difference modules of degree 0. We also study the triply periodic case [67] and the doubly periodic case [70] in the other papers. I hope that these studies will provide us with one of the starting points of a new interesting investigation for difference modules from the viewpoint of the non-abelian Hodge theory, as in the case of the equivalences for harmonic bundles, flat bundles and Higgs bundles. I am also partially motivated by the following basic question, which has attracted me for years. Let X1 be a compact space, and let X2 be a non-compact space. Then, how we can relate the asymptotic behaviour of good differential geometric objects on X1 × X2 with the asymptotic behaviour of good differential geometric objects on X2 which is obtained as the dimensional reduction? One of the purposes in this monograph is to pursue it in the case X1 = S 1 , X2 = R2 , monopoles on S 1 ×R2 and harmonic bundles on R2 . Relatedly, it was interesting for me to obtain equivalences between difference modules and mini-holomorphic bundles in the formal level, and that it is useful to compare the asymptotic conditions for monopoles and harmonic bundles. I originally studied the Nahm transforms between periodic monopoles and harmonic bundles on P1 [69], as a refinement of [13, 14], and I intended to prove the equivalences between periodic monopoles and parabolic difference modules as a consequence of the Kobayashi-Hitchin correspondence for wild harmonic bundles. However, the progress in the theory of Kobayashi-Hitchin correspondence on a non-compact space with infinite volume [66] allows us to go along a more direct route. Hence, I eventually omitted the Nahm transform from this monograph, but I still include an explanation that a parabolic structure of a filtered λ-flat bundle is transformed to a parabolic structure at finite place of a difference module, which was a little surprising to me. As mentioned above, it is my most ambitious hope that this monograph will lead us to a new aspect of the non-abelian Hodge theory. More modestly, I hope that this monograph will be useful in the mathematical study of monopoles and the differential geometric study of difference modules. I also expect that several concepts in this monograph will be significant for the further studies of monopoles and difference modules. For example, the notion of parabolic structure seems essential in the algebraic geometric study of the moduli space of difference modules. Kyoto, Japan November 2021

Takuro Mochizuki

Acknowledgements

It is my pleasure to thank Hiraku Nakajima whose excellent lectures attracted me to the study of monopoles. A part of this study was done during my stay at the University of Melbourne, and I am grateful to Kari Vilonen and Ting Xue for their excellent hospitality and support. I appreciate Carlos Simpson whose works provide the most important foundation with this study. During the study, I realized how deeply influenced I am by the ideas of Nigel Hitchin. This work is under the direct influence of the interesting works of Benoit Charbonneau and Jacques Hurtubise [11], and Sergey Cherkis and Anton Kapustin [13, 14]. I remember that Sergey asked me some questions in 2002 when we met at the Institute for Advanced Study, and I hope that this monograph is a partial answer after almost 20 years. I was inspired by a talk of Hurtubise in a conference held at the Tata Institute of Fundamental Research in 2009. I thank Maxim Kontsevich and Yan Soibelman for their comments and discussions. I am grateful to Claude Sabbah for his kindness and discussions on many occasions. I appreciate Masaki Tsukamoto for asking a question about parabolic structure of doubly periodic instantons, which encouraged me to study instantons and monopoles, not only harmonic bundles. I thank Szilard Szabo who first attracted my attention to Nahm transforms. I appreciate Motohico Mulase for discussions and encouragements. I am grateful to Masaki Yoshino for our discussions, which were particularly useful to keep my interest in monopoles. I am grateful to Indranil Biswas for inviting me several times To the Tata Institute of Fundamental Research, which was quite helpful for my study of monopoles. I appreciate Masa-Hiko Saito and Atsushi Moriwaki for their generous supports. I thank Yoshifumi Tsuchimoto and Akira Ishii for their constant encouragement. I am heartily grateful to the reviewers for their careful and patient readings and for their constructive comments to improve this manuscript. I thank the editorial board of Springer Lecture Notes in Mathematics and Ute McCrory for their helpful comments. Special thanks go to Pierre Deligne, Kenji Fukaya, William Fulton, David Gieseker, Akira Kono, Mikiya Masuda, Tomohide Terasoma and Michael Thaddeus for their supports in my early career.

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Acknowledgements

My interest in “Kobayashi-Hitchin correspondence” was renewed when I made a preparation for a talk in the 16th Oka Symposium, which drove me to this study. I thank the organizers, particularly Junichi Matsuzawa and Ken-ichi Yoshikawa. I wrote this monograph at the Research Institute for Mathematical Sciences, Kyoto University. I benefited from my visits to the University of Melbourne and Tata Institute of Fundamental Research for my study of monopoles. I thank the institutions for the excellent research environment. It was beneficial in improving this manuscript, to make preparations for the lecture series at Osaka University and Nagoya University, and for the talks at various conferences and workshops, for which I heartily thank the organizers. I am partially supported by the Grant-in-Aid for Scientific Research (S) (No. 17H06127), the Grant-in-Aid for Scientific Research (S) (No. 16H06335), the Grant-in-Aid for Scientific Research (A) (No. 21H04429), the Grant-in-Aid for Scientific Research (C) (No. 15K04843), and the Grant-in-Aid for Scientific Research (C) (No. 20K03609), Japan Society for the Promotion of Science.

Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Monopoles of GCK-Type .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Previous Works on Monopoles and Algebraic Objects .. . . . . . . . . . . . 1.3.1 SU(2)-Monopoles with Finite Energy on R3 .. . . . . . . . . . . . . 1.3.2 The Correspondence due to Charbonneau and Hurtubise . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.3 Remark .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Review of the Kobayashi-Hitchin Correspondences for λ-Flat Bundles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 Harmonic Bundles and Their Underlying λ-Flat Bundles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.2 Kobayashi-Hitchin Correspondences in the Smooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.3 Tame Harmonic Bundles and Regular Filtered λ-Flat Bundles . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.4 Wild Harmonic Bundles and Good Filtered λ-Flat Bundles . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Equivariant Instantons and the Underlying Holomorphic Objects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.1 Instantons and the Underlying Holomorphic Bundles . . . . 1.5.2 Instantons and Harmonic Bundles . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.3 Instantons and Monopoles . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.4 Instantons and Monopoles as Harmonic Bundles of Infinite Rank . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Difference Modules with Parabolic Structure .. .. . . . . . . . . . . . . . . . . . . . 1.6.1 Difference Modules . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6.2 Parabolic Structure of Difference Modules at Finite Place . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6.3 Good Parabolic Structure at ∞ . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6.4 Parabolic Difference Modules . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 3 4 5 7 8 9 9 10 11 12 14 14 14 15 16 18 18 19 19 20 xi

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1.6.5 1.6.6

1.7

1.8

2

Degree and Stability Condition . . . . . . . .. . . . . . . . . . . . . . . . . . . . Easy Examples of Stable Parabolic Difference Modules (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6.7 Easy Examples of Stable Parabolic Difference Modules (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Kobayashi-Hitchin Correspondences for Periodic Monopoles.. . . . 1.7.1 The Correspondence in the Case λ = 0 . . . . . . . . . . . . . . . . . . . 1.7.2 The Correspondences in the General Case . . . . . . . . . . . . . . . . Asymptotic Behaviour of Periodic Monopoles of GCK-Type . . . . . 1.8.1 Setting .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8.2 Decomposition of Mini-holomorphic Bundles .. . . . . . . . . . . 1.8.3 The Induced Higgs Bundles .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8.4 Asymptotic Orthogonality . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8.5 Curvature Decay .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8.6 The Filtered Extension in the Case λ = 0 . . . . . . . . . . . . . . . . . 1.8.7 The Filtered Extension for General λ . .. . . . . . . . . . . . . . . . . . . .

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Outline of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Mini-Complex Structures on 3-Manifolds . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Mini-Holomorphic Functions on R × C . . . . . . . . . . . . . . . . . . 2.2.2 Mini-Complex Structure on Three-Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.3 Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.4 Cotangent Bundles . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.5 Meromorphic Functions .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Mini-Holomorphic Bundles . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Mini-Holomorphic Bundles . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Metrics and the Induced Operators . . . .. . . . . . . . . . . . . . . . . . . . 2.3.3 Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.4 Scattering Maps. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.5 Dirac Type Singularity of Mini-Holomorphic Bundles .. . 2.3.6 Kronheimer Resolution of Dirac Type Singularity.. . . . . . . 2.3.7 Precise Description of Dirac Type Singularities . . . . . . . . . . 2.3.8 Subbundles and Quotient Bundles . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.9 Basic Functoriality . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Monopoles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 Monopoles and Mini-Holomorphic Bundles . . . . . . . . . . . . . . 2.4.2 Euclidean Monopoles . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.3 Dirac Type Singularity . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.4 Basic Functoriality . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Dimensional Reduction from 4D to 3D . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.1 Instantons Induced by Monopoles . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.2 Holomorphic Bundles and Mini-Holomorphic Bundles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

20 22 23 24 25 29 35 36 36 37 38 38 39 40 43 43 44 44 45 45 46 47 47 47 48 48 49 49 50 50 51 52 53 53 53 55 57 58 58 59

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2.6

60 60

Dimensional Reduction from 3D to 2D . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.1 Monopoles Induced by Harmonic Bundles.. . . . . . . . . . . . . . . 2.6.2 Mini-Holomorphic Bundles Induced by Holomorphic Bundles with a Higgs Field . . . . . . . . . . . . . . . . . 2.6.3 Mini-Holomorphic Sections and Monodromy . . . . . . . . . . . . 2.6.4 Appendix: Monopoles as Harmonic Bundles of Infinite Rank . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7 Twistor Families of Mini-Complex Structures on R × C and (R/T Z) × C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.1 Preliminary.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.2 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.3 Twistor Family of Complex Structures . . . . . . . . . . . . . . . . . . . . 2.7.4 Family of Mini-Complex Structures .. .. . . . . . . . . . . . . . . . . . . . 2.7.5 The Mini-Complex Coordinate System (t0 , β0 ). . . . . . . . . . . 2.7.6 The Mini-Complex Coordinate System (t1 , β1 ). . . . . . . . . . . 2.7.7 Coordinate Change . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.8 Compactification.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.9 Mini-Holomorphic Bundles Associated with Monopoles.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8 OMλ -Modules and λ-Connections .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.1 Dimensional Reduction from OMλ -Modules to λ-Flat Bundles . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.2 Comparison of Some Induced Operators .. . . . . . . . . . . . . . . . . 2.8.3 O λ -Modules and λ-Connections . . . .. . . . . . . . . . . . . . . . . . . . M 2.9 Curvatures of Mini-Holomorphic Bundles with Metric on Mλ . . . 2.9.1 Contraction of Curvature and Analytic Degree . . . . . . . . . . . 2.9.2 Chern-Weil Formula .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9.3 Another Description of G(h) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9.4 Change of Metrics . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9.5 Relation with λ-Connections .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9.6 Dimensional Reduction of Kronheimer . . . . . . . . . . . . . . . . . . . 2.9.7 Appendix: Ambiguity of the Choice of a Splitting . . . . . . . λ )-Modules . . . . . . . . . . . . . . . . . 2.10 Difference Modules and O λ (∗H∞ M \Z 2.10.1 Difference Modules with Parabolic Structure at Finite Place . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.10.2 Construction of Difference Modules from O

λ

M \Z

λ )-Modules . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . (∗H∞

2.10.3 Construction of O

λ

M \Z

(∗H λ )-Modules

60 61 62 63 63 63 63 65 65 66 67 67 68 68 68 72 74 77 77 77 78 81 82 85 87 89 89 90

from

Difference Modules . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.10.4 Appendix: Mellin Transform and Parabolic Structures at Finite Place . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.11 Filtered Prolongation of Acceptable Bundles .. .. . . . . . . . . . . . . . . . . . . . 2.11.1 Filtered Bundles on a Neighbourhood of 0 in C . . . . . . . . . .

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2.11.2 Acceptable Bundles on a Punctured Disc . . . . . . . . . . . . . . . . . 102 2.11.3 Global Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 104 3

Formal Difference Modules and Good Parabolic Structure . . . . . . . . . . 3.1 Outline of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Formal Difference Modules.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Formal Difference Modules of Level ≤1 .. . . . . . . . . . . . . . . . . 3.2.2 Formal Difference Modules of Pure Slope . . . . . . . . . . . . . . . . 3.2.3 Slope Decomposition of Formal Difference Modules . . . . 3.3 Good Filtered Bundles of Formal Difference Modules .. . . . . . . . . . . . 3.3.1 Filtered Bundles over C((yq−1 ))-Modules .. . . . . . . . . . . . . . . . . 3.3.2 Good Filtered Bundles over Formal Difference Modules .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.3 The Induced Endomorphisms on the Graded Pieces . . . . . . 3.4 Geometrization of Formal Difference Modules .. . . . . . . . . . . . . . . . . . . . 3.4.1 Ringed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.2 Some Formal Spaces . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.3 Difference Modules and OH∞,q (∗H∞,q )-Modules . . . . . . . 3.4.4 Lattices and the Induced Local Systems.. . . . . . . . . . . . . . . . . . 3.5 Filtered Bundles in the Formal Case . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.1 Pull Back and Descent of OH∞,p (∗H∞,p )-Modules .. . . . . 3.5.2 Filtered Bundles . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.3 Basic Filtered Objects with Pure Slope .. . . . . . . . . . . . . . . . . . . 3.5.4 Good Filtered Bundles over OH∞,q (∗H∞,q )-Modules with Level ≤1 . . . . . . . . . . . . . . . . . . 3.5.5 Good Filtered Bundles over OH∞,q (∗H∞,q )-Modules . . . 3.5.6 Global Lattices on the Covering Space .. . . . . . . . . . . . . . . . . . . 3.5.7 Local Lattices .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.8 Complement for Good Filtered Bundles with Level ≤1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Formal Difference Modules of Level ≤1 and Formal λ-Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.1 Formal λ-Connections .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . ∞,q . .. . . . . . . . . . . . . . . . . . . . 3.6.2 Some Sheaves of Algebras on H 3.6.3 From Formal λ-Connections to Formal Difference Modules . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.4 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.5 Example 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.6 Example 2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.7 Comparison of Good Filtered Bundles . . . . . . . . . . . . . . . . . . . . 3.6.8 Comparison of the Associated Graded Pieces.. . . . . . . . . . . . 3.6.9 Some Functoriality . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 Appendix: Pull Back and Descent in the R-Direction .. . . . . . . . . . . . . 3.7.1 Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

107 107 107 108 110 113 113 113 117 119 120 120 120 121 122 123 123 123 126 127 128 129 131 133 133 133 134 134 138 141 143 144 145 146 147 149

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Filtered Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Outline of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Filtered Bundles in the Global Case . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.1 Subbundles and Quotient Bundles . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Degree and Slope . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.3 Stability Condition . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.4 Good Filtered Bundles of Dirac Type and Parabolic Difference Modules . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Filtered Bundles on Ramified Coverings .. . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 The Case λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 Ramified Coverings for General λ . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 Filtered Bundles . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.4 Local Lattices and the Weight Filtration on the Graded Pieces . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.5 Convenient Frame . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Hermitian Metrics and Filtrations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Prolongation by Growth Conditions . . .. . . . . . . . . . . . . . . . . . . . 4.4.2 Norm Estimate for Good Filtered Bundles . . . . . . . . . . . . . . . . 4.4.3 Strong Adaptedness . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Comparison with λ-Connections . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.1 Some Sheaves on Yqλ cov and Yqλ . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.2 The Induced OBqλ -Modules from λ-Connections . . . . . . . . . 4.5.3 Norm Estimates for λ-Connections .. . .. . . . . . . . . . . . . . . . . . . . 4.5.4 Comparison of the Norm Estimates . . .. . . . . . . . . . . . . . . . . . . . 4.5.5 Non-integrable Case . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Basic Examples of Monopoles Around Infinity . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Examples of Monopoles with Pure Slope /p . .. . . . . . . . . . . . . . . . . . . . 5.1.1 Equivariance with Respect to the Z-Action . . . . . . . . . . . . . . . 5.1.2 The Underlying Mini-holomorphic Bundle at λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.3 The Underlying Mini-holomorphic Bundle at General λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.4 Complex Structures .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.5 The Induced Instanton and the Underlying Holomorphic Bundle . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.6 C ∞ -Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.7 Proof of Proposition 5.1.4 .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Examples of Monopoles Induced by wild Harmonic Bundles . . . . . 5.2.1 The Underlying Mini-holomorphic Bundle at λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 The Associated λ-Connection and the Induced Mini-holomorphic Bundle at λ. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.4 Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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151 151 151 153 154 155 156 158 159 159 161 162 163 164 164 165 166 167 167 168 170 171 173 177 177 178 179 179 180 181 182 183 184 185 186 187 188

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Examples of Monopoles Induced by Tame Harmonic Bundles (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.1 The Mini-holomorphic Bundle at λ = 0.. . . . . . . . . . . . . . . . . . 5.3.2 The Mini-holomorphic Bundle at λ. . . .. . . . . . . . . . . . . . . . . . . . Examples of Monopoles Induced by Tame Harmonic Bundles (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.1 Case of λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.2 Case of λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Asymptotic Behaviour of Periodic Monopoles Around Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Outline of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.2 Decomposition of Holomorphic Bundles with an Automorphism .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.3 Some Basic Mini-Holomorphic Bundles on Bq0∗ (R) . . . . . 6.2.4 Decomposition of Mini-Holomorphic Bundles . . . . . . . . . . . 6.3 Estimate of Periodic Monopoles Around Infinity.. . . . . . . . . . . . . . . . . . 6.3.1 Setting .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.2 Asymptotic Orthogonality of the Decomposition (6.5) .. . 6.3.3 Eigen Decomposition in the Level 1 . . .. . . . . . . . . . . . . . . . . . . . 6.3.4 Asymptotic Harmonic Bundles . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.5 Curvature .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.6 Another Equivalent Decay Condition .. . . . . . . . . . . . . . . . . . . . 6.4 Connections and Orthogonal Decompositions ... . . . . . . . . . . . . . . . . . . . 6.4.1 Statement.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.2 Preliminary.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.3 Step 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.4 Step 2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Some Lemmas from Linear Algebra . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5.1 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5.2 Almost Commuting Hermitian Matrix and Anti-Hermitian Matrix . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5.3 Decomposition of Finite Tuples in Metric Spaces (Appendix).. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Vector Bundles with a Connection on a Circle (I) . . . . . . . . . . . . . . . . . . 6.6.1 Statement.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6.2 Preliminary.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6.3 A Decomposition of Function Spaces .. . . . . . . . . . . . . . . . . . . . 6.6.4 Gauge Transformation .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6.5 Proof of Proposition 6.6.1 .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 Vector Bundles with a Connection on a Circle (II) . . . . . . . . . . . . . . . . . 6.7.1 Additional Assumption on the Eigenvalues of the Monodromy .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

189 189 190 191 191 192 195 195 195 195 196 197 198 199 199 200 201 203 204 205 205 205 206 208 211 212 212 214 215 216 216 217 218 218 221 221 221

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6.7.2 Gauge Transformations.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 222 6.7.3 Comparison of the Decompositions . . .. . . . . . . . . . . . . . . . . . . . 224 Proof of Theorem 6.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 224

7

The Filtered Bundles Associated with Periodic Monopoles .. . . . . . . . . . . 7.1 Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.1 Some Spaces and Morphisms . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.2 Neighbourhoods of Infinity . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.3 Norm of Differential Forms .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Meromorphic Prolongation . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.2 Proof.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Filtered Prolongation .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.1 Statement.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.2 Refined Statement . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.3 Norm Estimate . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.4 Step 0 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.5 Step 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.6 Step 2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.7 Step 3 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.8 Step 4 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.9 Step 5 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.10 Step 6 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.11 Proof of Proposition 7.3.5 .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Strong Adaptedness and the GCK-Condition .. .. . . . . . . . . . . . . . . . . . . . 7.4.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.2 Some Estimates for Tame Harmonic Bundles (1) .. . . . . . . . 7.4.3 Some Estimates for Tame Harmonic Bundles (2) .. . . . . . . . 7.4.4 λ-Connections . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.5 Proof of Proposition 7.4.3 .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.6 The Strong Adaptedness and the Norm Estimate . . . . . . . . . 7.4.7 Proof of Proposition 7.4.1 .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5 Some Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

227 227 227 228 229 229 229 230 231 231 231 233 233 234 235 236 237 238 240 241 243 243 244 247 248 251 253 253 257

8

Global Periodic Monopoles of Rank One . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 Ahlfors Type Lemma .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.2 Poisson Equation (1) . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.3 Poisson Equation (2) . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.4 Subharmonic Functions . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Global Periodic Monopoles of Rank One (1) . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 Reformulation.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Global Periodic Monopoles of Rank One (2) . . .. . . . . . . . . . . . . . . . . . . . 8.3.1 Construction of Mini-Holomorphic Bundles .. . . . . . . . . . . . . 8.3.2 Good Filtered Bundles . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.3 Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Global Periodic Monopoles of Rank One (3) . . .. . . . . . . . . . . . . . . . . . . .

259 259 259 261 262 263 264 265 266 266 267 268 268

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Contents

Global Periodic Monopoles and Filtered Difference Modules . . . . . . . . 9.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.1 Ambient Good Filtered Bundles with Appropriate Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.2 Degree of Filtered Subbundles .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.3 Analytic Degree of Subbundles .. . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Good Filtered Bundles Associated with Monopoles of GCK-Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Construction of Monopoles .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5 Smooth Parabolic 0-Difference Modules of Rank 2 .. . . . . . . . . . . . . . . 9.5.1 Smooth Parabolic 0-Difference Modules .. . . . . . . . . . . . . . . . . 9.5.2 Rank One Case . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5.3 Filtered Torsion-Free Sheaves of Rank One on an Integral Scheme . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5.4 Description of Parabolic Difference Modules of Rank 2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

10 Asymptotic Harmonic Bundles and Asymptotic Doubly Periodic Instantons (Appendix) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Formal λ-Connections Associated with Asymptotic Harmonic Bundles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.1 Asymptotic Harmonic Bundles . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.2 Simpson’s Main Estimate . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.3 The Associated Filtered Bundles and Formal λ-Connections . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.4 Formal Good Filtered λ-Flat Bundles .. . . . . . . . . . . . . . . . . . . . 10.1.5 Residues and KMS-Structure . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.6 Norm Estimate . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.7 Comparison of KMS-Structures . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.8 Appendix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Family of Vector Bundles on Torus with Small Curvature . . . . . . . . . 10.2.1 Preliminary.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.2 Partially Almost Holomorphic Frames . . . . . . . . . . . . . . . . . . . . 10.2.3 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.4 Additional Assumption on Spectra . . . .. . . . . . . . . . . . . . . . . . . . 10.2.5 Spaces of Functions . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.6 Some Estimates . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Estimates for Asymptotic Doubly Periodic Instantons . . . . . . . . . . . . . 10.3.1 Setting .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.2 Decomposition .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.3 Estimates .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

271 271 272

272 273 278 278 279 280 280 281 281 285 289 289 289 290 292 293 295 295 296 299 301 301 302 302 303 305 305 306 306 307 308

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 311 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 315

Chapter 1

Introduction

Abstract First, we shall explain the motivation of our study. In particular, we review some previous results for monopoles and some equivalences for harmonic bundles. Then, we shall closely explain the results of this monograph. Namely, we formulate parabolic difference modules, and how they are induced by periodic monopoles. It is the main result in this monograph that the procedure induces an equivalence between monopoles and parabolic difference modules. Finally, we shall also explain an outline of the study of the asymptotic behaviour of monopoles.

1.1 Background and Motivation One of the most interesting themes in complex differential geometry is to find an equivalence between objects in differential geometry and in algebraic geometry. An object in differential geometry is defined as a solution of a system of nonlinear partial differential equations. Therefore, it is usually difficult to prove even the existence of such an object. In contrast, it is much easier to construct objects in algebraic geometry. Moreover, we can expect to obtain the classification of such objects as an explicit description of the moduli space. It is one reason to study this kind of equivalences. Conversely, we expect to obtain strong consequences for algebraic objects from this type of equivalence. For example, it should imply that the moduli space of the algebraic objects is equipped with many interesting structures such as a hyperkähler structure, and that good properties of algebraic objects such as stability and semisimplicity are preserved by various operations such as the tensor product, the pull back and the push-forward. Therefore, there is a two-way benefit to obtain such an equivalence. An interesting trend is concerned with a good metric of an algebraic vector bundle on the side of differential geometry, and a good algebraic property of the same bundle on the side of algebraic geometry. The most classical is the theorem of Narasimhan-Seshadri [72]; an algebraic vector bundle on a compact Riemann surface of degree 0 has a flat metric if and only if it is stable. Here, a metric of an algebraic vector bundle is called flat if the associated Chern connection is flat, which is an object in differential geometry. The stability condition is a good © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Mochizuki, Periodic Monopoles and Difference Modules, Lecture Notes in Mathematics 2300, https://doi.org/10.1007/978-3-030-94500-8_1

1

2

1 Introduction

property for an algebraic vector bundle defined in an algebraic way. The higher dimensional generalization is known as “a holomorphic vector bundle on a compact Kähler manifold is equipped with a Hermitian-Einstein metric if and only if it is polystable”. It was first pursued by Kobayashi [45–47] and Hitchin [50] (see [35]). The “only if” part was established by Kobayashi [46, 47] and Lübke [55, 56], and the “if” part was established by Donaldson [20, 21], and Uhlenbeck and Yau [89]. Since then, such correspondences have been studied for vector bundles with an additional structure. One of the most fruitful is the equivalence between harmonic bundles on the side of differential geometry, and flat bundles, Higgs bundles and more generally λ-flat bundles (λ ∈ C) on the side of algebraic geometry. It was established by Corlette [15], Donaldson [22], Hitchin [34] and Simpson [79]. It is not only interesting in its own right, but also provides a starting point of an exciting and tremendous research area, so called the non-abelian Hodge theory. (We shall review briefly the equivalence for singular harmonic bundles on a compact Riemann surface in Sect. 1.4.) Nowadays, the non-abelian Hodge theory is so huge, and influential to various other research areas. We just mention that the equivalence for wild harmonic bundles is in particular essentially useful in the proof of the decomposition theorem for semisimple algebraic holonomic D-modules by the projective pushforward, which was an important problem in algebraic analysis (see [64]). In this monograph, we shall clarify the relationships between singular monopoles on S 1 ×R2 , and polystable parabolic difference modules, which is a new equivalence between differential geometric objects and algebraic geometric objects. As we shall explain (in particular, see Sect. 1.5), it can be regarded as a variant of equivalences between harmonic bundles and λ-flat bundles mentioned above. We also study the triply periodic case [67] and the doubly periodic case [70]. Hopefully, these studies will be one of the starting points of a new investigation for monopoles and difference modules from the viewpoint of the non-abelian Hodge theory. We should mention that there are many previous studies relating monopoles with algebraic geometric objects in various cases, by using ‘scattering maps”. The most classical is the work of Donaldson [19] and Hitchin [33] on the equivalence between SU(2)-monopoles on R3 with L2 -curvature and holomorphic maps from P1 to P1 . Note that Hurtubise [36] clarified the role of the scattering maps in this equivalence. More recently, inspired by the works of Kapustin and Witten [44] and Norbury [74], Charbonneau and Hurtubise [11] obtained an equivalence between singular monopoles on the product of S 1 and a compact Riemann surface Σ, and bundle pairs on Σ satisfying a stability condition, by using the scattering maps. We shall briefly review the works in Sect. 1.3. Our study is directly inspired by Charbonneau and Hurtubise [11], and the scattering maps are essentially used, too. However, there are also two main issues which we need to clarify in our particular situation. First, because we study singular monopoles on the non-compact space S 1 × R2 , it is fundamental to study their asymptotic behaviour around infinity. Note that it is essentially different from the asymptotic behaviour of monopoles on R3 . Roughly speaking, up to the pull back by a ramified covering, a monopole on S 1 × R2 is asymptotically close to a direct sum of the tensor product of monopoles of rank one and monopoles

1.2 Monopoles of GCK-Type

3

induced by wild harmonic bundles. (See Sect. 1.8 for a summary of our study of the asymptotic behaviour of monopoles on S 1 × R2 .) Second, we would like to emphasize that the equivalences depend on the twistor parameters. It is analogue to the fact that harmonic bundles are equivalent to flat bundles, Higgs bundles and more generally λ-flat bundles (λ ∈ C). We note that even in the case of SU(2)-monopoles on R3 with L2 -curvature, we obtain an equivalence for each choice of a twistor parameter λ. However, the method proving the equivalence is independent of λ, and hence the dependence on λ has not been attracted.√In the case of monopoles on S 1 × R2 , the corresponding algebraic objects are 2 −1λdifference modules, i.e., C(y)-modules √ M equipped with a C-linear automorphism Φ ∗ such that Φ ∗ (f s) = f (y + 2 −1λ)Φ ∗ (s) for any f ∈ C(y) and s ∈ M. In particular, the properties of the algebraic objects are quite different in the cases λ = 0 and λ = 0, and we need more involved arguments for non-zero λ to prove equivalences. Relatedly, there are many specific issues. For example, we need to formulate precisely the notion of parabolic structure for such difference modules, it is more convenient to generalize “scattering map” to the notion of mini-holomorphic structure, it is useful to clarify the relation between mini-holomorphic bundles and λ-flat bundles, etc. The author hopes that our study on these specific issues will be useful for the further investigations of difference modules and monopoles.

1.2 Monopoles of GCK-Type For T > 0, we set ST1 := R/T Z. Let (t, w) denote the standard local coordinate system on ST1 × C. We regard ST1 × C as a Riemannian manifold with the metric dt dt + dw dw. A periodic monopole is a monopole (E, h, ∇, φ) on ST1 × C. Namely, E is a vector bundle on ST1 × C with a Hermitian metric h, a unitary connection ∇ and an anti-Hermitian endomorphism φ satisfying the Bogomolny equation F (∇) − ∗∇φ = 0,

(1.1)

where F (∇) denote the curvature of ∇, and ∗ denotes the Hodge star operator. More precisely, we admit that the monopole may have isolated singularities at a finite subset Z ⊂ ST1 ×C, i.e., the monopole (E, h, ∇, φ) is defined on (ST1 ×C)\Z. We impose the following conditions on the behaviour of (E, h, ∇, φ) around Z and ST1 × {∞}. • Each point of Z is Dirac type singularity of (E, h, ∇, φ). • F (∇) → 0 and φ = O(log |w|) as |w| → ∞. In this paper, such monopoles are called of GCK-type (generalized CherkisKapustin type). Cherkis and Kapustin [13, 14] studied such monopoles under some more additional assumptions of genericity on the behaviour around Z and ST1 × {∞}. In particular, they studied the Nahm transforms between such periodic monopoles and

4

1 Introduction

harmonic bundles on (P1 , {0, ∞, p1 , . . . , pm }). Foscolo studied the deformation theory of such periodic monopoles in [25], and the gluing construction in [26]. More recently, Harland [30] classified SU(2)-monopoles in the case Z = ∅ in terms of line bundles with parabolic structure on spectral curves. See also [31, 57, 58] on the geometry of the moduli space of some type of periodic monopoles. In this monograph, we shall study periodic monopoles of GCK-type. The first goal is to clarify their asymptotic behaviour around ST1 × {∞}. Then, we shall establish that periodic monopoles are equivalent to difference modules. There are two origins of this study. One is the classifications of monopoles in terms of holomorphic objects. The other is the non-abelian Hodge theory for harmonic bundles on compact Riemann surfaces. Let us briefly recall them in Sects. 1.3 and 1.4.

1.3 Previous Works on Monopoles and Algebraic Objects There are several interesting and deep studies on the classification of monopoles in terms of algebraic objects. The most classical and pioneering work is due to Donaldson and Hitchin. Roughly speaking, they obtained an equivalence between SU(2)-monopoles (E, h, ∇, φ) on R3 with finite energy R3 |F (∇)|2 < ∞, and based holomorphic maps ϕ : P1 −→ P1 , where ϕ is called based if ϕ(0) = ∞. More precisely, Hitchin [33] established that the Nahm transform induces an equivalence between monopoles and solutions of the Nahm equation, and Donaldson [19] obtained an equivalence between the solutions of the Nahm equation and based holomorphic maps ϕ : P1 −→ P1 . Hurtubise [36] clarified a more direct approach to obtain based holomorphic maps from monopoles. (Atiyah [3] gave an analogue construction for monopoles on hyperbolic spaces.) We shall review a construction of a based holomorphic map from an SU(2)-monopole in Sect. 1.3.1. We remark that the periodic monopoles are excluded by the finite energy condition. The result was generalized by Hurtubise and Murray [37–39] and Jarvis [42, 43] to the context of a more general compact Lie group G rather than SU(2). Namely, they obtained a classification of G-monopoles in terms of holomorphic maps from P1 to flag varieties associated to G. The interest in monopoles was renewed by the work of Kapustin and Witten [44] on the geometric Langlands theory from a physics viewpoint. Inspired by their work, Norbury [74] established that any Hecke transform of a holomorphic bundle on a Riemann surface Σ is represented by a singular monopole on ]0, 1[×Σ satisfying the Dirichlet condition at t = 0 and the Neumann condition at t = 1. Here, for a < b, we set ]a, b[:= {a < t < b}. Charbonneau and Hurtubise [11] studied singular monopoles on S 1 × Σ, and they established an equivalence between such singular monopoles and holomorphic bundles with a meromorphic automorphism satisfying a stability condition on Σ. Because our study is directly influenced by the work of Charbonneau and Hurtubise, we shall briefly review it in Sect. 1.3.2.

1.3 Previous Works on Monopoles and Algebraic Objects

5

As noted above, we shall briefly recall the constructions of algebraic objects in the correspondences due to Donaldson-Hitchin, and Charbonneau-Hurtubise in Sects. 1.3.1 and 1.3.2, respectively. Though scattering maps are applied in the both cases, there are also different flavors. See also a remark in Sect. 1.3.3.

1.3.1 SU(2)-Monopoles with Finite Energy on R3 We recall an outline of the construction of based holomorphic maps from SU(2)monopoles with finite energy on R3 , where some of fundamental concepts have already appeared. We follow the explanation in [4, §2 and §16] and [36, 37] though we omit the detail. (See also [42].) Let (E, h, ∇, φ) be a monopole  2 with finite energy on R3 = {(x1 , x2 , x3 ) ∈ R3 } with the Euclidean metric dxi . Following [4], on the basis of the results in [41], we restrict ourselves to the case where the following asymptotic conditions are satisfied.  k • |φ|h = 1 − 2r xi2 , |φ|h = − 12 Tr(φ 2 ) and + O(r −2 ) as r → ∞, where r = k denotes a positive integer. ∂ −2 h • ∂|φ| ∂Ω = O(r ) as r → ∞, where ∂Ω denotes the angular derivative. −2 • |∇φ|h = O(r ) as r → ∞. √ We identify R3 with Rt ×Cz by setting t = x1 and z = x2 + −1x3 . Note that dt dt + dz dz = dxi2 . We set E t := E|{t }×C for t ∈ R. They are enhanced bundles by the differential operator ∇z = √ to holomorphic vector 1  ∈ R, we have the isomorphism E t E t  obtained (∇ + −1∇ ). For any t, t x2 x3 2 √ as the parallel transport with respect to the differential operator ∂t := ∇x1 − −1φ. The isomorphisms are called the scattering maps. Because the Bogomolny equation implies the commutativity 

 ∂t , ∇z = 0,

(1.2)

the scattering maps are holomorphic. This is one of the key facts in the study of the relationship between monopoles and holomorphic or algebraic objects, which goes back to [32]. Remark 1.3.1 In our study of periodic monopoles, it is useful to formulate the integrability condition (1.2) as a mini-complex structure which is an analogue of a complex structure. (See Sect. 2.2 for mini-complex structure.)  ± There exist unitary frames u± 1 , u2 of E such that the following holds. ± ± ± ± ± ± • ∇(u± = 0. Moreover, 1 , u2 ) = (u1 , u2 )(A2 dx2 + A3 dx3 ), i.e., ∇x1 ui ± ± A2 , A3 → 0 as x1 → ±∞. ± ± ± ± • Let Φ ± be the su(2)-valued functions determined by φ(u± 1 , u2 ) = (u1 , u2 )Φ . ± Then, as x1 → ±∞, Φ √ converge to a diagonal matrix whose (1, 1)-entry is √ −1 and (2, 2)-entry is − −1.

6

1 Introduction

Let (e1± , e2± ) denote the asymptotic gauge of E at x1 = ±∞ induced by ± ± (u1 , u2 ). They are well defined up to multiplications of complex numbers with absolute value 1 to ei± . As explained in [32], there exists a frame (v1 , v2 ) of E such that (i) ∂t vi = 0, (ii) et t −k/2 v1 → e1+ and e−t t k/2 v2 → e2+ as t → ∞. It turns out that ∇z v1 = 0. Indeed, by the commutativity (1.2), we obtain ∂t (∇z v1 ) = 0. Hence, we have the

expression ∇z v1 = α1 (z)v1 + β1 (z)v2 , which implies

∇z (et t −k/2 v1 ) = α1 (z)(et t −k/2 v1 ) + β1 (z)e2t t −k · (e−t t k/2 v2 ). Because ∇z (et t −k/2 v1 ) → 0 as t → ∞, we obtain α1 = β1 = 0. Similarly, ∇z v2 = α2 (z)v1 holds for a function α2 (z). For any t ∈ R, let Lt ⊂ E t denote t the subbundle generated by f+,1 := v1|{t }×C . It is characterized by the following property. t1 1 • For (t1 , z1 ) ∈ R × C and s ∈ Lt+|z ⊂ E|z , let  s denote the section of E|R×{z1 } 1 1 determined by the conditions ∂t s = 0 and  s|(t1 ,z1 ) = s. Then, | s|h → 0 as t → ∞. t t is a global Because ∇z f+,1 = 0, Lt+ is a holomorphic subbundle of E t , and f+,1 t holomorphic frame of Lt+ . Moreover, v2 induces a global holomorphic frame f+,2 



of E t /Lt+ . The scattering map E t E t induces the isomorphisms Lt+ Lt+ and   E t /Lt+ E t /Lt+ , which preserve the distinguished frames. By applying a similar consideration to the behaviour as t → −∞, we obtain a holomorphic subbundle Lt− ⊂ E t characterized by the following condition. t1 1 • For (t1 , z1 ) ∈ R × C and s ∈ Lt−|z ⊂ E|z , let  s denote the section of E|R×{z1 } 1 1 determined by the conditions ∂t s = 0 and  s|(t1 ,z1 ) = s. Then, | s|h → 0 as t → −∞.

Moreover, the line bundles Lt− and E t /Lt− are equipped with the distinguished t t frames f−,1 and f−,2 , respectively. The subbundles and the distinguished frames are preserved by the scattering maps. There exists the naturally defined morphism γ : Lt− −→ E t /Lt+ . By the frames t t f−,1 and f+,2 , we regard γ as an entire holomorphic function. It turns out that γ is a non-zero polynomial of degree k. As remarked in [4, §16], there uniquely t t t exists a holomorphic section f+,2 of E t such that (i) f+,2 induces f+,2 , (ii) for the t t t  expression f−,1 = γ f+,2 + δf+,1 , δ is a polynomial with deg δ < k. Thus, we obtain the based holomorphic map ϕ = δ/γ : P1 −→ P1 . Indeed, because of the ambiguity of the choice of an asymptotic gauge (e1+ , e2+ ), we obtain a family of based holomorphic maps parameterized by S 1 . Donaldson and Hitchin proved that this procedure induces an equivalence between SU(2)-monopoles with finite energy and based holomorphic maps. Remark 1.3.2 The construction  also depends on the choice of an R-linear isomordxi2 = dt dt + dw dw. The choice of such an phism R3 R × C such that

1.3 Previous Works on Monopoles and Algebraic Objects

7

isomorphism corresponds to a twistor parameter. Hence, for each twistor parameter, we obtain a family of based holomorphic maps parameterized by S 1 .  We may also formulate the associated algebraic object in terms of C[z]-modules, where C[z] denotes the polynomial ring of the variable z. Let M be a free C[z]module of rank 2. Let L± ⊂ M be free C[z]-submodules of rank 1 such that M/L± are also free C[z]-modules of rank 1. Let g±,1 be frames of L± , and let g2 be a frame of M/L+ . Assume that the induced map L− −→ M/L+ is expressed by a non-zero polynomial of degree k. From the above monopole (E, h, ∇, φ), we obtain such (M, L± , g±,1 , g2 ) as follows. Let OP1 (∗∞) denote the sheaf of meromorphic functions on P1 which may 0 ⊕ O (∗∞)f0 . It is equipped allow poles along ∞. We set E = OP1 (∗∞)f+,1 P1 +,2 0 0 ⊂ E. We with the filtrations L+ = OP1 (∗∞)f+,1 ⊂ E and L− = OP1 (∗∞)f−,1 0 0 0 1 0 1 set M := H (P , E), L± := H (P , L± ), g±,1 := f±,1 and g2 := f+,2 . It is easy to see that the associated based holomorphic map ϕ is obtained from the associated (M, L± , g±,1 , g2 ), and vice versa. Remark 1.3.3 In our study of periodic monopoles, we prefer C[z]-modules or Omodules as algebraic objects because we can easily add more structures such as parabolic structures. 

1.3.2 The Correspondence due to Charbonneau and Hurtubise We also recall the study of Charbonneau and Hurtubise [11], by omitting the details. Let Σ be a compact Riemann surface equipped with a Kähler metric gΣ . We set S 1 = R/T Z for some T > 0, which is equipped with the standard metric dt dt. Let g denote the induced Riemannian metric of S 1 × Σ. Let Z = {(ti , pi ) | i = 1, . . . , N} ⊂ S 1 × Σ be a finite subset. For simplicity, we assume that ti = tj and pi = pj if i = j . We also assume ti = 0. Let (E, h, ∇, φ) be a monopole on (S 1 × Σ) \ Z such that each point of Z is a Dirac type singularity of (E, h, ∇, φ). (We shall review the Dirac type singularity of monopoles in the Euclidean case in Sect. 2.4.3.) We recall the way to obtain an algebraic object in this context. Let pΣ : S 1 × Σ −→ Σ denote the projection. Let 0,1 −1 0,1 : E −→ E ⊗ pΣ ΩΣ denote the induced differential operator. We set ∂t := ∇|Σ √ 0,1 ∇t − −1φ. The Bogomolny equation implies the commutativity [∂t , ∇|Σ ] = 0. 1 Let Z1 ⊂ S and Z2 ⊂ Σ denote the image by the projections of Z. For any t ∈ S 1 \ Z1 , we obtain the vector bundle E t = E|{t }×Σ on Σ, which is enhanced 0,1 . As the parallel transport by ∂t , we obtain the to a holomorphic bundle by ∇|Σ 

t t

E|Σ\Z for t, t  ∈ S 1 \ Z1 , called the scattering isomorphisms Φt  ,t : E|Σ\Z 2 2

0,1 ] = 0, the maps are holomorphic. Under the maps. By the commutativity [∂t , ∇|Σ assumption that each point of Z is a Dirac type singularity of (E, h, ∇, φ), it turns out that the scattering maps are at most meromorphic along Z2 . Thus, we obtain a

8

1 Introduction

0 holomorphic vector bundle E 0 with a meromorphic automorphism ΦT ,0 : E|Σ\Z

2 0 T E|Σ\Z2 = E|Σ\Z2 . Such a pair is called a bundle pair in [11]. Let E be a holomorphic vector bundle on Σ of rank r equipped with a meromorphic automorphism ρ : E|Σ\Z2 E|Σ\Z2 . For each pi ∈ Z2 , there exist frames v1± , . . . , vr± of E around pi , and a decreasing sequence of integers ki,1 , . . . , ki,r such that ρ(vj− ) = zki,j vj+ , where z denotes a holomorphic coordinate around pi such that z(pi ) = 0. The tuple (ki,j | j = 1, . . . , r) is well defined for   each pi ∈ Z2 . We note that rj =1 N i=1 ki,j = 0. Then, Charbonneau and Hurtubise introduced the degree for such a bundle pair (E, ρ), depending on Z:

deg(E, ρ) = deg(E) −

N r ti ki,j . T i=1

j =1

  ) = E|Σ\Z . Then, we obtain the Let E  ⊂ E be a subbundle such that ρ(E|Σ\Z 2 2   induced meromorphic automorphism ρ of E , and deg(E  , ρ  ) is defined similarly. We say that (E, ρ) is stable if

deg(E  , ρ  ) deg(E, ρ) < rank E  rank E   ) = E|Σ\Z . We say that (E, ρ) for any proper subbundle E  ⊂ E such that ρ(E|Σ\Z 2 2

is poly-stable if it is a direct sum (E, ρ) = (E , ρ ) such that (i) each (E , ρ ) is stable, (ii) deg(E , ρ )/ rank(E ) = deg(E, ρ)/ rank(E) for any . Charbonneau and Hurtubise proved that if (E, ρ) is induced by a monopole (E, ∇, h, φ) with Dirac type singularity, then (E, ρ) is polystable of degree 0. By using the fundamental result of Simpson in [79], they proved that this procedure induces an equivalence between monopoles and polystable bundle pairs of degree 0.

Remark 1.3.4 Let (E, h, ∇, φ) be a vector bundle E on (S 1 × Σ) \ Z with a Hermitian metric h, a unitary connection ∇ and an anti-Hermitian endomorphism φ. As a generalization of the Bogomolny equation, Charbonneau and Hurtubise studied the Hermitian-Einstein-Bogomolny equation F (∇) −

√ −1cωΣ idE = ∗∇φ,

(1.3)

where ωΣ denotes the Kähler form associated with gΣ . If c = 0, it is the Bogomolny equation. 

1.3.3 Remark Though the scattering maps are efficiently used in the both constructions of algebraic objects in Sects. 1.3.1 and 1.3.2, there are also different flavors. Roughly

1.4 Review of the Kobayashi-Hitchin Correspondences for λ-Flat Bundles

9

speaking, in Sect. 1.3.1, we obtain a module with filtrations on Y from a monopole on R × Y , and in Sect. 1.3.2, we obtain a module with an automorphism on Y from a monopole on S 1 × Y . In this monograph, we construct a difference module with parabolic structure from a monopole depending on the twistor parameter λ. In the case λ = 0, our construction is exactly an analogue of that in Sect. 1.3.2 in the context of Σ = C though we need to study the additional issues caused by the non-compactness of Σ. In the case λ = 0, our construction is again similar to that in Sect. 1.3.2, but it is more complicated. We also note that in the case λ = 0 it is also natural to expect to obtain another algebraic object as the associated “Betti” object, and the construction should be in some sense similar to the construction in Sect. 1.3.1 though we shall not discuss it in this monograph. Indeed, in our study of doubly periodic monopoles [70], two kinds of the constructions appear. From a monopole on the product of R and an elliptic curve, we construct a qλ -difference module with parabolic structure depending on a twistor parameter λ [70, Theorem 1.5]. If |qλ | = 1, or equivalently |λ| = 1, by the Riemann-Hilbert correspondence for qλ -difference modules in the local case due to van der Put and Reversat [90] and Ramis, Sauloy and Zhang [77] and in the global case due to Kontsevich and Soibelman, the monopole also induces a locally free sheaf with some filtrations on the elliptic curve T λ obtained as the quotient of C∗ by the action of (qλ )Z [70, Theorem 1.7]. The construction of the qλ difference module from the monopole is similar to the construction in Sect. 1.3.2, and the construction of the locally free module with filtrations on T λ is similar to the construction in Sect. 1.3.1.

1.4 Review of the Kobayashi-Hitchin Correspondences for λ-Flat Bundles Recall that Simpson established the Kobayashi-Hitchin correspondence for tame harmonic bundles on non-compact curves [80]. It is our main goal to develop an analogue theory in the context of periodic monopoles. Hence, let us begin with a brief review on the theory of harmonic bundles on curves.

1.4.1 Harmonic Bundles and Their Underlying λ-Flat Bundles Let C be any complex curve. Let (E, ∂ E ) be a holomorphic vector bundle on C. Let θ be a Higgs field, i.e., a holomorphic section of End(E)⊗ΩC1 . Let h be a Hermitian metric of E. We obtain the Chern connection ∇h = ∂ E + ∂E,h determined by ∂ E and h, and the adjoint θh† of θ with respect to h. The metric h is called harmonic if

10

1 Introduction

the Hitchin equation [∂ E , ∂E,h ] + [θ, θh† ] = 0

(1.4)

is satisfied. A Higgs bundle with a harmonic metric is called a harmonic bundle. The Hitchin equation implies that the connection ∂ E + θh† + ∂E,h + θ is flat. More generally, for any complex number λ, there exists the associated flat λ-connection. Indeed, we set Dλ := ∂ E + λθ † + λ∂E + θ . As explained in [82, 83], it is a λconnection in the sense of Deligne, i.e., it satisfies a twisted Leibniz rule Dλ (f s) = (λ∂C + ∂ C )f · s + f Dλ (s) for any f ∈ C ∞ (C, C) and s ∈ C ∞ (C, E). The Hitchin equation means that Dλ is flat, i.e., Dλ ◦ Dλ = 0. Hence, harmonic bundles have the underlying λ-flat bundles. We may define the concept of harmonic metrics for λ-flat bundles. A λ-flat bundle with a harmonic metric is equivalent to a Higgs bundle with a harmonic metric. See [63], for example.

1.4.2 Kobayashi-Hitchin Correspondences in the Smooth Case  Suppose that C is projective and connected. We set deg(F ) = C c1 (F ) for any vector bundle F on C. A λ-flat bundle (V , ∇ λ ) is called stable (semistable) if we obtain deg(V  )/ rank V  < (≤) deg(V )/ rank V for any λ-flat subbundle (V  , ∇ λ ) ⊂ (V , ∇ λ ) with 0 < rank(V  ) < rank(V ). A λ-flat

bundle (V , ∇ λ ) is called polystable if it is a direct sum of stable λ-flat bundles (Vi , ∇ λ ) such that deg(V )/ rank V = deg(Vi )/ rank Vi for any i. Note that deg(V ) = 0 always holds if λ = 0. The following is a fundamental theorem in the study of harmonic bundles on projective curves, due to Diederich-Ohsawa [17], Donaldson [22] and Hitchin [34], in the rank 2 case, and Corlette [15] and Simpson [79, 81, 82] in the higher rank case, or even in the higher dimensional case. (See also [63] for the case of general flat λ-connections.) Theorem 1.4.1 Suppose that C is projective and connected. A λ-flat bundle (V , ∇ λ ) is polystable of degree 0 if and only if (V , ∇ λ ) has a harmonic metric. If (V , ∇ λ ) has metrics hj (j = 1, 2), there exists a decomposition

two harmonic (V , ∇ λ ) = (Vi , ∇ λ ) which is orthogonal with respect to both h1 and h2 , such that h1|Vi = ai · h2|Vi for some constants ai > 0. In particular, if (V , ∇ λ ) is stable, it has a unique harmonic metric up to the multiplication of positive constants.  Corollary 1.4.2 For each λ, there exists a natural bijective correspondence between the equivalence classes of polystable λ-flat bundles of degree 0 and the equivalence classes of polystable Higgs bundles of degree 0, through harmonic bundles. 

1.4 Review of the Kobayashi-Hitchin Correspondences for λ-Flat Bundles

11

The Kobayashi-Hitchin correspondence in the theorem provides us with an interesting equivalence between objects in differential geometry and objects in algebraic geometry. It is an origin of the hyperkähler property of the moduli spaces. It is a starting point of the non-abelian Hodge theory of Simpson. (See [82].)

1.4.3 Tame Harmonic Bundles and Regular Filtered λ-Flat Bundles Simpson studied a generalization of Theorem 1.4.1 to the context of harmonic bundles on quasi-projective curves in [80]. To state his result, let us recall the concepts of tame harmonic bundles and regular filtered λ-flat bundles. Let Y be a neighbourhood of 0 in C. Let (E, ∂ E , θ, h) be a harmonic bundle on Y \ {0}. We obtain the holomorphic endomorphism f of E such that θ = f dz/z. The harmonic bundle is called tame on (Y, {0}) if the following holds.  E−1 • Let det(t idE −f ) = t rank E + rank aj (z)t j be the characteristic polynomial j =0 of f . Then, aj are holomorphic at z = 0. Let us explain the notion of regular filtered λ-flat bundles on (Y, {0}). Let OY (∗0) denote the sheaf of meromorphic functions on Y which may have poles along 0. Recall that a filtered bundle on (Y, {0}) is a locally free OY (∗0)-module V of finite rank with an increasing sequence of locally free OY -submodules Pa V ⊂ V (a ∈ R) satisfying the following conditions. • Pa V (a ∈ R) are lattices of V, i.e., Pa V(∗{0}) = V. • Pa+n V = Pa V(n{0}) for any a ∈ R and n ∈ Z. • For any a ∈ R, there exists  > 0 such that Pa+ V = Pa V. A regular filtered λ-flat bundle is a filtered bundle P∗ V with a λ-connection ∇ λ : V −→ V ⊗ Ω 1 such that ∇ λ is logarithmic with respect to P∗ V in the sense of ∇ λ Pa V ⊂ Pa V ⊗ Ω 1 (log{0}) for any a ∈R.  Note that we obtain the finite dimensional C-vector spaces GrP (V) := P (V) a a b 0. Thus, we obtain an increasing sequence of OY -submodules Pah E λ ⊂ P h E λ (a ∈ R). Simpson proved that (P∗h E λ , Dλ ) is a regular filtered λ-flat bundle in [80].

12

1 Introduction

Remark 1.4.3 We prefer to consider filtered λ-flat bundle (P∗h E λ , Dλ ) on (Y, 0) rather than (E λ , Dλ ) on Y \ {0} to keep the information of the behaviour of h around 0.  Let C be a smooth connected projective curve with a finite subset D ⊂ C. The concept of tame harmonic bundles on (C, D) is defined in a natural way. Let OC (∗D) denote the sheaf of meromorphic functions on C which may have poles along D. A filtered bundle on (C, D) is a locally free OC (∗D)-module V of finite rank with an increasing sequence of OC -locally free submodules Pa V (a = (aP | P ∈ D) ∈ RD ) such that the following holds. • For any P ∈ D, take a small neighbourhood UP of P in C. Then, Pa V|UP (P ) depends only on aP , which we denote by PaP (V|UP ). (P ) • P∗ (V|UP ) is a filtered bundle on (UP , P ) in the above sense. A regular filtered λ-flat bundle is a filtered bundle P∗ V with a λ-connection ∇ λ such that ∇ λ Pa V ⊂ Pa V ⊗ ΩC1 (log D) for any a ∈ RD . Moreover, for any regular filtered λ-flat bundle (P∗ V, ∇ λ ) on (C, D), we define deg(P∗ V) := deg(P0 V) −





b dimC GrP b

(P )

(V|UP ).

P ∈D −1 0. We may regard U1 = U/Γ as an open subset of

16

1 Introduction

(R/T Z) × R2x3 ,x4 . Let p1 : U −→ U1 denote the projection. If (E, h, ∇) is a Γ -equivariant tuple of a vector bundle E with a Hermitian metric and a unitary connection ∇ on U , we obtain the vector bundle E1 on U1 with a Γ -equivariant isomorphism p1∗ (E1 ) E. We obtain the induced Hermitian metric h1 of E1 . There exist the unitary connection ∇1 and the anti-Hermitian endomorphism φ of (E1 , h1 ) such that ∇ = p1∗ (∇) + p1∗ (φ) dx1. Then, the ASD equation (1.6) holds if and only if (E, h, ∇, φ) is a monopole. We shall introduce the notion of mini-complex structure of a 3-dimensional manifold (see Sect. 2.2 for mini-complex structure). We also introduce the notion of mini-holomorphic bundle on a mini-complex manifold, which is essentially a vector bundle equipped with two commuting differential operators as in Sects. 1.3.1 and 1.3.2. (See Sect. 2.3 for mini-holomorphic bundles.) In our situation, we obtain the mini-complex manifold U1λ as U1 equipped with the mini-complex structure induced by the complex structure of U λ . (See Sect. 2.7.1.) A Γ -equivariant holomorphic vector bundle on U λ is equivalent to a mini-holomorphic bundle on U1λ . Moreover, the Γ -equivariant holomorphic bundle (E λ , ∂ E λ ) underlying (E, h, ∇) is equivalent to the mini-holomorphic bundle (E1λ , ∂ E λ ) underlying the 1 monopole (E1 , h1 , ∇1 , φ): equivariant instantons on

monopoleson

1

equivariant holomorphic vector bundles on mini-holomorphic . bundles on 1

Very roughly, when U1 is the complement of a finite subset in (R/T Z) × R2 , our Kobayashi-Hitchin correspondence for periodic monopoles are equivalences between monopoles on U1 and mini-holomorphic bundles on U1λ satisfying some conditions, and the latter objects are translated to difference modules. We shall explain our correspondences for more details in Sects. 1.6 and 1.7.

1.5.4 Instantons and Monopoles as Harmonic Bundles of Infinite Rank 1.5.4.1 Instantons as Harmonic Bundles of Infinite Rank If U = Cz × U0 ⊂ Cz × Cw , Cz -equivariant harmonic bundles on U are equivalent to harmonic bundles on U0 , as mentioned in Sect. 1.5.2. It also implies that we may regard instantons on U as harmonic bundles of infinite rank on U0 , as we shall explain below. Let (E, ∇, h) be an instanton on U . Let E ∞ denote the sheaf of C ∞ -sections of E on U . Let CU∞0 denote the sheaf of C ∞ -functions on U0 . Let p :

1.5 Equivariant Instantons and the Underlying Holomorphic Objects

17

U −→ U0 denote the projection, and let p! denote the proper push-forward functor for sheaves with respect to p. We obtain the CU∞0 -module p! E ∞ on U0 . We regard p! E ∞ as a C ∞ -vector bundle on U0 of infinite rank. The metric h and the integration induces a Hermitian metric p! h on p! E ∞ . We may regard ∇w as a holomorphic structure ∂p! E ∞ ,w of p! E ∞ . We also have the differential operators ∂p! E ∞ ,w on p! E ∞ induced by ∇w . Then, ∂p! E ∞ ,w dw + ∂p! E ∞ ,w dw is a connection of p! E ∞ which is unitary with respect to p! h. The differential operator f = ∇z on p! E ∞ is CU∞0 linear. Because [∇z , ∇w ] = 0, we may regard f as a holomorphic endomorphism of (p! E ∞ , ∂p! E ∞ ,w ), and we obtain the Higgs field θ = f dw. We also have the CU∞0 -linear endomorphism f † = −∇z . We may regard f † as the adjoint of f with respect to p! h. We set θ † = f † dw. Let F denote the curvature of the connection ∂p! E ∞ ,w dw + ∂p! E ∞ ,w dw. Then, the equation [∇w , ∇w ] + [∇z , ∇z ] = 0 is exactly the equation F + [θ, θ † ] = 0. In this sense, (p! E ∞ , θ, p! h) is a harmonic bundle of infinite rank on U0 . 1.5.4.2 The Underlying λ-Flat Bundles of Infinite Rank Let us pursue this analogy further. For a complex number λ, if (ξ, η) = (z+λw, w− λz), the following holds: (1 + |λ|2 )∂ξ = λ∂w + ∂z ,

(1 + |λ|2 )∂η = ∂w − λ∂z .

Hence, for any section s of p! E ∞ and g ∈ CU∞0 , we have the following equalities: (1 + |λ|2 )∇ξ (gs) = g(1 + |λ|2 )∇ξ (s) + λ∂w (g) · s, (1 + |λ|2 )∇η (gs) = g(1 + |λ|2 )∇η (s) + ∂w (g) · s. We consider the differential operator ∇ λ = (1 + |λ|2 )∇ξ dw + (1 + |λ|2 )∇η dw on p! E ∞ , which is naturally regarded as a λ-connection. The commutativity of ∇ξ and ∇η is equivalent to the flatness of the λ-connection ∇ λ , i.e., ∇ λ ◦ ∇ λ = 0. Moreover, we have ∇ λ = ∂p! E ∞ ,w dw + λθ † + λ∂p! E ∞ ,w dw + θ . Hence, we may regard the induced holomorphic bundle (E λ , ∂ E λ ) on U λ as the λ-flat bundle of infinite rank on U0 associated with the harmonic bundle (p! E ∞ , θ, p! h) of infinite rank. 1.5.4.3 Monopoles as Harmonic Bundles of Infinite Rank √ Let us consider the case where Γ = {(x1 + −1nT , 0) | x1 ∈ R, n ∈ Z} ⊂ Cz ×Cw . Suppose that U = Cz × U0 for an open subset U0 ⊂ Cw . Set U1 = U/Γ ⊂ (R/T Z) × Cw . Because (R/T Z)-equivariant monopoles on U1 are equivalent to Cz -equivariant instantons on U , they are equivalent to harmonic bundles on U0 . It means that we may regard monopoles on U1 as harmonic bundles on U0 of infinite

18

1 Introduction

rank. We shall explain the more precise relation between monopoles on U1 and harmonic bundles on U0 in Sect. 2.6, the relation of the mini-holomorphic bundles on U1λ and λ-flat bundles on U0 in Sect. 2.8.1 (see also Sects. 2.8.3, 3.6, and 4.5), and the compatibility of the relations in Sect. 2.8.2. We shall give a complement on the compatibility in Sect. 2.9.5.

1.6 Difference Modules with Parabolic Structure As an algebraic objects corresponding to periodic monopoles, we introduce the concept of parabolic structure for difference modules. Let C(y) denote the field of rational functions of the variable y. Let C[y] denote the ring of polynomials. For y0 ∈ C, C[[y of the   −∞y0 ]] denote thej ring  formal power series of y − y0 , i.e.,  aj ∈ C . Let C((y − y0 )) denote the fraction C[[y − y0 ]] = j =0 aj (y − y0 ) field of C[[y − y0 ]], i.e., C((y − y0 )) is the field of formal Laurent power series of y − y0 . Similarly, let C[[y −1 ]] denote the ring of the formal power series of y −1 , and let C((y −1 )) denote the ring of the formal Laurent power series of y −1 .

1.6.1 Difference Modules Take  ∈ C. Let Φ ∗ be the automorphism of the field C(y) induced by Φ ∗ (y) = y + . We prefer to regard it as the pull back of functions by the automorphism Φ of C or P1 . Definition 1.6.1 In this monograph, a difference module is a finite dimensional C(y)-vector space V with a C-linear automorphism Φ ∗ such that Φ ∗ (g s) = Φ ∗ (g) · Φ ∗ (s) for any g(y) ∈ C(y) and s ∈ V . When we emphasize , it is called a -difference module. 

It is naturally a C-vector space. We Remark 1.6.2 We set A := n∈Z C[y](Φ ∗ )n .  ∗ )n · ∗ )m = define the multiplication A × A −→ A by a (y)(Φ b (y)(Φ n m  ∗ m+n an (y)bm (y + n)(Φ ) . Thus, A is an associative C-algebra. As a difference module, it would be more natural to consider a finitely generated left A-module M such that (i) M is a torsion-free C[y]-module, (ii) dimC(y) C(y) ⊗C[y] M < ∞. The data of such finitely generated A-modules are contained in the parabolic structure at finite place, which will be explained below. Indeed, let V be a difference module in the sense of Definition 1.6.1. For any C[y]-free submodule V ⊂ V such that C(y) ⊗C[y] V = V , we obtain the A-module A · V ⊂ V which satisfies the above finiteness conditions. 

1.6 Difference Modules with Parabolic Structure

19

Remark 1.6.3 If  = 0, the difference module is just a finite dimensional C(y)vector space with a C(y)-linear automorphism. 

1.6.2 Parabolic Structure of Difference Modules at Finite Place Let V be a difference module. Definition 1.6.4 (Definition 2.10.1) A parabolic structure of V at finite place is a tuple as follows. • A C[y]-free submodule V ⊂ V such  that V ⊗C[y] C(y) = V . • A function m : C −→ Z≥0 such that x∈C m(x) < ∞. We assume V ⊗C[y] C[y]D = (Φ ∗ )−1 (V ) ⊗C[y] C[y]D , where we set D := {x ∈ C | m(x) > 0}, and let C[y]D denote the localization of C[y] with respect to y − x (x ∈ D). (1) (m(x)) • A sequence of real numbers 0 ≤ τx < · · · < τx < 1 for each x ∈ C. If m(x) = 0, the sequence is assumed to be empty. The sequence is denoted by τ x . • Lattices Lx,i ⊂ V ⊗C[y] C((y − x)) for x ∈ C and i = 1, . . . , m(x) − 1. We formally set Lx,0 := V ⊗C[y] C[[y − x]] and Lx,m(x) := (Φ ∗ )−1 V ⊗ C[[y − x]]. The tuple of lattices is denoted by Lx . Note that V ⊗ C[[y − x]] = (Φ ∗ )−1 (V ) ⊗ C[[y − x]] if m(x) = 0.  Remark 1.6.5 We may regard a parabolic structure of finite place of a difference module as a reincarnation of a part of an ordinary parabolic structure of λ-flat bundles. See Sect. 2.10.4. 

1.6.3 Good Parabolic Structure at ∞ Note that the automorphism Φ ∗ of C(y) induces an automorphism of C((y −1 )),  := V ⊗C(y) C((y −1 )), which is a finite which is also denoted by Φ ∗ . We set V −1 dimensional C((y ))-vector space with the induced C-linear automorphism Φ ∗  . Such (V , Φ ∗ ) such that Φ ∗ (f s) = Φ ∗ (f )Φ ∗ (s) for f ∈ C((y −1 )) and s ∈ V is called a formal difference module.     p−1 −j/p  a ∈ C . According to the For any p ∈ Z≥1 , we set S(p) := j j =1 aj y classification of formal difference modules [12, 76, 86], there exist p ∈ Z≥1 and a decomposition of the formal difference module  ⊗C((y −1 )) C((y −1/p )) = V

ω∈p −1 Z α∈C∗ a∈S(p)

  p,ω,α,a , Φ ∗ , V

(1.7)

20

1 Introduction

 p,ω,α,a has a C[[y −1/p ]]-lattice Lp,ω,α,a satisfying such that each V

 α −1 y ω Φ ∗ − (1 + a) id Lp,ω,α,a ⊂ y −1 Lp,ω,α,a .

We note that in the case  = 0 we can obtain the decomposition (1.7) from the generalized eigen decomposition of a linear map. Definition 1.6.6 (Definition 3.3.14, Definition 3.3.16) A good parabolic structure  over V  which is good with respect to Φ ∗ in the of V at ∞ is a filtered bundle P∗ V following sense.

) =  p,ω,α,a , where ϕp∗ (P∗ V ) P∗ V • There exists the decomposition ϕp∗ (P∗ V −1/p  . )) obtained as the pull back of P∗ V is the filtered bundle over V ⊗ C((y (See Sect. 1.4.4 or Sect. 3.3.1 for the pull back of filtered bundles via a ramified covering.) 

 p,ω,α,a ⊂ y −1 Pa V  p,ω,α,a for any a ∈ R and for • α −1 y ω Φ ∗ − (1 + a) id Pa V any (ω, α, a). 

1.6.4 Parabolic Difference Modules Definition 1.6.7 A parabolic difference module means a difference module equipped with a parabolic structure at finite place and a good parabolic structure at infinity.  Note that in the decomposition (1.7), the numbers r(ω) :=



 p,ω,α,a rank V

(1.8)

α,a

are independent of p, and

well defined for ω ∈ Q. Indeed, there exists the  =  as explained in Sect. 3.2 for which we slope decomposition V V ω∈Q Sω

−1   obtain Sω (V ) ⊗C((y −1 )) C((yp )) = α,a V p,ω,α,a for any p as above. Hence,  . For P∗ V  as in Definition 1.6.6, we obtain the decomposition r(ω) =

dimC Sω V  ). = P∗ V P∗ Sω (V

1.6.5 Degree and Stability Condition Consider a parabolic difference module  ). V ∗ = (V , (V , m, (τ x , Lx )x∈C), P∗ V

1.6 Difference Modules with Parabolic Structure

21

Let FV be the OP1 (∗∞)-module associated with V . We obtain the filtered bundle  . If m(x) > 0, for each i = 1, . . . , m(x), we set P∗ FV over FV induced by P∗ V

  deg(Li,x , Li−1,x ) := dimC Li,x /(Li,x ∩ Li−1,x ) − dimC Li−1,x /(Li,x ∩ Li−1,x ) . Then, we define deg(V ∗ ) := deg(P∗ FV ) +

m(x) 

 ω r(ω). 1 − τx(i) deg Li,x , Li−1,x + 2

x∈C i=1

ω∈Q

(1.9) Here, r(ω) are defined in (1.8). We also define  μ(V ∗ ) := deg(V ∗ ) dimC(y) V .

(1.10)

Let V  be any difference submodule of V , i.e., V  is a C(y)-subspace of V such that Φ ∗ (V  ) = V  . By setting V  := V  ∩V and Li,x := Li,x ∩(V  ⊗C(y) C((y −x))), 

we obtain a parabolic structure V  , m, (τ x , Lx )x∈C at finite place of V  . By setting   := C((y −1 )) ⊗C(y) V  and Pa V   ∩ Pa V   := V  , we obtain a good parabolic V   structure P∗ V at ∞. Thus, we obtain the following induced parabolic difference module from V  :   

 . V ∗ = V  , V  , m, (τ x , Lx )x∈C , P∗ V Definition 1.6.8 A parabolic difference module V ∗ is called stable (resp. semistable) if μ(V ∗ ) < μ(V ∗ ) (resp. μ(V ∗ ) ≤ μ(V ∗ )) for any difference submodule V  of V with 0 < dimC(y) V  < dimC(y) V . It is called polystable if it is semistable and a direct sum of stable ones.  Remark 1.6.9 There exists the relation m(x) x∈C i=1

deg(Li,x , Li−1,x ) +



ω · r(ω) = 0.

(1.11)

ω∈Q

Indeed, by the comparison of V and (Φ ∗ )−1 (V ) in V , we obtain deg(P∗ (F(Φ ∗ )−1 (V ) )) − deg(P∗ FV ) =



deg(Li,x , Li−1,x ).

x,i

There exists the isomorphism of OP1 (∗∞)-modules Φ ∗ (F(Φ ∗ )−1 (V ) ) FV induced

 = ), it induces Φ ∗ (Pa Sω (V  )) Sω (V by Φ ∗ . For the slope decomposition V

22

1 Introduction

 ). Hence, we obtain Pa−ω Sω (V deg(P∗ FV ) = deg(P∗ F(Φ ∗ )−1 (V ) ) +



r(ω) · ω.

ω∈Q



Thus, we obtain (1.11).

1.6.6 Easy Examples of Stable Parabolic Difference Modules (1) Let us mention some easy examples of parabolic difference modules. Needless to say, we can easily construct many more interesting examples. For simplicity, we consider the case  = 0.  We take a non-empty finite subset S ⊂ C and a function  : S −→ Z such that = 0. We also assume that one of (a) is an odd integer. We set P (y) :=  a∈S |(a)| (a) ∈ C(y). We set V := C(y)e1 ⊕C(y)e2 and V := C[y]e1 ⊕C[y]e2 . a∈S (y −a) We define the C(y)-automorphism Φ ∗ of V by Φ ∗ (e1 , e2 ) = (e1 , e2 ) ·



 0 P (y) . 1 0

Let m : C −→ Z≥0 be given by m(a) = 1 (a ∈ S) and m(a) = 0 (a ∈ S). We take 0 ≤ τa < 1 for a ∈ S. Thus, we obtain a parabolic structure of V at finite place. Note the following equality for any a ∈ S: deg(La,1 , La,0 ) = −(a). −

As for good parabolic structures at infinity, there are two cases where (∞) :=  a∈S (a) is even or odd.

1.6.6.1 The Case Where (∞) Is Even Suppose (∞) = −2k for k ∈ Z. There exists τ ∈ C((y −1 )) such that τ 2k = P (y). Note that τ/y ∈ C[[y −1 ]] and that it is invertible. We set v1 := τ k e1 + e2 and v2 := τ k e1 − e2 . Because Φ ∗ (v1 ) = τ k v1 and Φ ∗ (v2 ) = −τ k v2 , we obtain the  = C((y −1 ))v1 ⊕ C((y −1 ))v2 compatible with the action of Φ ∗ . decomposition V  over V  is good with respect to Φ ∗ if and only if it satisfies A filtered bundle P∗ V



 −1   is determined P∗ V = P∗ C((y ))v1 ⊕ P∗ C((y −1 ))v2 . Hence, such good P∗ V by the numbers  }. di = degP (vi ) := min{c ∈ R | vi ∈ Pc V

1.6 Difference Modules with Parabolic Structure

23

We can easily see that V has no non-trivial difference submodule, under the ) assumption that one of (a) is odd. Hence, V ∗ = (V , (V , m, τ x , Lx )x∈C , P∗ V is a stable parabolic difference module. It is easy to see deg(P∗ FV ) = k − d1 − d2 , and hence deg(V ∗ ) = −d1 − d2 − (1 − τa )(a). (1.12) a∈S

1.6.6.2 The Case Where (∞) Is Odd Let us consider the case where (∞) is odd. There exists τ ∈ C((y −1/2 )) such that τ −2(∞) = P (y) and τ/y 1/2 ∈ C[[y −1/2]]. We set v1 := τ −(∞) e1 + e2 and v2 := τ −(∞) e1 − e2 . Because Φ ∗ (v1 ) = τ −(∞) v1 and Φ ∗ (v2 ) = −τ −(∞) v2 , we  ⊗C((y −1/2 )) = C((y −1/2))v1 ⊕C((y −1/2 ))v2 compatible obtain the decomposition V ∗  over V  is good with respect to Φ ∗ with the action of Φ . A filtered bundle P∗ V

⊗  ⊗ C((y −1/2 )) satisfies P∗ V if and only if the induced filtered bundle over V



  C((y −1/2)) = P∗ C((y −1/2))v1 ⊕ P∗ C((y −1/2 ))v2 . Hence, it is determined by    ⊗ C((y −1/2 ))) . As in the previous case, the numbers di := min c ∈ R  vi ∈ P∗ (V  ) is stable. V ∗ = (V , (V , m, (τ x , Lx )), P∗ V  ⊗ C((y −1/2))) is preserved by the natural action of the Galois Because P∗ (V group of C((y −1/2 ))/C((y −1 )), we obtain d1 = d2 =: d. It is easy to see deg(P∗ FV ) = −d − (∞)/2, and hence deg(V ∗ ) = −d −



(1 − τa )(a).

(1.13)

a∈S

1.6.7 Easy Examples of Stable Parabolic Difference Modules (2) Take polynomials P (y) and Q(y) such that the following holds. 2 − 4pdeg(P ) = 0 • 2 deg(Q) ≥ deg(P ). If 2 deg(Q) = deg(P ), we assume qdeg(Q) deg(Q)  deg(P ) j j for the expressions Q = j =0 qj y and P = j =0 pj y . We assume that (deg(P ), deg(Q)) = (0, 0). • P has simple zeroes.

We set V := C(y)e1 ⊕ C(y)e2 , and we define the C(y)-automorphism Φ ∗ by 

 0 P Φ (e1 , e2 ) = (e1 , e2 ) . −1 Q ∗

24

1 Introduction

Because P has simple zeroes, it is easy to see that there is no difference submodule V  ⊂ V such that dimC(y) V  = 1.

 We put α1 = 2−1 Q + Q(1 − 4P Q−2 )1/2 ∈ C((y −1 )) and α2 = α1−1 P . If 2 deg(Q) > deg(P ), we choose (1 − 4P Q−1 )1/2 ∈ C[[y −1 ]] such that (1 − 4P Q−1 )1/2 − 1 ∈ y −1 C[[y −1]]. Note that α1 = α2 . Then, αi (i = 1, 2) are the roots of the characteristic polynomial T 2 − QT + P of the automorphism. Note that y − deg Q α1 and y − deg P +deg Q α2 are invertible in C[[y −1 ]]. We set V := C[y]e1 ⊕ C[y]e2. Set S := {a ∈ C | P (a) = 0}. Let m : C −→ Z≥0 be the function determined by m(a) = 1 (a ∈ S) and m(a) = 0 (a ∈ S). We take 0 ≤ τa < 1 (a ∈ S). Then, the tuple (V , m, (τa )a∈S ) determines a parabolic structure at finite place of V . Set vi := P e1 + αi e2 , for which Φ ∗ (vi ) = αi vi holds. Then, a good filtered  over V  with respect to Φ ∗ is determined by di := degP (vi ). We bundle P∗ V obtain deg(P∗ FV ) := deg(P ) + deg(Q) − d1 − d2 . The degree of V ∗ =  )) is as follows: (V , (V , m, {τa }a∈S , P∗ V deg(V ∗ ) = deg(P ) + deg(Q) − d1 − d2 −

deg(Q) deg(P ) − deg(Q) − (1 − τa )(a) − 2 2 a∈S

1 = deg(P ) + deg(Q) − d1 − d2 − (1 − τa ). 2

(1.14)

a∈S

For any given {τa }a∈S , we may choose (d1 , d2 ) ∈ R2 such that

(1.14) vanishes. Note that the numbers n := deg(P ) and k := 2−1 deg(Q) − (deg(P ) − deg(Q)) + n = deg(Q) correspond to the number of singularity and the nonabelian charge in [14, Page 5] in the context of periodic monopoles.

1.7 Kobayashi-Hitchin Correspondences for Periodic Monopoles Kobayashi-Hitchin correspondences between periodic monopoles and parabolic difference modules are stated as follows. Theorem 1.7.1 (Corollary 9.1.4) For any T > 0 and λ ∈ C, there exists a natural bijective correspondence between the isomorphism classes of the following objects. • Periodic monopoles of GCK type on ST1 × Cw . √ • Polystable parabolic (2 −1λT )-difference modules of degree 0. √ We shall explain an outline to obtain a parabolic 2 −1λT -difference module from a periodic monopole for λ = 0 in Sect. 1.7.1, and for general λ in

1.7 Kobayashi-Hitchin Correspondences for Periodic Monopoles

25

Sect. 1.7.2. The constructions are given by way of mini-holomorphic bundles on a 3-dimensional manifolds, inspired by the work of Charbonneau-Hurtubise [11], which we briefly recalled in Sect. 1.3.2. Corollary 1.7.2 In Sect. 1.6.6, if we choose (d1 , d2 ) ∈ R2 (resp. d ∈ R) such that (1.12) (resp. (1.13)) vanishes, we obtain the periodic monopoles of GCK-type on ST1 × C for any T > 0, corresponding to the stable parabolic difference modules. Similarly, in Sect. 1.6.7, if we choose (d1 , d2 ) ∈ R2 such that (1.14) vanishes, we obtain the periodic monopoles of GCK-type on ST1 ×C for any T > 0, corresponding to the stable parabolic difference modules.  Note that even the existence of periodic monopoles was non-trivial, which was studied by Foscolo in [26]. Remark 1.7.3 A special case of this type of correspondences is independently studied by Elliott and Pestun in [24], where rich studies of the related subjects are described.  Remark 1.7.4 In [30], Harland studied a classification of periodic monopoles of rank 2 without any singularity, by using the Nahm transform between harmonic bundles on C∗ and periodic monopoles. It is given in terms of spectral curves with a parabolic line bundle, which corresponds to the classification at λ = 0 in Theorem 1.7.1.  Remark 1.7.5 In [70], we studied similar correspondences between q-difference modules and doubly periodic monopoles, i.e., monopoles on the product of R and a 2-dimensional torus. In [67], we studied similar correspondences between difference modules on elliptic curves and triply periodic monopoles, i.e., monopoles on a 3dimensional torus. The triply periodic case is much easier because we do not have to study the asymptotic behaviour of monopoles around infinity.  Remark 1.7.6 Kontsevich and Soibelman proposed a non-abelian Hodge theory in the context of their holomorphic Floer theory [49]. Correspondences for monopoles may be regarded as 1-dimensional examples of their theory. 

1.7.1 The Correspondence in the Case λ = 0 We first explain our construction in the case λ = 0, which is conceptually and technically easier than the case λ = 0.

1.7.1.1 Mini-complex Structure Let κ denote the Z-action on Rt × Cw defined by κn (t, w) = (t + nT , w) (n ∈ Z). Let M denote the quotient space of Rt × Cw by the Z-action κ, i.e., M = ST1 × Cw . We note that the complex vector fields ∂t and ∂w are naturally defined on M.

26

1 Introduction

We have the naturally defined mini-complex structure on Rt × Cw . (See Sect. 2.2 for mini-complex structure.) When we emphasize to consider this mini-complex structure, we use the notation M0 , i.e., let M0 denote the 3-dimensional manifold M equipped with the mini-complex structure induced by (t, w). A C ∞ -function f on an open subset U ⊂ M0 is called mini-holomorphic if ∂t f = ∂w f = 0. Let OM0 denote the sheaf of mini-holomorphic functions on M0 . 1.7.1.2 Mini-holomorphic Bundles Associated with Monopoles Let Z be a finite subset of M. Let (E, h, ∇, φ) be a monopole on M \ Z. Recall that the Bogomolny equation implies that [∇t −

√

−1φ, ∇w ] = 0.

√ Hence, by considering the sheaf of local sections s of E such that (∇t − −1φ)s = ∇w s = 0, we obtain a locally free OM0 \Z -module E 0 . 1.7.1.3 Dirac Type Singularity Let P ∈ Z. We take a lift P= (t 0 , w0 ) ∈  Rt × Cw of P . Let  > 0 and δ > 0 be small, and consider U− := (t 0 − , w)  |w| < δ and U := (t 0 + , w)  |w| <  δ . By taking the restriction of E 0 to U± , we obtain locally free OU± -modules 0 ∗ := U 0 0 E|U . Set U± ± \ {(t ± , w )}. As in the previous studies recalled in ± √ Sects. 1.3.1 and 1.3.2, by the parallel transport with respect to ∇t − −1φ, we 0 0 obtain the isomorphism ΠP : E|U ∗ E|U ∗ , called the scattering map. If P ∈ Z is −



Dirac type singularity of the monopole (E, h, ∇, φ), it is easy to see that E 0 is of Dirac type at P in the sense that ΠP is meromorphic. (See Sect. 2.4.3 for Dirac type singularity of monopoles.)

1.7.1.4 Meromorphic Extension and Filtered Extension at Infinity We have the partial compactification Rt × P1w of Rt × Cw . The Z-action κ on Rt × Cw naturally extends to a Z-action on Rt × P1w , which is also denoted by κ. The 0

quotient space M is naturally equipped with the mini-complex structure, and it is 0 := M0 \ M0 S 1 . Let O 0 a compactification of M0 . Let H∞ 0 (∗H∞ ) denote T 0

M

0 . (See the sheaf of meromorphic functions on M whose poles are contained in H∞ Sect. 2.2.5 for meromorphic functions in the context of mini-complex structures.)

Remark 1.7.7 In the study of tame or wild harmonic bundles we prefer to consider meromorphic or filtered Higgs bundles on (C, D) rather than Higgs bundles on C \

1.7 Kobayashi-Hitchin Correspondences for Periodic Monopoles

27

D. (See Remark 1.4.3.) Similarly, we would like to consider meromorphic objects 0 0 ) to keep the information of the growth orders with or filtered objects on (M , H∞ respect to h, rather than the transcendental object on M0 . It is the reason why we 0 consider the compactification M .  0

Let U be any open subset in M \ Z. Let P h E 0 (U ) denote the space of sections 0 satisfying the following condition. of E 0 on U \ H∞ 0 , there exists a neighbourhood U of P in U such that • For any P ∈ U ∩ H∞ P  N for some N. |s|UP \H∞ 0 |h = O |w|

Thus, we obtain the O

0

M \Z

0 )-module P h E 0 . We shall prove the following. (∗H∞

0 )Proposition 1.7.8 (Proposition 7.2.1) P h E 0 is a locally free O 0 (∗H∞ M \Z module. 0

0

Let π 0 : M −→ ST1 denote the map induced by (t, w) −→ t. We set M t = 0

(π 0 )−1 (t) and M0 t = M0 ∩ M t. We also set Zt := (π 0 )−1 (t) ∩ Z. By taking the restriction, we obtain the locally free OCw \Zt -module E|0M0 t \Z . As in

 Sects. 1.4.3 and 1.4.4, we obtain an increasing sequence Pah E|0M0 0\Z (a ∈ R)

of OP1 \Zt -modules by considering local sections s satisfying |s|h = O(|w|a+ ) for any  > 0.  

Theorem 1.7.9 (Theorem 7.3.1) The tuple P∗h E|0M0 t \Z  t ∈ ST1 is a good

filtered bundle over P h E 0 in the sense of Definitions 3.5.16 and 4.2.3. (See Theorem 7.3.4 for a more detailed description.)

  As we shall see in Proposition 7.3.5, the tuple P∗h E|0M0 t \Z  t ∈ ST1

0 up to boundedness. We shall determines the behaviour of the metric h around H∞ also show that the compatibility with such filtrations implies the GCK condition around infinity. (See Sect. 7.4.1.)

h 0 Remark

0 Although   we1 shall often use the abbreviation 1P∗ E to denote the

1.7.10 h tuple P∗ E|M0 t \Z  t ∈ ST , the filtrations depend on t ∈ ST as basic examples show. (See Sect. 5.1.) 

1.7.1.5 Kobayashi-Hitchin Correspondence in the Case λ = 0 0 )-module. Suppose that it is of Dirac type at Let V be a locally free O 0 (∗H∞ M \Z

each point of Z, and that  it is equipped with a tuple of filtered bundles P∗ V = P∗ (V 0 ) | t ∈ ST1 over V which is good in the sense of Definitions 3.5.16 |M t \Z and 4.2.3. It is denoted by P∗ V, and called a good filtered bundle of Dirac type over 0 0 ). (M ; Z, H∞

28

1 Introduction 0

0 For any good filtered bundle of

 Dirac type P∗ V over (M1 ; Z, H∞ ), we note that the numbers deg P∗ (V 0 ) are well defined for t ∈ ST \ π(Z), which induces |M t \Z

affine functions on the connected components of ST1 \ π(Z). We define deg(P∗ V) :=

1 T



T 0

deg P∗ (V

0

|M t \Z

 ) dt.

(1.15)

We define the stability condition for good filtered bundles of Dirac type over 0 0 ) by using the degree as in the standard way. (M ; Z, H∞ The following theorem is a natural analogue of Theorem 1.4.5 in the case λ = 0, in the context of periodic monopoles, for which we apply the Kobayashi-Hitchin correspondence for analytic stable bundles studied in [66]. Theorem 1.7.11 (Theorem 9.1.2) The construction from (E, h, ∇, φ) to P∗ E 0 induces a bijective correspondence between the equivalence classes of monopoles of GCK-type on M \ Z and the equivalence classes of polystable good filtered bundles 0 0 ). of Dirac type with degree 0 on (M ; Z, H∞ Remark 1.7.12 Let Σ be a compact Riemann surface, and let D ⊂ Σ denote a finite subset. Let gΣ\D be a Kähler metric of Σ \ D. Suppose that for each P ∈ D there exists a neighbourhood ΣP such that ΣP \{P } with gΣ\D is isomorphic to {|z| > R} with the metric dz dz. Let Z denotes a finite subset of ST1 ×(Σ \D). Theorem 1.7.22 can be generalized to correspondences for monopoles on (ST1 × (Σ \ D)) \ Z with  the metric dt dt + gΣ\D satisfying similar conditions around (ST1 × D) ∪ Z. 1.7.1.6 O

0

M \Z

0 )-Modules and C(w)-Modules (∗H∞

with an Automorphism Let V be a locally free O

0

M \Z

0 )-module of Dirac type at Z. Let D denote the (∗H∞

image of Z by the projection pw : M0 = ST1 × Cw −→ Cw . Take a sufficiently small positive number . By the scattering map along the loop − + s (0 ≤ s ≤ T ), we obtain the automorphism ρ of the OP1 (∗(D ∪ {∞}))-module V 0 . |M −

1  0 , and V := C(w) ⊗C[w] V . It is equipped with a We set V := H P , V 0 |M − C(w)-linear automorphism ρ, i.e., and (V , ρ) is a 0-difference module.  −1 (x) ∩ Z . If m(x) > 0, we obtain 0 ≤ For any x ∈ C, we set m(x) := pw −1 (x) ∩ Z = {(t (i) , x)}. We set tx(1) < tx(2) < · · · < txm(x) < T defined by pw x (i) (i) (i) (i+1) τx := tx /T . For i = 1, . . . , m(x) − 1, by choosing tx < si < tx , we set Lx,i as V(si ,x) ⊗ C[[w − x]], where V(si ,x) denotes the stalk of V at (si , x). Thus, we obtain a parabolic structure at finite place (V , {τ x , Lx }) of V .

1.7 Kobayashi-Hitchin Correspondences for Periodic Monopoles

29

Let (P∗ (V 0 ) | t ∈ ST1 ) be a good filtered bundle over V. By the definition |M t  of good filtered bundles over V (Definition 3.5.16), P∗ V 0 induces a good |M 0 parabolic structure of V at ∞. By this correspondence, the degree (1.15) for

P∗ V is translated to the degree   . (See (1.9) for the parabolic difference module V ∗ = V , V , m, (τ x , Lx )x∈C , P∗ V Lemma 4.2.8.) Hence, the stability condition for P∗ V is equivalent to the stability condition for V ∗ . Thus, we obtain the following proposition. Proposition 1.7.13 (Proposition 4.2.16, Proposition 4.2.17) The above construction induces an equivalence between (stable, polystable) good filtered bundle of Dirac type of degree 0 and (stable, polystable) parabolic 0-difference modules of degree 0.  Theorem 1.7.1 in the case λ = 0 follows from Theorem 1.7.11 and Proposition 1.7.13.

1.7.2 The Correspondences in the General Case Let us outline the construction for general λ in a way as parallel to the case λ = 0 as possible.

1.7.2.1 Preliminary Consideration In Sect. 1.7.1, we explained the construction of a 0-difference module from a monopole √ of GCK-type on M \ Z, where one of the keys is the integrability [∇t − −1φ, ∇w ] = 0 as in the previous works in Sect. 1.3. If we choose another R-linear isomorphism Rt × Cw Rt 0 × Cβ 0

(1.16)

√ such that dt dt + dw dw = dt0 dt0 + dβ0 dβ 0 , we also obtain [∇t0 − −1φ, ∇β 0 ] = 0. It is natural to ask what objects we could obtain by using the mini-complex structure induced by (t0 , β0 ) instead of (t, w). (See Sect. 2.2 for mini-complex structure.)

1.7.2.2 Mini-complex Structure Corresponding to the Twistor Parameter λ We can easily observe that isomorphisms (1.16) satisfying dt dt + dw dw = dt0 dt0 + dβ0 dβ 0 are naturally parameterized by S 2 = P1 . Conceptually, such a decomposition is induced by a complex structure underlying the hyperkähler

30

1 Introduction

 manifold R4 with the standard Euclidean metric 4i=1 dxi2 . We choose a complex √ √ structure  2 (z, w) = (x1 + −1x2 , x3 + −1x4 ) for which we have dz dz+dw dw =2 dxi . We can regard the mini-complex manifold Rt × Cw as the quotient of C by the Rs -action defined as s • (z, w) = (z + s, w). (See Sect. 2.7.1.) In other words, we may regard that the coordinate system (t, w) on Rt × Cw is induced by (t, w) = (Im z, w). For the twistor parameter λ ∈ C, we have the corresponding complex structure of (R4 , dxi2), for which a complex coordinate system is given by (ξ, η) = (z + λw, w − λz). It induces a mini-complex structure on Rt × Cw , which is different from the mini-complex structure induced by (t, w) unless λ = 0. To emphasize the mini-complex structure depending on λ, we use the notation (Rt × Cw )λ . Note that the Rs -action is expressed as s • (ξ, η) = (ξ + s, η − λs) in terms of (ξ, η). Let (α0 , β0 ) be the complex coordinate system defined as (α0 , β0 ) =

1 (ξ − λη, η + λξ ). 1 + |λ|2

Because the Rs -action is expressed as s • (α0 , β0 ) = (α0 + s, β0 ), we obtain the induced mini-complex coordinate system (t0 , β0 ) = (Im α0 , β0 ) on (Rt ×Cw )λ , i.e., t0 =

1 − |λ|2 2 t+ Im(λw), 2 1 + |λ| 1 + |λ|2

β0 =

√  1 w + 2 −1λt + λ2 w . 2 1 + |λ| (1.17)

We note dt dt +dw dw = dt0 dt0 +dβ0 dβ 0 . The Z-action κ is described as follows in terms of (t0 , β0 ): √ 2 −1λ  , . κn (t0 , β0 ) = (t0 , β0 ) + nT · 1 + |λ|2 1 + |λ|2  1 − |λ|2

(1.18)

Because the complex vector fields ∂t0 and ∂β 0 are κ-invariant, we obtain the induced complex vector fields on M, which are also denoted by ∂t0 and ∂β 0 . Let Mλ denote the 3-dimensional manifold obtained as the (Rt × Cw )λ by κ.  Though Mλ = M = Mλ as C ∞ -manifolds, the mini-complex structures of Mλ  and Mλ are different if λ = λ . A C ∞ -function f on an open subset U ⊂ Mλ is called mini-holomorphic if ∂t0 f = ∂β 0 f = 0. Let OMλ denote the sheaf of mini-holomorphic functions on Mλ . Remark 1.7.14 Once the twistor parameter λ is fixed, the mini-complex coordinate system (t0 , β0 ) is uniquely determined by dt√0 dt0 + dβ0 dβ 0 = dt dt + dw dw up  to the multiplication of a complex number e −1ϕ (ϕ ∈ R) to β0 .

1.7 Kobayashi-Hitchin Correspondences for Periodic Monopoles

31

1.7.2.3 Another Coordinate System and the Compactification of Mλ As recalled in Sect. 1.3, it has been standard to use efficiently the mini-complex coordinate system like (t0 , β0 ) in the study of monopoles, which induces the orthogonal decomposition of the 3-dimensional Euclidean space. However, in our study of periodic monopoles, we also use another convenient coordinate system (t1 , β1 ) given as follows: √ 

 (t1 , β1 ) = t0 + Im(λβ0 ), (1 + |λ|2 )β0 = t + Im(λw), w + 2λ −1t + λ2 w . (1.19) We note that a C ∞ -function f on U ⊂ Mλ is mini-holomorphic if and only if ∂t1 f = ∂β 1 f = 0. The Z-action κ is described as √ κn (t1 , β1 ) = (t1 , β1 ) + n · (T , 2 −1λT ).

(1.20)

We have the partial compactification (Rt × Cw )λ = Rt1 × Cβ1 ⊂ Rt1 × P1β1 . By the same formula, we define the Z-action κ on Rt1 × P1β1 . The quotient space λ

M is naturally equipped with the induced mini-complex structure, and it is a λ := Mλ \ Mλ S 1 . Let O λ compactification of Mλ . Let H∞ λ (∗H∞ ) denote T λ

M

λ. the sheaf of meromorphic functions on M whose poles are contained in H∞

Remark 1.7.15 As in the case of λ = 0 or the case of harmonic bundles (see Remark 1.7.7), we would like to consider meromorphic objects or filtered objects λ λ ) to keep the information of the growth orders with respect to h, on (M , H∞ rather than the transcendental object on Mλ . It is the reason why we consider the λ  compactification M . Remark 1.7.16 When we regard the coordinate change from (t1 , β1 ) to (t0 , β0 ) as a diffeomorphism Rt1 × Cβ1 Rt0 × Cβ0 , it does not extend to a diffeomorphism Rt1 × P1β1 Rt0 × P1β0 . Similarly, the diffeomorphism Rt1 × Cβ1 Rt × Cw does not extend to a diffeomorphism Rt1 × P1β1 Rt × P1w . Note that we may regard Rt1 × P1β1 as the quotient of Cξ × P1η by the naturally induced Rs -action.  There are two reasons to use (t1 , β1 ) instead of (t0 , β0 ). It is one reason that for the coordinate system (t0 , β0 ) the Z-action in the Rt0 -direction is trivial if |λ| = 1. (Compare (1.18) with (1.20).) See Remark 1.7.20 for another reason. Clearly, we have (t0 , β0 ) = (t1 , β1 ) in the case λ = 0. Remark 1.7.17 As explained above, we obtain (ti , βi ) (i = 0, 1) from (ξ, η). If λ = 0, there exists another pair of mini-complex coordinate systems (ti† , βi† ) (i = 0, 1) obtained from (ξ † , η† ) = (λ−1 (w − λz), λ−1 (z + λw)) = (λ−1 η, λ−1 ξ ) which we may use to obtain difference modules from monopoles. See Remark 2.7.3. 

32

1 Introduction

1.7.2.4 Mini-holomorphic Bundles Associated with Monopoles Let Z be a finite subset of M. Let (E, h,√ ∇, φ) be a monopole on M\Z. Because the Bogomolny equation implies that [∇t0 − −1φ, ∇β 0 ] = 0, we obtain the locally free √ OMλ \Z -module E λ as the sheaf of local sections s of E such that (∇t0 − −1φ)s = ∇β 0 s = 0. If P ∈ Z is a Dirac type singularity of (E, h, ∇, φ), then E λ is of Dirac type at P as in Sect. 1.7.1.3.

1.7.2.5 Meromorphic Extension and Filtered Extension at Infinity λ

Let U be any open subset in M \ Z. As in Sect. 1.7.1.4, let P h E λ (U ) denote the λ satisfying the following. space of sections of E λ on U \ H∞ λ , there exists a neighbourhood U of P in U such that • For any P ∈ U ∩ H∞ P  N |s|UP \H∞ for some N. λ |h = O |w|

Thus, we obtain the O

λ

M \Z

λ )-module P h E λ . We shall prove the following. (∗H∞

λ )Proposition 1.7.18 (Proposition 7.2.1) P h E λ is a locally free O λ (∗H∞ M \Z module.

Let π λ : M λ

λ

−→ ST1 denote the map induced by (t1 , β1 ) −→ t1 . We set λ

M t1  = (π λ )−1 (t1 ) and Mλ t1  = Mλ ∩ M t1  for t1 ∈ ST1 . By taking the restriction, we obtain the locally free OMλ t1 \Z (∗∞)-module P h (E|λMλ t \Z ). As

1 in Sects. 1.4.3 and 1.4.4, we obtain an increasing sequence Pah E|λMλ t \Z (a ∈ R) 1 of O λ -modules by considering local sections s satisfying |s|h = O(|w|a+ ) M t1 \Z for any  > 0.

  Theorem 1.7.19 (Theorem 7.3.1) The tuple P∗h E|λMλ t \Z  t1 ∈ ST1 is a good 1

filtered bundle over P h E λ in the sense of Definitions 3.5.16 and 4.2.3. (See Theorem 7.3.4 for a more detailed description.)

  As we shall see in Proposition 7.3.5, the tuple P∗h E|λMλ t \Z  t1 ∈ ST1 1

λ up to boundedness. We shall determines the behaviour of the metric h around H∞ also show that the compatibility with such filtrations implies the GCK condition around infinity. (See Sect. 7.4.1.)

  We shall often use the abbreviation P∗h E λ to denote P∗h E|λMλ t \Z  t1 ∈ ST1 , 1

but we remark that the filtrations depend on t1 ∈ ST1 as basic examples show (see Sect. 5.1).

Remark 1.7.20 In the proof of Proposition 1.7.18 and Theorem 1.7.19, one of the key facts is that the holomorphic vector bundle with a Hermitian metric (E λ , h)|Mλ t1 \Z is acceptable (Lemma 7.2.2). (See Sect. 2.11.2 for the acceptability

1.7 Kobayashi-Hitchin Correspondences for Periodic Monopoles

33

condition.) Hence, we may apply the general result for acceptable bundles in [80] λ to extend (E λ , h)|Mλ t1 \Z across ∞ ∈ M t1  P1β1 as a filtered bundle. See Sect. 2.9.7 for a more detailed explanation. Let π0λ : Mλ −→ ST1 denote the projection induced by (t0 , β0 ) −→ t0 . We note that (E λ , h)|(π λ )−1 (t0 )\Z is not acceptable, in general. 0 We may use (t1 , β1 ) not only (t0 , β0 ) thanks to the√notion of mini-complex structure instead of a pair of differential operators ∇t0 − −1φ and ∇β 0 satisfying √ the integrability condition [∇t0 − −1φ, ∇β 0 ]. This is a merit to consider minicomplex structure.  Remark 1.7.21 To define the concept of good filtered bundle in Theorem 1.7.19, λ )-modules of finite we need to know the classification of locally free OHλ (∗H∞ ∞

λ is the formal space obtained as the completion of M along ∞ rank, where H λ H∞ . Because such modules are naturally equivalent to formal difference modules (see Sect. 3.4.3), we may apply the classical results on the classification of formal difference modules mentioned in Sect. 1.6.3. For our purpose, it is also convenient to use another equivalence of formal difference modules of level ≤ 1 and formal differential modules whose Poincaré rank is strictly smaller than 1, which will be explained in Sect. 3.6.  λ

1.7.2.6 Kobayashi-Hitchin Correspondence of Periodic Monopoles of GCK Type λ )-module. Suppose that it is of Dirac type at Let V be a locally free O λ (∗H∞ M \Z each point of Z, and that  it is equipped with a tuple of filtered bundles P∗ V =

P∗ (V λ ) | t1 ∈ ST1 over V which is good in the sense of Definitions 3.5.16 |M t1  and 4.2.3. We denote such object by P∗ V, and call a good filtered bundle of Dirac λ λ ). type over (M ; Z, H∞ λ λ ), we note that For any good filtered bundleof Dirac type P∗ V over (M ; Z, H∞ 1 λ the numbers deg P∗ (V λ ) are well defined for t1 ∈ ST \π (Z), which induces |M t1 

affine functions on the connected components of ST1 \ π λ (Z). We define 1 deg(P∗ V) := T



T 0

deg P∗ (V

λ

|M t1 

 ) dt1 .

(1.21)

We define the stability condition for good filtered bundles of Dirac type over λ λ ) by using the degree as in the standard way. (M ; Z, H∞ The following theorem is a generalization of Theorem 1.7.11 to the case of general λ, and it is a natural analogue of Theorem 1.4.5 in the context of periodic monopoles. We again apply the Kobayashi-Hitchin correspondence for analytic stable bundles studied in [66].

34

1 Introduction

Theorem 1.7.22 (Theorem 9.1.2) The construction from (E, h, ∇, φ) to P∗ E λ induces a bijective correspondence between the equivalence classes of monopoles of GCK-type on M \ Z and the equivalence classes of polystable good filtered bundles λ λ ). of Dirac type with degree 0 on (M ; Z, H∞ The following is an analogue of Corlette-Simpson correspondence between flat bundles and Higgs bundles, which is an immediate consequence of Theorem 1.7.22. Corollary 1.7.23 (Corollary 9.1.3) For any λ ∈ C, there exists the natural bijective correspondence of the following objects through periodic monopoles of GCK-type: 0

0 ). • Polystable good filtered bundles of Dirac type with degree 0 on (M ; Z, H∞ • Polystable good filtered bundles of Dirac type with degree 0 on λ λ ). (M ; Z, H∞ 

1.7.2.7 Difference Modules and O

λ

M \Z

Let V be a locally free O P1β1 −→ Rt1 and p2 :

λ

M \Z Rt1 × P1β1

λ )-Modules (∗H∞

λ )-module of Dirac type at Z. Let p : R × (∗H∞ 1 t1 1  −→ P denote the projections. Let Z ⊂ Rt1 × P1 β1

be the pull back of Z by the projection

λ

:

Rt1 × P1β1

λ

β1

−→ M . We put



  ∩ p−1 {0 ≤ t1 < T } . D := p2 Z 1 Take a sufficiently small positive number . By the scattering map, we obtain the following isomorphism of OP1 (∗(D ∪ {∞}))-modules: ( λ )∗ (V)|p−1 (−) (∗D) ( λ )∗ (V)|p−1 (T −) (∗D). 1

(1.22)

1

We also have the following natural isomorphism Φ ∗ ( λ )∗ (V)|p−1 (T −) ( λ )∗ (V)|p−1 (−) . 1

(1.23)

1

 We set V := H 0 P1 , ( λ )∗ (V)|p−1 (−) . Let C[β1 ]D denote the localization of 1  C[β1 ] by x∈D (β1 − x). By (1.22) and (1.23), we obtain the C-linear isomorphism Φ ∗ : V ⊗C[β1 ] C[β1 ]D −→ V ⊗C[β1 ] C[β1 ]Φ −1 (D) .

(1.24)

We set V := C(β1 ) ⊗ V . It is equipped with a C-linear automorphism Φ ∗ induced by (1.24), and (V , Φ ∗ ) is a difference module.

1.8 Asymptotic Behaviour of Periodic Monopoles of GCK-Type

35

  . If m(x) > 0, we obtain 0 ≤ For any x ∈ C, we set m(x) := p2−1 (x) ∩ Z (1) (2) m(x)  We set τx(i) := t (i) /T . For i = t1,x < t1,x < · · · < t1,x < T as p2−1 (x) ∩ Z. 1,x (i)

(i+1)

1, . . . , m(x) − 1, by choosing t1,x < si < t1,x , we set Lx,i as V(si ,x) ⊗ C[[β1 − x]], where V(si ,x) denotes the stalk of V at (si , x). Thus, we obtain a parabolic structure at finite place (V , {τ x , Lx }) of V .  Let P∗ (V λ ) | t ∈ ST1 be a good filtered bundle over V. We obtain |M t1 \Z 1  the filtered bundle P∗ ( λ )−1 (V)|p−1 (0) over ( λ )−1 (V)|p−1 (0) induced by the 1 1 filtered bundle P∗ (V λ ). By the definition of good filtered bundles over V |M 0 (Definition 3.5.16), it induces a good parabolic structure of V at ∞. By this correspondence, the degree in (1.21) for P∗ V is translated to the degree   . (See (1.9) for the parabolic difference module V ∗ = V , V , m, (τ x , Lx )x∈C , P∗ V Lemma 4.2.8.) Hence, the stability condition for P∗ V is equivalent to the stability condition for V ∗ . Thus, we obtain the following proposition. Proposition 1.7.24 (Proposition 4.2.16, Proposition 4.2.17) The above construction induces an equivalence between (stable, polystable) good filtered bundle of Dirac type of degree 0 and (stable, polystable) difference modules of degree 0.  Theorem 1.7.1 for general λ follows from Theorem 1.7.22 and Proposition 1.7.24.

1.8 Asymptotic Behaviour of Periodic Monopoles of GCK-Type In the theory of wild harmonic bundles (E, ∂ E , θ, h), the first task is to study the asymptotic behaviour of h and θ around the singularity, called Simpson’s main estimate. See [80, Theorem 1] for the tame case, and [64, §7.2] for the wild case. The estimate is fundamental not only for the study of the filtered extension (P∗h E, θ ) of the Higgs bundle (E, ∂ E , θ ) but also for the study of the filtered extensions (P∗h E λ , Dλ ) (λ ∈ C) of the λ-flat bundles (E λ , Dλ ) underlying the harmonic bundle. We pursue a similar route in our study of periodic monopoles by following the idea that periodic monopoles are regarded as harmonic bundles of infinite rank. (See Sect. 1.5.4.) Indeed, for the results in Sects. 1.7.1.4 and 1.7.1.5 and Sects. 1.7.2.5 and 1.7.2.6, it is fundamental to understand the asymptotic behaviour of monopoles (E, h, ∇, φ) on B ∗ (R) = ST1 × {w ∈ C | |w| > R} such that F (∇) → 0 and |φ|h = O(log |w|) as |w| → ∞, which we call GCK-condition. We shall briefly describe the results in this section.

36

1 Introduction

1.8.1 Setting It is more convenient to study monopoles on a ramified covering. Namely, we set ∗ (R ) := {w ∈ C | |w q | > R } for q ∈ Z Uw,q 1 q 1 ≥1 and for R1 > 0. There exists the q q ∗ natural map Uw,q (R1 ) −→ Cw defined by wq −→ wq . We consider monopoles ∗ (R ) with respect to the Riemannian metric (E, h, ∇, φ) on Bq∗ (R1 ) = ST1 × Uw,q 1 2 2(q−1) dwq dwq . In the rest of this section, we dt dt + dw dw = dt dt + q |wq | impose the GCK-condition, i.e., F (∇) → 0 and |φ|h = O(log |wq |) as |wq | → ∞. For each λ ∈ C, Bq∗ (R1 ) has the mini-complex structure induced by the covering map Bq∗ (R1 ) −→ B ∗ (R) and the mini-complex structure of B ∗ (R) as an open subset of Mλ . (See Sect. 1.7.2.2 for Mλ .) When we emphasize the mini-complex structure depending on λ, we use the notation Bqλ∗ (R1 ). Let (E λ , ∂ E λ ) denote the mini-holomorphic vector bundle on Bqλ∗ (R1 ) underlying the monopole (E, h, ∇, φ). (See Sect. 2.3 for mini-holomorphic bundles.) As in the case of harmonic bundles, we begin with the analysis of (E 0 , ∂ E 0 ) with h, and we prove Theorem 1.8.2 below which is an analogue of the asymptotic orthogonality for wild harmonic bundles contained in Simpson’s main estimate (see [64, Theorem 8.2.1]). It is fundamental even for the study of (E λ , ∂ E λ , h) as in the case of harmonic bundles. To show it, we outline our proof of Proposition 1.7.18 and Theorem 1.7.19.

1.8.2 Decomposition of Mini-holomorphic Bundles By considering the monodromy along ST1 × {wq }, we obtain the automorphism 0 ∗ M(wq ) of E|{0}×U ∗ (R ) . The eigenvalues of M(wq ) (wq ∈ Uw,q ) determine a 1 w,q

∗ (R ) × C∗ , called the spectral curve of the periodic complex curve Sp(E 0 ) in Uw,q 1 ∗ (R ) ∪ {∞} in P1 . Under the monopole. (See [13, 14].) Set Uw,q (R1 ) := Uw,q 1 wq

GCK-condition, the closure Sp(E 0 ) of the spectral curve Sp(E 0 ) in Uw,q (R1 )×P1 is also a complex analytic curve. (See Proposition 6.3.13.) Hence, after taking a ramified covering, we may assume that there exists the following decomposition: Sp(E 0 ) =



Si .

i∈Λ

Here, each Si is a graph of a meromorphic function gi on (Uw,q , ∞). There exist − (i , αi ) ∈ Z × C∗ such that gi ∼ αi wq i . We obtain Sp(E 0 ) =

 (,α)∈Z×C∗



 i∈Λ (i ,αi )=(,α)

 Si .

1.8 Asymptotic Behaviour of Periodic Monopoles of GCK-Type

37

We obtain the corresponding decomposition of mini-holomorphic bundles: (E 0 , ∂ E 0 ) =



(E,α , ∂ E,α ).

(1.25)

(,α)

1.8.3 The Induced Higgs Bundles 1.8.3.1 Preliminary (1) ∗ (R ) -module with an endomorphism g. Let Ψq : Let V be a locally free OUw,q 1 ∗ (R ) denote the projection. Let V  be the C ∞ -bundle on Bq∗ (R1 ) Bq∗ (R1 ) −→ Uw,q 1 obtained as the pull back of V by Ψq . We obtain the naturally defined operator ∂V,w  determined by ∂V,w (f Ψq−1 (s)) = ∂w (f )Ψq−1 (s) + f Ψq−1 (∂V ,w s) for any on V ∗ (R ). We also have the operator f ∈ C ∞ (Bq∗ (R1 )) and C ∞ section s of V on Uw,q 1  determined by the following condition for f and s as above: ∂V,t on V

√ ∂V,t (f Ψq−1 (s)) = ∂t (f )Ψq−1 (s) − 2 −1f Ψq−1 (gs).  on Bq0∗ (R1 ). We set Thus, we obtain a mini-holomorphic structure ∂ V on V , ∂ ). Ψq∗ (V , g) := (V V √ Remark 1.8.1 The monodromy along ST1 × {wq } is exp(2 −1T g(wq )).  1.8.3.2 Preliminary (2) For each (, α) ∈ Z × C∗ , there exists a basic example of monopole (L∗q (, α), hL,q,,α , ∇L,q,,α , φL,q,,α )

(1.26)

√ on Bq∗ (R1 ), for which the monodromy along ST1 × {wq } with respect to ∇t − −1φ is given as the multiplication of αwq− . (See Sect. 5.2.4.) Let L0∗ q (, α) denote the underlying mini-holomorphic bundle on Bq0∗ (R1 ). 1.8.3.3 The Induced Higgs Bundles It is not difficult to see that, for each (E,α , ∂ E,α ) in (1.25), there exist a ∗ (R ) holomorphic vector bundle with an endomorphism (V,α , ∂ V,α , g,α ) on Uw,q 1 and an isomorphism ∗ (E,α , ∂ E,α ) L0∗ q (, α) ⊗ Ψq (V,α , ∂ V,α , g,α ).

38

1 Introduction

We may choose g,α such that the eigenvalues of g,α|wq goes to 0 as |wq | → ∞. By setting θ,α := g,α dw, we obtain Higgs bundles (V,α , ∂ V,α , θ,α ).

1.8.4 Asymptotic Orthogonality Let h,α be the restriction of h to E,α . We obtain the metric h,α ⊗ h−1 L,q,,α of

Ψq∗ (V,α , ∂ V,α , g,α ). There exists the Fourier expansion of h,α ⊗ h−1 L,q,,α along 1 the fibers ST × {wq }, and it turns out that the invariant part induces a metric hV ,,α of V,α . We obtain the following metric of E: h :=



 hL,q,,α ⊗ Ψq−1 hV ,,α .

(,α)

Let s be the automorphism of E determined by h = h · s. The following theorem implies that the difference of h and h decays rapidly. Theorem 1.8.2 (Theorem 6.3.4, Proposition 6.3.8) For any m ∈ Z≥0 , there exist positive constants Ci (m) (i = 1, 2) such that  

  q  ∇κ1 ◦ · · · ◦ ∇κm (s − idE ) ≤ C1 (m) exp − C2 (m)|wq | h

for any (κ1 , . . . , κm ) ∈ {t, w, w}m . As a consequence of Theorem 1.8.2, (V,α , ∂ V,α , θ,α , hV ,,α ) asymptotically satisfies the Hitchin equation. Namely, let F (hV ,,α ) denote the curvature of the † Chern connection determined by ∂ V,α and hV ,,α , and let θ,α be the adjoint of θ,α with respect to hV ,,α . Then, we obtain the following decay for some  > 0:    q †  F (hV ,,α ) + θ,α , θ,α = O exp(−|wq |) . We also obtain similar estimates for any derivatives of the left hand side. In this sense, (V,α , ∂ V,α , θ,α , hV ,,α ) is an asymptotic harmonic bundle.

1.8.5 Curvature Decay Many of the estimates for harmonic bundles can be generalized to estimates for asymptotic harmonic bundles. (See Sect. 10.1.1 and [65, §5.5].) It allows us to obtain estimates for periodic monopoles of GCK-type. For example, we obtain the estimate for the curvature F (∇) of the monopole (E, h, ∇, φ) of GCK-type.

1.8 Asymptotic Behaviour of Periodic Monopoles of GCK-Type

39

Corollary 1.8.3 (Corollary 6.3.12) For the expression F (∇) = F (∇)w,w dw dw + F (∇)w,t dw dt + F (∇)w,t dw dt, the following estimates hold:     F (∇)ww  = O |wqq |−2 (log |wq |)−2 , h   F (∇)wt  = O(|wqq |−1 ), h

  F (∇)wt  = O(|wqq |−1 ). h

The estimates in Corollary 1.8.3 are useful in the study of (E λ , ∂ E λ ) for any λ. Let (t1 , β1 ) be the mini-complex coordinate system as in Sect. 1.7.2.3. Note that the complex vector fields ∂β 1 are defined on Bq∗ (R1 ). We obtain the differential operator ∂E λ ,β 1 on E λ induced by ∂β 1 and the mini-holomorphic structure ∂ E λ . We also obtain ∂E λ ,h,β1 induced by h and ∂E λ ,β 1 in the standard way. According to Proposition 2.9.5, there exists the following relation: 

 ∂E λ ,h,β1 , ∂E λ ,β 1 =

1 F (∇)w,w . 1 + |λ|2

√ We also note that β1 = (1 + |λ|2 )w + 2λ −1t1 . Hence, we obtain the following estimate as |β1 | → ∞ where t1 varies in a compact set:       ∂E λ ,h,β1 , ∂E λ ,β 1  = O |β1 |−2 (log |β1 |)−2 . h

(1.27)

This means the acceptability mentioned in Remark 1.7.20, which is useful in the proof of Proposition 1.7.18 and Theorem 1.7.19.

1.8.6 The Filtered Extension in the Case λ = 0 Let us explain an outline of the proof of Proposition 1.7.8 and Theorem 1.7.9 in the case λ = 0, which are easier than the claims for general λ (Proposition 1.7.18 and Theorem 1.7.19). We discuss it in the ramified case under the setting in Sect. 1.8.1. ∗ =U Let Uw,q be a neighbourhood of ∞ in P1wq , and Uw,q w,q \ {∞}. We assume ∗ ∗ that Uw,q ⊂ Uw,q (R1 ). We have the natural partial compactification Bq0 = ST1 × ∗ . Let H 0 0 0∗ Uw,q of the mini-complex manifold Bq0∗ = ST1 × Uw,q ∞,q := Bq \ Bq = ST1 × {∞}. Let πq0 : Bq0 −→ ST1 denote the projection induced by (t, wq ) −→ t. For t ∈ ST1 , we set Bq0 t = (πq0 )−1 (t) and Bq0∗ t = B 0∗ ∩ Bq0 t. 0 )-module P h E 0 by the procedure in Sect. 1.7.1.4 from We define the OBq0 (∗H∞,q 0 )-module by the E 0 with h. It is easy to prove that P h E 0 is a locally free OBq0 (∗H∞,q

40

1 Introduction

acceptability (1.27) as in Proposition 1.7.8. Moreover, we obtain a tuple of filtered bundles {P∗h (E|0B0∗ t  ) | t ∈ ST1 }. To obtain Theorem 1.7.9, we need to prove that the q

tuple is good. By using the estimates for the asymptotic harmonic bundles, we obtain the filtered bundle P∗ V,α on (Uw,q , ∞) from the holomorphic bundle (V,α , ∂ V,α ) ∗ with the metric hV ,,α . (See Sect. 10.1.) Note that the projection Ψq : Bq0∗ −→ Uw,q given by (t, wq ) −→ wq naturally extends to Ψq0 : Bq0 −→ Uw,q . We can naturally generalize the construction in Sect. 1.8.3.1 to the construction of filtered bundle on 0 ) from a filtered bundle with endomorphism on (U (Bq0 , H∞,q w,q , ∞). Hence, we 0 ∗ ∗ obtain a filtered extension (Ψq ) (P∗ V,α , g,α ) of Ψq (V,α , g,α ), which is good (see Proposition 4.5.2 which goes back to Proposition 3.6.15). It is easy to see that Ψq∗ (P∗ V,α , g,α ) is equal to the filtered extension of Ψq∗ (V,α , g,α ) with respect to Ψq−1 (hV ,,α ).    By explicit computations in Sect. 5, we obtain that P∗ L0∗ q (, α)|Bq0∗ t  t ∈  ST1 is good. Because

 ∗ P∗h (E|0B0∗ t  ) = P∗ L0∗ q (, α)|Bq0∗ t  ⊗ Ψq (P∗ V,α , g,α )|Bq0∗ t  q

(t ∈ ST1 )

we obtain that {P∗h (E|0B0∗ t  ) | t ∈ ST1 } is good. This is an outline of the proof of q

Theorem 1.7.9 in the case λ = 0.

1.8.7 The Filtered Extension for General λ 1.8.7.1 Ramified Covering Space λ

We continue to use the notation in Sect. 1.8.6. There M −→ P1w √ exists the map 2 −1 λ induced by (t1 , β1 ) −→ (1 + |λ| ) (β1 − 2 −1t1 ). Let Bq denote the fiber λ

product of M and Uw,q over P1w . Let Ψqλ : Bqλ −→ Uw,q denote the projection. λ ∗ ). Because the induced := (Ψqλ )−1 (∞) and Bqλ∗ := (Ψqλ )−1 (Uw,q We set H∞,q map Bqλ∗ −→ Mλ is a local diffeomorphism, Bqλ∗ inherits the locally Euclidean metric and the mini-complex structure. Note that Bqλ∗ = Bq0∗ as Riemannian manifolds because Mλ = M0 as Riemannian manifolds. We can observe that the mini-complex structure of Bqλ∗ uniquely extends to a mini-complex structure of Bqλ . (See Sect. 4.3.2.) We also have the naturally defined map πqλ : Bqλ −→ ST1 which is induced by the projection (t1 , β1 ) −→ t1 where (t1 , β1 ) denotes the mini-complex coordinate system in Sect. 1.7.2.3. We set Bqλ t1  = (πqλ )−1 (t1 ) and Bqλ∗t1  = Bqλ t1  ∩ Bqλ∗. We obtain the mini-holomorphic bundle (E λ , ∂ E λ ) on Bqλ∗ as in Sect. 1.8.1. We λ )-module P h (E λ ) from (E λ , ∂ , h) by the procedure in obtain the OBqλ (∗H∞,q Eλ

1.8 Asymptotic Behaviour of Periodic Monopoles of GCK-Type

41

λ )-module by the Sect. 1.7.2.5. We can prove that it is a locally free OBqλ (∗H∞,q acceptability (1.27), 1.7.18. Moreover, we obtain the tuple of  as in Proposition   filtered bundles P∗h E|λBλ∗ t   t1 ∈ ST1 . To obtain Theorem 1.7.19, we need to q

1

prove that the tuple is good.

1.8.7.2 Approximation As an analogue of the Chern connection for holomorphic vector bundles with  a Hermitian metric, we construct ∂E 0 and φ  from (E 0 , ∂ E 0 ) with the metric

q   h . By Theorem 1.8.2, we obtain ∇ − (∂ E 0 + ∂E 0 ) = O exp(−|wq |) and

 q φ − φ  = O exp(−|wq |) for some  > 0. We construct the differential operator 





∂ E λ of E by using the connection ∂ E 0 + ∂E 0 and φ  as in Sect. 7.3.4. Then, ∂ E λ is

   asymptotically a mini-holomorphic structure, i.e., ∂ E λ ◦ ∂ E λ = O exp(−|wq |q ) .

  We also have ∂ E λ − ∂ E λ = O exp(−|wq |q ) . Let CB∞λ denote the sheaf of q

 on Bqλ . It turns out that ∂ E λ induces  CB∞λ ⊗O λ P h (E λ ), and ∂ E λ and ∂ E λ induce the Bq q λ . completion of P h (E λ ) along H∞,q

C ∞ -functions

a

C ∞ -differential

operator on

same operator on the formal

1.8.7.3 Formal Completion of Asymptotic Harmonic Bundles at Infinity Let (V,α , ∂ V,α , θ,α , hV ,,α ) be the asymptotic harmonic bundle in Sect. 1.8.3. Let ∂ V,α + ∂V,α denote the Chern connection associated with hV ,,α . We obtain † λ the holomorphic vector bundle V,α = (V,α , ∂ V,α + λθ,α ). It turns out that λ λ on (V,α , hV ,,α ) is acceptable, and hence we obtain the filtered extension P∗ V,α † + λ∂V,α + θ,α of V,α . (Uw,q , ∞). We have the λ-connection Dλ,α = ∂ V,α + λθ,α

q  Though it is not necessarily flat, we have Dλ,α ◦ Dλ,α = O exp(−|wq |) . It turns out that as the formal completion at infinity, we obtain a formal good filtered λ-flat λ λ ,  bundle (P∗ V ,α D,α ) as in the case of wild harmonic bundles. (See Sects. 10.1.3 and 10.1.4.)

1.8.7.4 The Formal Structure of P h E λ at Infinity λ λ . ∞,q as the formal completion of Bqλ along H∞,q We can consider the formal space H λ t  = B λ ∩ H λ , and let H λ t  denote the formal ∞,q (See Sect. 3.4.) We set H∞,q 1 1 q ∞,q λ λ completion of Bq t1  along H∞,q t1 .

42

1 Introduction λ λ We obtain the OH∞,q induced by P(E λ ). We also λ (∗H∞,q )-module P(E )|H λ ∞,q

have the tuple of filtered bundles P∗ (E λ )|Hλ

∞,q t1 

(t1 ∈ ST1 ).

λ∗ We have the mini-holomorphic bundles Lλ∗ q (, α) on Bq underlying the λ )-modules monopoles (1.26). As their filtered extension, we obtain OBqλ (∗H∞,q    Lλq (, α) and the tuples of filtered bundles P∗ Lλq (, α)|Bqλ t1   t1 ∈ ST1 . As the λ , we obtain O λ λ formal completion along H∞,q λ (∗H∞,q )-modules Lq (, α)|H λ ∞,q ∞,q H    λ 1  and the good tuple of the filtered bundles P∗ Lq (, α)|H∞,q λ t  t1 ∈ ST . 1 As we shall explain in Sect. 3.6, from the formal good filtered λ-flat bundles λ λ ∗ ,α ,  ,α ,  (P∗ V Dλ,α ), good filtered OH∞,q Dλ,α ) are λ (∗H∞,q )-modules (Ψq ) (P∗ V obtained. (See Proposition 3.6.15.) By the approximation in Sect. 1.8.7.3, and by the compatibility as in Propoλ )sition 2.8.3 and Lemma 4.5.12, there exists an isomorphism of OHλ (∗H∞,q ∞,q modules ,α ,  P(E λ )|H∞,q

Lλq (, α)|H∞,q ⊗ (Ψqλ )∗ (V Dλ,α ), λ λ

which induces an isomorphism of the tuples of filtered bundles P(E λ )|H∞,q λ t  1



λ ∗ ,α ,  P∗ Lλq (, α)|H∞,q Dλ,α )|H∞,q λ t  ⊗ (Ψq ) (P∗ V λ t  . 1 1

   1 is good. This is an outline  Thus, we obtain that the tuple P∗ (E λ )|H∞,q λ t  t1 ∈ ST 1 of our proof of Theorem 1.7.19.

Chapter 2

Preliminaries

Abstract We introduce some basic notions such as mini-holomorphic bundles which are useful in our study of monopoles. We also recall the dimensional reductions to relate instantons, monopoles, harmonic bundles and their underlying holomorphic objects.

2.1 Outline of This Chapter In Sect. 2.2, we introduce the notion of mini-complex structure of three-dimensional manifolds. In Sect. 2.3, we introduce the notion of mini-holomorphic vector bundles on mini-complex three-dimensional manifolds. The notions of mini-complex structure and mini-holomorphic bundle have been implicitly used in the study of monopoles. In Sect. 2.4, we recall the notion of monopole as mini-holomorphic bundles equipped with a Hermitian metric satisfying (2.3). We also recall the notion of Dirac type singularity of monopoles [51], and a characterization which easily follows from [71]. In Sect. 2.5, we recall the dimensional reduction of instantons to monopoles. We also explain the underlying dimensional reduction of holomorphic bundles to mini-holomorphic bundles. In Sect. 2.6, we explain the dimensional reduction of monopoles to harmonic bundles. We also explain the underlying dimensional reduction of mini-holomorphic bundles to Higgs bundles in Sect. 2.6.2. In Sect. 2.7, we study the twistor family of mini-complex structures on M = (R/T Z) × C for T > 0. For λ ∈ C, we obtain the mini-complex manifold Mλ as M equipped with a mini-complex structure corresponding to λ. We introduce two convenient mini-complex local coordinate systems (t0 , β0 ) and (t1 , β1 ) for Mλ . We λ introduce a compactification M of Mλ by using the coordinate system (t1 , β1 ). λ := Mλ \ Mλ . Set H∞ In Sect. 2.8, we study the dimensional reduction from OMλ -modules to λ-flat bundles on Cw for general λ, and its compatibility with the dimensional reduction from monopoles to harmonic bundles. The case of λ = 0 is already essentially explained in Sect. 2.6. The case of λ = 0 is also useful for our study. We also explain © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. Mochizuki, Periodic Monopoles and Difference Modules, Lecture Notes in Mathematics 2300, https://doi.org/10.1007/978-3-030-94500-8_2

43

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the dimensional reduction from O λ -modules to an OP1w -modules equipped with a M meromorphic λ-connection. In Sect. 2.9, we shall introduce a section G(h) of End(E) for a mini-holomorphic bundle (E, ∂ E ) with a Hermitian metric h on an open subset of Mλ . It is an analogue of the contraction of the curvature of the Chern connection for a holomorphic vector bundle with a Hermitian metric on a Kähler manifold. In Sects. 2.9.2– 2.9.4, we explain some general formulas for G(h) which are analogues of standard and useful formulas in the study of Hermitian-Einstein metrics [79]. We give a complement to Sect. 2.8 in Sect. 2.9.5, which is mainly a preliminary for the proof of Proposition 7.4.3. In Sect. 2.9.6, we recall the dimensional reduction of instantons with respect to the Hopf fibration due to Kronheimer, and prove a formula which will be useful in the proof of Proposition 9.2.2. In Sect. 2.10, we explain an equivalence between difference modules with parabolic structure at finite place and locally free OMλ \Z (∗Hλ∞ )-modules for finite subsets Z. In Sect. 2.11, we review filtered bundles on punctured curves, and we recall the general construction of a filtered bundle from a holomorphic bundle with a Hermitian metric satisfying the acceptability condition.

2.2 Mini-Complex Structures on 3-Manifolds 2.2.1 Mini-Holomorphic Functions on R × C Let t and w be the standard coordinates of R and C, respectively. The orientation of R × C is given as the product of the orientation of R and C. Let U be an open subset in R × C. A C ∞ -function f on U is called mini-holomorphic if ∂t f = 0 and ∂w f = 0. Let Ui (i = 1, 2) be open subsets in R × C. Let F : U1 −→ U2 be a diffeomorphism. It is called mini-holomorphic if (i) F preserves the orientations, (ii) F ∗ (f ) is mini-holomorphic for any mini-holomorphic function f on U2 . Lemma 2.2.1 Let F = (Ft (t, w), Fw (t, w)) : U1 −→ U2 be a diffeomorphism. Then, F is mini-holomorphic if and only if we have ∂t Ft > 0, ∂t Fw = 0 and ∂w Fw = 0. Proof Suppose that F is mini-holomorphic. Because F ∗ (w) = Fw is miniholomorphic, we obtain ∂t Fw = ∂w Fw = 0. Because F is orientation preserving, we obtain ∂t Ft · |∂w Fw |2 > 0, which implies ∂t Ft > 0. The converse is also easily proved. 

2.2 Mini-Complex Structures on 3-Manifolds

45

2.2.2 Mini-Complex Structure on Three-Dimensional Manifolds Let us define the notion of mini-complex structure for Three-dimensional manifolds as in the case of smooth structure. (For example, see [53]). Let M be an oriented three-dimensional C ∞ -manifold. A mini-complex atlas on M is a family of open subsets Uλ (λ ∈ Λ) with an orientation-preserving embedding ϕλ : Uλ −→ R × C satisfying the following conditions.  • M = λ∈Λ Uλ . • The coordinate change ϕλ (Uλ ∩ Uμ ) −→ ϕμ (Uλ ∩ Uμ ) is mini-holomorphic. Such (Uλ , ϕλ ) is called a mini-complex chart. We shall often use (t, w) instead of ϕλ . Two mini-complex atlas {(Uλ , ϕλ ) | λ ∈ Λ} and {(Vγ , ψγ ) | γ ∈ Γ } are defined to be equivalent if the union is also a mini-complex atlas. There exists the partial order on the family of mini-complex atlases defined by inclusions. Each equivalence class of mini-complex atlases has a unique maximal mini-complex atlas. A minicomplex structure on M is defined to be a maximal mini-complex atlas on M. When M is equipped with a mini-complex structure, a C ∞ -function f on M is called mini-holomorphic if its restriction to any mini-complex chart is miniholomorphic. Let OM denote the sheaf of mini-holomorphic functions.

2.2.3 Tangent Bundles Suppose that M is equipped with a mini-complex structure. Let (U ; t, w) be a minicomplex chart. The real vector field ∂t determines an oriented subbundle TS U of T U . The quotient bundle TQ U is equipped with the complex structure J , where J is an automorphism of TQ U such that J 2 = −1. Because the mini-complex coordinate change preserves the subbundles, we obtain a globally defined subbundle TS M of T M of rank 1. We also obtain the quotient bundle TQ M, which is equipped with the complex structure. Let us consider T C M := T M ⊗R C. There exists the decomposition of complex √ vector bundles TQC M = TQ1,0 M ⊕ TQ0,1 M, where TQ1,0 M is the −1-eigen bundle √ of J , and TQ0,1 M is the − −1-eigen bundle of J . We obtain the exact sequence of complex vector bundles on M: −−−−→

−−−−→

−−−−→

−−−−→

1,0 0,1 Let ΘM and ΘM denote the inverse image of TQ1,0 M and TQ0,1 M by a1 , respec1,0 0,1 tively. Note that ΘM ∩ ΘM = TSC M. The complex conjugate on C induces the

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2 Preliminaries

1,0 0,1 complex conjugate on T M ⊗ C for which we obtain ΘM = ΘM . It also induces

the complex conjugate on TQ1,0 M ⊕ TQ0,1 M for which TQ1,0 M = TQ0,1 M holds. Lemma 2.2.2 Let F : M1 −→ M2 be a diffeomorphism of oriented threedimensional manifolds. Suppose that Mi are equipped with mini-complex structure, and that the following holds: • F is orientation preserving. • dF (TS M1 ) = TS M2 . • The induced isomorphism TQ M1 TQ M2 is C-linear. Then, F preserves the mini-complex structures. Proof It is enough to study the claim locally around any point of Mi . Let Ui (i = 1, 2) be open subsets of R × C which are naturally equipped with mini-complex structures. Let F : U1 −→ U2 be an orientation preserving diffeomorphism such that (i) dF (TS U1 ) = TS U2 , (ii) the induced isomorphism TQ U1 TQ U2 is Clinear. We have the expression F = (Ft (t, w), Fw (t, w)). By the condition (i), we obtain ∂t Fw (t, w) = 0. By the condition (ii), we obtain ∂w Fw (t, w) = 0. Hence, F is mini-holomorphic. 

2.2.4 Cotangent Bundles 1,0 0,1 Let ΩQ and ΩQ denote the complex dual bundles of TQ1,0 M and TQ0,1 M, 1,0 0,1 1,0 0,1 respectively. Let ΩM and ΩM denote the complex dual bundle of ΘM and ΘM . We obtain the following exact sequences: 1,0 0,1 ⊕ ΩQ −→ T ∗ M ⊗ C −→ (TSC M)∨ −→ 0, 0 −→ ΩQ 1,0 1,0 −→ ΩM −→ (TSC M)∨ −→ 0, 0 −→ ΩQ 0,1 0,1 −→ ΩM −→ (TSC M)∨ −→ 0, 0 −→ ΩQ 1,0 0,1 −→ T ∗ M ⊗ C −→ ΩM −→ 0, 0 −→ ΩQ 0,1 1,0 −→ T ∗ M ⊗ C −→ ΩM −→ 0. 0 −→ ΩQ

2 0,1 0,1 1,0 0,2 2,0 = rankC ΩM = 2. We set ΩM := ΩM and ΩM := Note rankC ΩM 2 1,0 ΩM . For any C ∞ -vector bundle E on M, let C ∞ (M, E) denote the space of C ∞ sections of E.

2.3 Mini-Holomorphic Bundles

47

The exterior derivative d : C ∞ (M, C) −→ C ∞ (M, T ∗ M ⊗ C) induces the following differential operators: 0,1 ∂ M : C ∞ (M, C) −→ C ∞ (M, ΩM ),

1,0 ∂M : C ∞ (M, C) −→ C ∞ (M, ΩM ).

We alsoobtain the following operators induced by d : C ∞ (M, T ∗ M ⊗ C) −→ C ∞ (M, 2 T ∗ M ⊗ C): 0,1 0,2 ∂ M : C ∞ (M, ΩM ) −→ C ∞ (M, ΩM ), 1,0 2,0 ) −→ C ∞ (M, ΩM ). ∂M : C ∞ (M, ΩM

We have ∂ M ◦ ∂ M = 0 and ∂M ◦ ∂M = 0. It is easy to see that a C ∞ -function f on M is mini-holomorphic if and only if ∂ M f = 0.

2.2.5 Meromorphic Functions Let M be a three-dimensional manifold with a mini-complex structure. Let H ⊂ M be a one-dimensional submanifold such that the tangent bundle T H is contained in (TS M)|H . Let U be an open subset of M. A holomorphic function f on U \ H is called meromorphic of at most order N along H if the following holds. • Let P be any point of H ∩U . We take a mini-complex coordinate neighbourhood (UP , t, w) around P . Note that H is described as {w = w0 }. Then, (w − w0 )N f|UP gives a mini-holomorphic function on UP . Let OM (NH ) denote the sheaf of meromorphic functions of order N along H . We obtain the sheaf OM (∗H ) := limN OM (NH ). A local section of OM (∗H ) is called − → a meromorphic function on M whose poles are contained in H .

2.3 Mini-Holomorphic Bundles 2.3.1 Mini-Holomorphic Bundles Let M be a three-dimensional manifold with a mini-complex structure. Let E be a C ∞ -bundle on M of finite rank. Let us consider a differential operator 0,1 ∂ E : C ∞ (M, E) −→ C ∞ (M, E ⊗ ΩM )

satisfying ∂ E (f s) = ∂ M (f ) s + f ∂ E (s) for any f ∈ C ∞ (M, C) and s ∈ C ∞ (M, E). As in the ordinary case of vector bundles on complex manifolds, it

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0,1 0,2 induces ∂ E : C ∞ (M, E ⊗ ΩM ) −→ C ∞ (M, E ⊗ ΩM ). Such a differential operator ∂ E is called a mini-holomorphic structure of E if ∂ E ◦ ∂ E = 0. In terms of a mini-complex coordinate system (t, w), a mini-holomorphic structure is equivalent to a pair of differential operators ∂E,t and ∂E,w on C ∞ (E) such that ∂E,t (f s) = ∂t (f )s + f ∂E,t (s) and ∂E,w (f s) = ∂w (f )s + f ∂E,w (s) for f ∈ C ∞ (M, C) and s ∈ C ∞ (M, E), satisfying the commutativity condition [∂E,t , ∂E,w ] = 0. Indeed, the operators ∂E,w and ∂E,t are induced by the inner product of ∂w and ∂t with ∂ E (s) (s ∈ C ∞ (U, E)), respectively. A local section s of E is called mini-holomorphic if ∂ E (s) = 0. By considering the sheaf of mini-holomorphic sections of E, we obtain a locally free OU -module, which is often denoted by the same notation E. The following is standard.

Lemma 2.3.1 The above procedure induces an equivalence of mini-holomorphic  bundles of finite rank and locally free OM -modules of finite rank.

2.3.2 Metrics and the Induced Operators Let (E, ∂ E ) be a mini-holomorphic bundle on M. Let h be a C ∞ -metric of a miniholomorphic bundle E. As in the case of complex differential geometry (see [48]), we obtain the induced differential operator ∂E,h : C ∞ (M, E) −→ C ∞ (M, E ⊗ 1,0 ΩM ) such that ∂E,h (f s) = ∂M (f )s + f ∂E,h (s) for f ∈ C ∞ (M, C) and s ∈ ∞ C (M, E), determined by the condition ∂ M h(u, v) = h(∂ E u, v) + h(u, ∂E,h v) for any u, v ∈ C ∞ (M, E). We have ∂E,h ◦ ∂E,h = 0.

2.3.3 Splittings Let TQ M −→ T M be a splitting of the exact sequence 0 −→ TS M −→ T M −→ TQ M −→ 0. We obtain the induced splitting (TSC M)∨ −→ (T C M)∨ , and the decomposition: 1,0 0,1 ⊕ ΩQ . (T C M)∨ = (TSC M)∨ ⊕ ΩQ

We also obtain the decompositions: 1,0 1,0 = ΩQ ⊕ (TSC M)∨ , ΩM

0,1 0,1 ΩM = ΩQ ⊕ (TSC M)∨ .

(2.1)

Let (E, ∂ E ) be a mini-holomorphic bundle with a metric h. We obtain the Q S Q S according to (2.1). decompositions ∂ E = ∂ E + ∂ E and ∂E,h = ∂E,h + ∂E,h

2.3 Mini-Holomorphic Bundles

49

We obtain the unitary connection ∇h and the anti-self-adjoint section φh of End(E) ⊗ (TS M ⊗ C)∨ : 1 S Q Q S ), ∇h := ∂ E + ∂E,h + (∂ E + ∂E,h 2

√ φh =

−1 S S (∂ E − ∂E,h ). 2

They are called the Chern connection and the Higgs field. Note that the construction of the Chern connection and the Higgs field depend on the choice of the splitting TQ M −→ T M. Remark 2.3.2 If M is equipped with a Riemannian metric, the orthogonal complement of TS M in T M is naturally isomorphic to TQ M. Hence, we obtain a splitting, which we shall use without mention. Moreover, by the Riemannian metric, TS M is identified with the product bundle R×M. Hence, we regard φh as an anti-Hermitian section of End(E). 

2.3.4 Scattering Maps Let (E, ∂ E ) be a mini-holomorphic bundle on M. Let γ : [0, 1] −→ M be a path such that T γ (∂s ) is contained in TS M. Then, the mini-holomorphic structure of E induces a connection of γ ∗ E. Hence, we obtain the isomorphism Eγ (0) −→ Eγ (1) as the parallel transport of the connection, called the scattering map [11]. Let U be an open subset of M with a mini-complex coordinate ϕ such that ϕ(U ) ⊃] − 2, 2[×Uw , where Uw is an open subset in C. Here, for a < b, we set ]a, b[:= {t ∈ R | a < t < b}. We regard [−, ] × Uw as a subset of M. By the mini-holomorphic structure, (E t , ∂ E t ) := (E, ∂ E )|{t }×Uw is naturally a holomorphic vector bundle on Uw for any t ∈ [−, ]. By  considering the scattering map along the path γw : [0, 1] −→ t1 + s(t2 − t1 ), w (w ∈ Uw ), we obtain the isomorphism (E t1 , ∂ E t1 ) (E t2 , ∂ E t2 ) because of the commutativity [∂E,t , ∂E,w ] = 0.

2.3.5 Dirac Type Singularity of Mini-Holomorphic Bundles Let U be a neighbourhood of (0, 0) in R × C. We take  > 0 and δ > 0 such that ] − 2, 2[×{|w| < δ} is contained in U . Set Uw := {|w| < δ} and Uw∗ := Uw \ {0}. Let (E, ∂ E ) be a mini-holomorphic bundle on U \ {(0, 0)}. For any t ∈ R with 0 < |t| < 2, we obtain the holomorphic vector bundle (E t , ∂ E t ) on Uw as the −  restriction of (E, ∂ E ). We have the scattering map Ψ : E|U ∗ E|U ∗ . Recall that w

w

(0, 0) is called Dirac type singularity of (E, ∂ E ) if Ψ is meromorphic at 0. (See [71].) In that case, we say that (E, ∂ E ) is of Dirac type on (U, (0, 0)).

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Let M be a three-dimensional manifold with a mini-complex manifold. Let Z be a discrete subset in M. Let (E, ∂ E ) be a mini-holomorphic bundle on M \ Z. We say that (E, ∂ E ) is of Dirac type on (M, Z) if the following holds. • For any P ∈ Z, we take a mini-complex chart (MP , ϕP ) such that (i) ϕP (P ) = (0, 0), (ii) the induced mini-holomorphic bundle on ϕP (MP \ {P }) is of Dirac type on (ϕP (MP ), (0, 0)). We can easily see that the condition is independent of the choice of a mini-complex chart (MP , ϕP ).

2.3.6 Kronheimer Resolution of Dirac Type Singularity Let U be a neighbourhood of (0, 0) in R × C. Let (E, ∂ E ) be a mini-holomorphic bundle on U ∗ := U \ {(0, 0)}. Let ϕ : C2 −→ R × C be the map given by ϕ(u1 , u2 ) = (|u1 |2 − |u2 |2 , 2u1 u2 ).

(2.2)

∗ := U  \ {(0, 0)}. The induced map U ∗ −→ U ∗ is also  := ϕ −1 (U ) and U We set U −1   denoted by ϕ. We set E := ϕ (E). We obtain the holomorphic structure ∂ E on E determined by the following condition: −1 −1 −1 ∂E,u  1 ϕ (s) = u1 ϕ (∂E,t s) + 2u2 ϕ (∂E,w s), −1 −1 −1 ∂E,u  2 ϕ (s) = −u2 ϕ (∂E,t s) + 2u1 ϕ (∂E,w s).

Here, s denotes a C ∞ -section of E. (See [11, 71].) Lemma 2.3.3 ([11, 71]) (0, 0) is Dirac type singularity of (E, ∂ E ) if and only if  ∂ E) extends to a holomorphic vector bundle (E 0 , ∂ E ) on U . (E,  0 0 , ∂ E ) the Kronheimer resolution of (E, ∂ E ) at (0, 0). We call such (E 0 Remark 2.3.4 Note that there exists a natural morphism ϕ −1 (OU ∗ ) −→ OU∗ . Let E denote the OU ∗ -module obtained as the sheaf of mini-holomorphic sections of  ∂ E) is naturally E. It is easy to see that the sheaf of holomorphic sections of (E, −1 isomorphic to OU∗ ⊗ϕ −1 (OU ) ϕ (E). 

2.3.7 Precise Description of Dirac Type Singularities Let us describe mini-holomorphic bundles of Dirac type more precisely. For simplicity, we assume that U =] − 2, 2[×Uw , where Uw is as above. We put U ∗ := U \ {(0, 0)}. We set H>0 :=]0, 2[×{0}.

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51

Lemma 2.3.5 For any mini-holomorphic bundle E of rank r,

there exist a tuple r of integers 1 ≤ 2 ≤ · · · ≤ r and an isomorphism E i=1 OU ∗ (i H>0 ). The tuple of the integers (1 , . . . , r ) is called the weight of E at the Dirac type singularity (0, 0). Proof Let E(±,0) denote the stalks of E at (±, 0), which are free OC,0 -modules. We have the isomorphism E(−,0)(∗0) E(,0)(∗0). It is

a standard fact that there exists a frame v1 , . . . , vr of E(−,0) such that E(,0)(∗0) = ri=1 OC,0 w−i vi . Then, the claim follows.  Lemma 2.3.6 Let E be as in Lemma 2.3.5. We set S2 = {(t, w) | t 2 + |w|2 =  2 } ⊂ U \ {(0, 0)} as the boundary of {(t, w) | t 2 + |w|2 ≤  2 }. Then,  with the orientation r we have S 2 c1 (E) = i=1 i . 

Proof It is enough to check the claim in the case rank E = 1. There exists a frame v of E(−,0) such that w− v is a frame of E(,0) under the isomorphism E(−,0)(∗0) E(,0). In the case E  = OU · u, we clearly have S 2 c1 (E  ) = 0. There exists a 

complex structure of S2 such that (i) it is compatible with the orientation, (ii) w induces a holomorphic coordinate

around  (, 0). Then, by the correspondence v →  (, 0) . Hence, we obtain S 2 c1 (E) = . u, we can identify E|S2 with E|S  2 



We set H0 ) OU ∗ (1 H>0 − 2 H0 )

OU (H≥0 ) ( < 0), OU (H>0 ) ( ≥ 0).

Let ι0 denote the inclusion {0} × Uw −→ U . We obtain

 ∗ ι−1 0 j∗ OU (H>0 )



OUw ({0}) ( < 0), OUw ( ≥ 0).

2.3.8 Subbundles and Quotient Bundles Let U be as in Sect. 2.3.7. Let E be a mini-holomorphic bundle of Dirac type on (U, 0).

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Lemma 2.3.7 • Let E  be an OU ∗ -submodule of E which is also locally free. Then, (0, 0) is Dirac type singularity of E  . • Let E  be a quotient OU ∗ -module of E which is also locally free. Then, (0, 0) is Dirac type singularity of E  . Proof It is easy to see that the scattering maps of E  and E  are meromorphic at 0. Hence, (0, 0) is Dirac type singularity.  Remark 2.3.8 Let 0 −→ E  −→ E −→ E  −→ 0 be an exact sequence of miniholomorphic bundles of Dirac type on U ∗ . We have the Kronheimer resolutions E , E and E at (0, 0), as in Sect. 2.3.6. Note that 0 −→ E −→ E −→ E −→ 0 is not necessarily exact at (0, 0). 

2.3.9 Basic Functoriality Let (Ei , ∂ Ei ) (i = 1, 2) be a three-dimensional mini-complex manifolds. Let (E1 , ∂ E1 ) ⊕ (E2 , ∂ E2 ) denote the mini-holomorphic bundle (E1 ⊕ E2 , ∂ E1 ⊕E2 ), 0,1 ) where the operator ∂ E1 ⊕E2 : C ∞ (M, E1 ⊕ E2 ) −→ C ∞ (M, (E1 ⊕ E2 ) ⊗ ΩM is determined by ∂ E1 ⊕E2 (s1 ⊕ s2 ) = ∂ E1 (s1 ) ⊕ ∂ E2 (s2 ) for si ∈ C ∞ (M, Ei ). Let (E1 , ∂ E1 ) ⊗ (E2 , ∂ E2 ) denote the mini-holomorphic bundle (E1 ⊗ E2 , ∂ E1 ⊗E2 ), 0,1 where the operator ∂ E1 ⊗E2 : C ∞ (M, E1 ⊗ E2 ) −→ C ∞ (M, (E1 ⊗ E2 ) ⊗ ΩM ) satisfies ∂ E1 ⊗E2 (s1 ⊗ s2 ) = ∂ E1 (s1 ) ⊗ s2 + s1 ⊗ ∂ E2 (s2 )

 for si ∈ C ∞ (M, Ei ). Let Hom (E1 , ∂ E1 ), (E2 , ∂ E2 ) denote the mini-holomorphic bundle (Hom(E1 , E2 ), ∂ Hom(E1 ,E2 ) ), where the operator 0,1 ∂ Hom(E1 ,E2 ) : C ∞ (M, Hom(E1 , E2 )) −→ C ∞ (M, Hom(E1 , E2 ) ⊗ ΩM )

satisfies ∂ E2 (f (s1 )) = ∂ Hom(E1 ,E2 ) (f )(s1 ) + f (∂ E1 s1 ) for f ∈ C ∞ (M, Hom(E1 , E2 )), and s1 ∈ C ∞ (M, E1 ). For a mini-holomorphic bundle (E, ∂ E ) on M, the dual (E, ∂ E )∨ is defined as Hom((E, ∂ E ), (C × M, ∂ M )).

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53

2.4 Monopoles 2.4.1 Monopoles and Mini-Holomorphic Bundles Let M be an oriented three-dimensional manifold with a Riemannian metric gM . Definition 2.4.1 Let E be a complex vector bundle with a Hermitian metric h, a unitary connection ∇ and an anti-self-adjoint endomorphism φ. The tuple (E, h, ∇, φ) is called a monopole on M if the Bogomolny equation F (∇) − ∗∇φ = 0, where F (∇) denotes the curvature of ∇, and ∗ denotes the Hodge star operator  with respect to gM . Suppose that M is equipped with a mini-complex structure such that the ⊥ orthogonal decomposition T M = TS M √ ⊕ (TS M) induces a splitting of T M −→ TQ M, and that the multiplication of −1 on TQ M is an isometry with respect to the induced metric of TQ M. We identify TS M with the product bundle R × M. Let (E, ∂ E ) be a mini-holomorphic bundle on M. For any Hermitian metric h of E, the Chern connection ∇h and the Higgs field φh are associated. Definition 2.4.2 We say that (E, ∂ E , h) is a monopole if (E, h, ∇h , φh ) is a monopole. In the case, we say that (E, ∂ E ) is the mini-holomorphic bundle underlying the monopole. 

2.4.2 Euclidean Monopoles In this paper, we study monopoles on spaces which are locally isomorphic to R × C with the natural metric and the natural mini-complex structure. Let us look at the condition more explicitly in terms of local coordinate systems. Let U be an open subset of R × C. It is equipped with the metric dt dt + dw dw. It is also equipped with the mini-complex structure. Let (E, ∂ E ) be a miniholomorphic bundle on U . Let h be a Hermitian metric of E. We obtain the differential operator ∂E,h,w : E −→ E determined by the condition





 ∂w h u, v = h ∂E,w u, v + h u, ∂E,h,w v .  We also obtain the differential operator ∂E,h,t : E −→ E determined by the condition  ∂t h(u, v) = h(∂E,t u, v) + h(u, ∂E,h,t v).

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We set ∇h,x := ∂E,w + ∂E,w , ∇h,t

√  ∇h,y := − −1 ∂E,w − ∂E,w ,

 1  , = ∂E,t + ∂E,h,t 2

√  −1  ∂E,t − ∂E,h,t φh = . 2

Thus, from the metric h, we obtain the unitary connection ∇h (s) = ∇h,x (s) dx + ∇h,y (s) dy + ∇h,y (s) dt, and the anti-self-adjoint section φh of End(E). Note that the mini-holomorphic structure is recovered by ∂E,w =

√ 1 (∇h,x + −1∇h,y ), 2

∂E,t = ∇h,t −

√

−1φh .

We also write ∂E,w and ∂E,h,w as ∇h,w and ∇h,w . Let F (h) denote the curvature of ∇h , which is expressed as F (h) = F (h)w,t dw dt+F (h)w,t dw dt+F (h)w,w dw dw. √ √ Lemma 2.4.3 F (h)w,t − −1∇h,w φh = 0 and F (h)w,t + −1∇h,w φh = 0 hold. Proof We have the commutativity [∂E,w , ∂E,t ] = 0, which implies 

∇h,w , ∇h,t −

√ √  −1φh = F (h)w,t − −1∇h,w φh = 0.

As the adjoint, we obtain F (h)w,t +

√

−1∇h,w φh = 0.



Corollary 2.4.4 A mini-holomorphic bundle (E, ∂ E ) with a Hermitian metric h is a monopole, i.e., the associated tuple (E, h, ∇h , φh ) is a monopole if and only if the following equation is satisfied. √ −1 ∇h,t φh + F (h)w,w = 0. (2.3) 2 √ √ Proof Because ∗(dw) = − −1dw dt, ∗(dw) = −1dw dt, and ∗(dt) = √ − 2−1 dw dw, the claim follows from Lemma 2.4.3.  Conversely, let E be a C ∞ -vector bundle on U with a Hermitian metric h, a unitary connection√∇ and an anti-Hermitian endomorphism φ. We set ∂E,w := ∇w and ∂E,t := ∇t − −1φ. We obtain the differential operator ∂ E : C ∞ (U, E) −→ C ∞ (U, E ⊗ ΩU0,1) by ∂ E (s) = ∂E,w (s) dw + ∂E,t (s) dt for any s ∈ C ∞ (U, E). In general, (E, ∂ E ) is not necessarily mini-holomorphic. If (E, h, ∇, φ)  is a monopole, the Bogomolny equation implies the commutativity ∂E,w , ∂E,t = 0, and hence (E, ∂ E ) is a mini-holomorphic bundle, which is called the underlying mini-holomorphic bundle of the monopole (E, h, ∇, φ). Note that (E, h, ∇, φ) is recovered from (E, ∂ E ) with h.

2.4 Monopoles

55

Remark 2.4.5 Let (E, ∂ E ) be a mini-holomorphic bundle with a Hermitian metric Chern connection and the Higgs field. We also have the h. Let ∇h and φh be the √  operator ∂E,h,t = ∇h,t + −1φh . By the construction, we have  [∇h,w , ∂E,h,t ] = 2[∇h,w , ∇h,t ],

[∇h,w , ∂E,t ] = 2[∇h,w , ∇h,t ].

We also have √ √ −1 −1  ∇h,w φh = − [∇h,w , ∂E,h,t ], ∇h,w φh = [∇h,w , ∂E,t ], 2 2 √ −1  [∂E,t , ∂E,h,t ∇h,t (φh ) = − ]. 2 

2.4.3 Dirac Type Singularity We recall the notion of Dirac type singularity of monopoles. It is originally introduced by Kronheimer [51], and it was later generalized by Pauly [75] to the context of general three-dimensional Riemannian manifolds, and by Charbonneau and Hurtubise [11] in the higher rank case. Let ϕ : C2 −→ R × C be the map (2.2). Let U be a neighbourhood of (0, 0) in R × C. We set U ∗ := U \ {(0, 0)}. Let (E, h, ∇, φ) be a monopole on U ∗ . We set ξ := −u1 du1 + u1 du1 − −1  u2 du2 + u2 du2 . We √ put∗(E, h) := ϕ (E, h). We obtain the unitary connection ∗    is an instanton ∇ := ϕ (∇) + −1ϕ (φ) ⊗ ξ . As proved in [51], (E, h, ∇) −1 ∗ on ϕ (U ). Recall that (0, 0) is called Dirac type singularity of (E, h, ∇, φ)   extends to an instanton on ϕ −1 (U ). It particularly implies that the if (E, h, ∇) underlying mini-holomorphic bundle (E, ∂ E ) of (E, h, ∇, φ) is of Dirac type on (U, (0, 0)). We have the following simple characterization of Dirac type singularity Theorem 2.4.6 ([71, Theorem 1]) (0, 0)  is a Dirac type singularity of (E, h, ∇, φ)

if and only if |φ|h = O (t 2 + |w|2 )−1/2 .  We also have the following characterization which easily follows from [71, Theorem 2]. Proposition 2.4.7 Let (E, ∂ E , h) be a monopole on U ∗ . It has a Dirac type singularity at (0, 0) if and only if the following holds. • (E, ∂ E ) is a mini-holomorphic bundle of Dirac type on (U, (0, 0)). In particular, 0 , ∂  ) on ϕ −1 (U ). (See Sect. 2.3.6.) we obtain the Kronheimer resolution (E E0

56

2 Preliminaries

0 . Then, • Let h1 be a C ∞ -metric of E such that ϕ −1 (h1 ) induces a C ∞ -metric on E there exist positive constants C1 > 1 and N1 such that C1−1 (|t|2 +|w|2 )−N1 ·h1 ≤  h ≤ C1 (|t|2 + |w|2 )N1 · h1 .

2.4.3.1 Dirac Monopoles (Examples) The fundamental examples are Dirac monopoles. Let us recall the description from [71, §5.2] with a minor correction. We set U = R × C, U ∗ = U \ {(0, 0)}, and A± := U \ {(t, 0) | ± t ≤ 0}. We have U ∗ = A+ ∪ A− . We regard U as a mini-complex manifold in a natural way. For any integer m, let L(m) be the mini(m) holomorphic line bundle on U ∗ equipped with mini-holomorphic frames σ± of (m) (m) (m) L|A± such that σ− = (w/2)m σ+ on A+ ∩ A− . Let h(m) be the Hermitian metric of L(m) determined by h(m) (σ+ , σ+ ) = 2m (t + R)−m , (m)

(m)

h(m) (σ− , σ− ) = 2−m (−t + R)−m , (m)

(m)

 where R = t 2 + |w|2 . We obtain the Chern connection ∇ (m) and the Higgs field φ (m) associated with (L(m) , ∂ L(m) , h(m) ). We obtain the holomorphic line bundle  L(m) on C2 \ {(0, 0)} = ϕ −1 (U ∗ ) as in Sect. 2.3.6. It is equipped with a holomorphic frame e(m) such that e(m) = (m) −1 (m) −1 (m) = u−m ϕ −1 (σ (m) ) on ϕ −1 (A ). Let  um h0 be − − 1 ϕ (σ+ ) on ϕ (A+ ), and e 2 (m) (m) defined as  the Hermitian metric of L h0 (e(m) , e(m) ) = 1. It is easy to check (m) (m) that  h0 is the pull back of h . Because ( L(m) ,  h(m) ) is an instanton, we obtain (m) (m) (m) (m) that (L , h , ∇ , φ ) is a monopole. (m) (m) Let us compute ∇ (m) and φ (m) explicitly. ∂L(m) ,w σ+ = ∂L(m),t σ+ = 0. We have ∂L(m) ,w σ+(m) = σ+(m) ∂w log h(m) (σ+(m) , σ+(m) ) = σ+(m)

−mw . 2R(t + R)

We also have ∂L (m) ,t σ+

(m)

(m)

(m)

(m)

(m) −m

= σ+ ∂t log h(m) (σ+ , σ+ ) = σ+

R

.

Therefore, we obtain ∇ (m) σ+(m)

=

σ+(m)



m  mw dw − dt , − 2R(t + R) 2R

Remark 2.4.8 In [71, §5.2], φ (m) =

√

−1m R

φ

(m)

√ −1m . = 2R

should be corrected to φ (m) =

√ −1m 2R .



2.4 Monopoles

57

2.4.4 Basic Functoriality Let M be a three-dimensional mini-complex manifold with a Riemannian metric. Assume that it is locally isomorphic to R × C with the canonical mini-complex structure and the Euclidean metric. Let (Ei , hi , ∇i , φi ) (i = 1, 2) be monopoles on M. The underlying mini-holomorphic bundle is denoted by (Ei , ∂ Ei ). The vector bundle E1 ⊕ E2 is equipped with the naturally induced Hermitian metrics hE1 ⊕E2 , the unitary connection ∇E1 ⊕E2 , and the Higgs field φE1 ⊕E2 = φE1 ⊕ φE2 . The tuple (E1 , h1 , ∇1 , φ1 ) ⊕ (E2 , h2 , ∇2 , φ2 ) = (E1 ⊕ E2 , hE1 ⊕E2 , ∇E1 ⊕E2 , φE1 ⊕E2 ) is a monopole. The underlying mini-holomorphic bundle is naturally isomorphic to (E1 , ∂ E1 ) ⊕ (E2 , ∂ E2 ). The vector bundle E1 ⊗ E2 is equipped with the naturally induced Hermitian metrics hE1 ⊗E2 , the unitary connection ∇E1 ⊗E2 , and the Higgs field φE1 ⊗E2 = φE1 ⊗ idE2 + idE1 ⊗φE2 . The tuple (E1 , h1 , ∇1 , φ1 ) ⊗ (E2 , h2 , ∇2 , φ2 ) = (E1 ⊗ E2 , hE1 ⊗E2 , ∇E1 ⊗E2 , φE1 ⊗E2 ) is a monopole. The underlying mini-holomorphic bundle is naturally isomorphic to (E1 , ∂ E1 ) ⊗ (E2 , ∂ E2 ). The vector bundle Hom(E1 , E2 ) is equipped with the naturally induced Hermitian metrics hHom(E1 ,E2 ) , the unitary connection ∇Hom(E1 ,E2 ) , and the Higgs field φHom(E1 ,E2 ) given as φHom(E1 ,E2 ) (s) = φ2 ◦ s − s ◦ φ1 . The tuple   Hom (E1 , h1 , ∇1 , φ1 ), (E2 , h2 , ∇2 , φ2 ) = (Hom(E1 , E2 ), hHom(E1 ,E2 ) , ∇Hom(E1 ,E2 ) , φHom(E1 ,E2 ) )

(2.4)

is a monopole. The underlying

 mini-holomorphic bundle is naturally isomorphic to Hom (E1 , ∂ E1 ), (E2 , ∂ E2 ) . The product line bundle C × M on M is equipped with the metric h0 defined by h0 (1, 1) = 1, and the trivial unitary connection ∇0 and the Higgs field 0, and the tuple (C × M, h0 , ∇0 , 0) is a monopole. For any monopole (E, h, ∇, φ) on M, we obtainthe induced monopole (E, h, ∇, φ)∨ =  Hom (E, h, ∇, φ), (C × M, h0 , ∇0 , 0) as the dual. The underlying miniholomorphic bundle of (E, h, ∇, φ)∨ is naturally isomorphic to the dual of the mini-holomorphic bundle underlying (E, h, ∇, φ).

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2 Preliminaries

2.5 Dimensional Reduction from 4D to 3D 2.5.1 Instantons Induced by Monopoles Monopoles are regarded as the one-dimensional reduction of instantons. Namely, monopoles are equivalent to R-equivariant instantons. Let us recall the construction explicitly in terms of coordinate systems. Let U be√an open subset of R × C. We use real and complex coordinates t and w = x + −1y for √ R and C, respectively. Let p : C × C −→ R × C be the map given by (s + −1t, w) −→ (t, w). We use the standard Euclidean metrics dt dt + dw dw on R × C, and dz dz + dw dw on C × C. Let E be a C ∞ -bundle on U with a Hermitian metric h, a unitary connection  ∇E and an anti-self-adjoint endomorphism φ of E. We set (E, h) := p∗ (E, h) and ∗ ∗ −1 ∇E := p ∇E + p φ ds on p (U ).  h, ∇E) Proposition 2.5.1 (Hitchin) (E, h, ∇E , φ) is a monopole if and only if (E, is an instanton. Proof Let ∗3 denote the Hodge star operator of R3 with the orientation dt ∧dx ∧dy. For example, we have ∗3 (dx ∧ dy) = dt and ∗3 (dx) = dy ∧ dt. Note that ∗3 ◦ ∗3 = id. Let ∗4 denote the Hodge star operator on R4 with the orientation ds∧dt ∧dx∧dy. For example, we have ∗4 (dx ∧ dy) = ds ∧ dt and ∗4 (dx ∧ ds) = −dy ∧ dt. By the construction of ∇E, we obtain F (∇E) = p∗ F (∇E ) + p∗ (∇E φ) ds, which implies ∗4 F (∇E) = −p∗ (∗3 F (∇E )) ds − p∗ (∗3 ∇E φ). Hence, we obtain  

∗4 F (∇E) + F (∇E) = p∗ F (∇E ) − ∗3 ∇E φ + p∗ ∇E φ − ∗3 F (∇E ) ds. Therefore, ∗4 F (∇E) + F (∇E) = 0 holds if and only if F (∇E ) = ∗3 ∇E φ holds.  Remark 2.5.2 Let us consider an R-action on p−1 (U ) by T · (z, w) = (z + T , w). Then, the proposition means that R-equivariant instantons on p−1 (U ) correspond to monopoles on U .  The following lemma is clear by the construction.

√ ∗ Lemma 2.5.3 Let f be any section of E. Then, (∇E,s −1∇E,t  +  )p f = 0 √ holds if and only if (∇E,t − −1φ)f = 0 holds. In other√words, R-invariant z -holomorphic sections on p−1 (U ) correspond to (∇E,t − −1φ)-flat sections ∇ on U .  The following lemma follows from the estimate for instantons due to Uhlenbeck [87].

2.6 Dimensional Reduction from 3D to 2D

59

  Lemma 2.5.4 Let (E, h, ∇E , φ) be a monopole on U . Suppose ∇E φ h ≤ , or   equivalently F (∇E )h ≤  for a positive number . Let K be any relatively compact subset of U . If  is sufficiently small, the higher derivatives of ∇E φ and F (∇E ) are dominated by CK  on K for some CK > 0, where CK depends on the order of derivatives and K, but independent of (E, h, ∇E , φ).  −1 −1 We write some formulas. We clearly have ∇E,w  (p (u)) = p (∇E,w u) and −1 −1 ∞ ∇E,w  (p (u)) = p (∇E,w u) for any u ∈ C (U, E). As for the derivative in the z-direction, we have

√ √  −1 −1 ∇E,z p (∇t − −1φ)u ,  (p (u)) = 2 √ √  −1 −1 −1 p (∇t + −1φ)u . ∇E,z  (p (u)) = − 2 √ For the real coordinate system (s, t) induced by z = s + −1t, we obtain −1 −1 −1 −1 ∇E,s  p (u) = p (φu) and ∇E,t  p (u) = p (∇t u). −1

2.5.2 Holomorphic Bundles and Mini-Holomorphic Bundles We have the corresponding procedure to construct a holomorphic bundle from a mini-holomorphic bundle. Let E be a C ∞ -vector bundle on U with a miniholomorphic structure ∂ E : C ∞ (U, E) −→ C ∞ (U, E ⊗ Ω 0,1). We have the  := operators ∂E,w and ∂E,t on C ∞ (U, E) satisfying [∂E,w , ∂E,t ] = 0. We set E p−1 (E). We have the holomorphic structure ∂ E satisfying the following formula for any C ∞ -section v of E:

∂ E p

−1



(v) = dw ∧ p

−1



∂w v + dz ∧

√

−1 −1  p ∂t v . 2

(2.5)

 We can Let h be a Hermitian metric of E. It induces a Hermitian metric  h of E. easily check that (E, ∂ E , h) is a monopole on U with respect to the Euclidean metric  ∂ E,  dt dt + dw dw if and only if (E, h) is an instanton on p−1 (U ) with respect to the Euclidean metric dz dz + dw dw. This is compatible with the construction in Sect. 2.4.1.

60

2 Preliminaries

2.6 Dimensional Reduction from 3D to 2D 2.6.1 Monopoles Induced by Harmonic Bundles Let us recall that the concept of harmonic bundle was discovered by Hitchin [34] as the two-dimensional reduction of instantons. Because monopoles are regraded as the one-dimensional reduction of instantons, harmonic bundles are the one-dimensional reduction of monopoles. We recall the construction in an explicit way. Let U be a Riemann surface equipped with a holomorphic 1-form ϕ which is nowhere vanishing on U . Let (V , ∂ V , hV , θV ) be a harmonic bundle on U . We obtain the holomorphic endomorphism f of V determined by θV = f ϕ. Let Y be a real one-dimensional manifold equipped with a closed real 1-form τ which is nowhere vanishing on Y . The product Y × U is naturally equipped with a miniholomorphic structure and the metric τ τ + ϕ ϕ. Let p2 : Y × U −→ U be the projection. We set (E, hE ) := p2∗ (V , hV ),

∇E = p2∗ ∇V −

√

−1p2∗ (f + f † ) τ,

φE = p2∗ (f − f † ).

(2.6) Here, p2∗ ∇V denotes the connection of E induced as the pull back of ∇V . Then, (E, hE , ∇E , φE ) is a monopole on Y × U with the metric τ τ + ϕϕ. We set Hit32 (V , ∂ V , θV , hV ) := (E, hE , ∇E , φE ). Let v be the vector field of Y such that τ, v = 1. It naturally defines a vector field on Y × U , which is also denoted by v. Let Lv be the differential operator of E defined by Lv (g p2∗ v) = v(g)p2∗ v for any g ∈ C ∞ (Y × U ) and any C ∞ -section v of V . Then, we have ∇E,v −

√ √ √ √ −1φE = Lv − −1p2∗ (f + f † ) − −1p2∗ (f − f † ) = Lv − 2 −1p2∗ f.

2.6.2 Mini-Holomorphic Bundles Induced by Holomorphic Bundles with a Higgs Field We have the corresponding procedure to construct a mini-holomorphic bundle on Y × U from a holomorphic bundle V with an endomorphism f on U . We use the 0,1 C ∨ natural splitting ΩY0,1 ×U = (T Y ) ⊕ ΩU . Let (V , ∂ V , θV ) be a Higgs bundle on U . We have the expression θV = f ϕ. We set E := p2−1 (V ),

√ ∂ E := p2∗ (∂ V ) − 2 −1p2∗ f τ.

2.6 Dimensional Reduction from 3D to 2D

61

Here, p2∗ (∂ V ) is the mini-holomorphic structure of E obtained as the pull back of ∂ V . We obtain a mini-holomorphic bundle (E, ∂ E ) on Y × U . We denote it by p2∗ (V , ∂ V , θV ) or p2∗ (V , ∂ V , f ). If V is equipped with a Hermitian metric hV , we have the induced metric h = p2∗ (hV ) on p2∗ (V , ∂ V , θV ). The Chern connection and the Higgs fields are induced by the formula (2.6). If hV is a harmonic metric of the Higgs bundle (V , ∂ V , θV ), then h satisfies the Bogomolny equation for p2∗ (V , ∂ V , θV ) with respect to the metric τ τ + ϕϕ. This is compatible with the construction in Sect. 2.6.1. Remark 2.6.1 The above procedure describes the dimensional reduction of the underlying objects of monopoles and harmonic bundles at the twistor parameter 0. There exists a similar procedure for each twistor parameter λ which we shall describe in Sect. 2.8. 

2.6.3 Mini-Holomorphic Sections and Monodromy Let us consider Y1 = R equipped with the nowhere vanishing closed 1-form τ1 = dt, where t is the standard coordinate of R. Let p2,1 : Y1 × U −→ U denote the projection. For any Higgs bundle (V , ∂ V , θV ) on U , we obtain the mini-holomorphic ∗ (V , ∂ , θ ) on Y × U as explained in Sect. 2.6.2. The following lemma bundle p2,1 V V 1 is clear by the construction. Lemma 2.6.2 For any holomorphic section v of V on U ,

√  ∗ ∗ exp 2 −1tp2,1 (f ) p2,1 (v) ∗ (V , ∂ , θ ). is a mini-holomorphic section of p2,1 V V



Let Y2 = R/T Z equipped with τ2 = dt. Let p2,2 : Y2 × U −→ U denote the projection. For any Higgs bundle (V , ∂ V , θV ) on U , we obtain the mini-holomorphic ∗ (V , ∂ , θ ) on Y × U . bundle p2,2 V V 2 For P ∈ U , let γP denote the loop [0, T ] −→ Y × U given by s −→ (s, P ). We identify V|P and E|(0,P ). By the mini-holomorphic structure, we obtain the parallel transport along γP which induces the monodromy automorphism M(γP ) of V|P . We obtain the following lemma from the previous one. √ Lemma 2.6.3 M(γP ) = exp(2 −1Tf|P ). 

62

2 Preliminaries

2.6.4 Appendix: Monopoles as Harmonic Bundles of Infinite Rank Let Y × U with τ τ + ϕϕ be as in Sect. 2.6.1. Let E be a C ∞ -vector bundle on Y × U equipped with a Hermitian metric h, a unitary connection ∇ and an anti-Hermitian endomorphism φ. Let E ∞ denote the sheaf of C ∞ -sections of E. Let CU∞ denote the sheaf of C ∞ -functions on U . We obtain the CU∞ -module p2! E ∞ , where p2! denote the proper push-forward with respect to p2 . It is equipped with the Hermitian metric p2! h defined as  p2! h(s1 , s2 ) =

h(s1 , s2 )τ Y

for sections si of p2! E ∞ , which we can naturally regard as C ∞ -sections of E Q whose supports are proper over U . By using the Ω 0,1 (U )-part ∂ E of ∇ in the decomposition T ∗ (Y × U ) ⊗ C = T ∗ (Y ) ⊗ C ⊕ Ω 1,0 (U ) ⊕ Ω 0,1 (U ), we obtain the Q holomorphic structure ∂ p2! E ∞ of p2! E ∞ by ∂ p2! E ∞ (s) = ∂ E (s). Similarly, by using Q the Ω 1,0(U )-part ∂E of ∇, we obtain the differential operator ∂p2! E ∞ of p2! E ∞ by ∂p2! E ∞ (s) = ∂EQ (s). It is easy to check that ∂ p2! E ∞ + ∂p2! E ∞ is unitary with respect to p2! h. We define the CU∞ -endomorphisms f and f † of p2! E ∞ by √ √ −1 (∇v − −1φ)s, f (s) = 2

√ √ −1 (∇v + −1φ)s. f (s) = 2 †

We set θ = f ϕ and θ † = f † ϕ, which are mutually adjoint with respect to p2! h. It is easy to see that (E, h, ∇, φ) is a monopole if and only if (p2! E ∞ , ∂ p2! E ∞ , θ, p2! h) is a harmonic bundle of infinite rank in the sense that ∂ p2! E ∞ (f ) = 0 and [∂ p2! E ∞ , ∂p2! E ∞ ] + [θ, θ † ] = 0 are satisfied. Let (E1 , ∂ E1 ) be a mini-holomorphic bundle on Y × U with the natural minicomplex structure. Similarly, let E1∞ denote the sheaf of C ∞ -sections of E1 , and we obtain the CU∞ -module p2! E1∞ . It is equipped with the holomorphic structure ∂ p2! E1∞ induced by the Ω 0,1 (U )-component of ∂ E1 . Let ∂E1 ,v be the differential operator of E1 induced by ∂ E1 and v. We obtain the endomorphism f1 of p2! E1∞ by f1 (s) = ∂E1 ,v (s), which is holomorphic with respect to ∂ p2! E1∞ . We obtain the Higgs field θ1 = f1 ϕ. Thus, we obtain a Higgs bundle (p2! E1∞ , ∂ p2 !E1∞ , θ1 ) of infinite rank on U . Clearly, if the mini-holomorphic bundle (E1 , ∂ E1 ) underlies a monopole, the Higgs bundle (p2! E1∞ , ∂ p2 !E1∞ , θ1 ) underlies the corresponding harmonic bundle of infinite rank.

2.7 Twistor Families of Mini-Complex Structures on R × C and (R/T Z) × C

63

2.7 Twistor Families of Mini-Complex Structures on R × C and (R/T Z) × C 2.7.1 Preliminary Let X be a two-dimensional complex vector space. Let L ⊂ X be an oriented onedimensional real vector space. Let M := X/L be the quotient three-dimensional real vector space. Note that M is equipped with a naturally induced mini-complex structure. Indeed, there exists a C-linear isomorphism ϕ : X C2 = {(z, w)} such that ϕ(L) = {(s, 0) | s ∈ R} in a way compatible with the orientations. Let t be the imaginary part of z. Then, by the mini-complex coordinate system (t, w), we obtain the isomorphism M R × C which induces a mini-complex structure on M. We can check the following claim by a direct computation. Lemma 2.7.1 The mini-complex structure is independent of a choice of ϕ.



Remark 2.7.2 Let LC ⊂ X denote the one-dimensional complex vector space generated by L. There exists the natural exact sequence of R-vector spaces 0 −→ LC /L −→ M −→ X/LC −→ 0.

(2.7)

Once we fix isomorphisms LC /L R and X/LC C, a choice of linear mini-complex coordinate system (t, w) as above induces a splitting of the exact sequence (2.7). 

2.7.2 Spaces Let X be a two-dimensional C-vector space. We take a C-linear isomorphism X C2 = {(z, w)}, and consider the hyperkähler metric gX := dz dz + dw dw. We consider the R-subspace L = {(s, 0) | s ∈ R} ⊂ X with the natural orientation. We set M := X/L. Let t := Im(z). We obtain the mini-complex coordinate system (t, w) on M. We take T > 0. We consider the Z-action κ on M given by κn (t, w) = (t + T n, w) (n ∈ Z). The quotient space is denoted by M.

2.7.3 Twistor Family of Complex Structures For any complex number λ, we have the complex structure on X whose twistor parameter is λ. We use the notation Xλ to denote the complex manifold X equipped with the complex structure corresponding to λ. We define the complex coordinate

64

2 Preliminaries

system (ξ, η) on Xλ as follows: ξ = z + λw,

η = w − λz.

(2.8)

The inverse transform is described as follows: z=

1 (ξ − λη), 1 + |λ|2

w=

1 (η + λξ ). 1 + |λ|2

The metric dz dz + dw dw is equal to (1 + |λ|2 )−2 (dξ dξ + dη dη). We obtain the following relations of complex vector fields: ∂ξ =

1 (∂z + λ∂w ), 1 + |λ|2

∂η =

1 (∂w − λ∂z ), 1 + |λ|2

∂ξ =

1 (∂z + λ∂w ), 1 + |λ|2

∂η =

1 (∂w − λ∂z ). 1 + |λ|2

The R-subspace L and the Z-action κ are described as follows in terms of the coordinate system (ξ, η):    L = s(1, −λ)  s ∈ R , κn (ξ, η) = (ξ +

√ √ √ √ −1T n, η + λ −1T n) = (ξ, η) + nT · ( −1, λ −1).

Remark 2.7.3 Recall that the space of the twistor parameters is P1 . Let P1 = Cλ ∪ Cμ be the covering where λ and μ are related by λμ = 1. Let X†μ denote the complex manifold X equipped with the complex structure corresponding to μ. Let M †μ denote the mini-complex manifold obtained as X† μ /L. We have X†μ = Xλ and M †μ = M λ if λμ = 1. At μ = 0, i.e., λ = ∞, the complex structure of X† 0 is induced by (z† , w† ) = (−z, w). We note that L = {(s, 0)} with respect to (z† , w† ), but that the orientations of L induced by z and z† = −z are mutually reversed. The complex structure of X† μ is induced by (ξ † , η† ) = (z† + μw† , w† − μz† ) = (λ−1 (w − λz), λ−1 (z + λw)) = (λ−1 η, λ−1 ξ ). We shall explain how to obtain difference modules from monopoles behind which the complex coordinate system (ξ, η) is implicitly used. For λ = μ−1 = 0, we may also obtain another difference module from the monopole similarly by using (ξ † , η† ). It is analogue to the situation that a harmonic bundle (E, ∂ E , θ, h) on a complex manifold Y induces a λ-flat bundle (E, ∂ E + λθ † + λ∂E + θ ) on Y and a μ-flat bundle (E, ∂E + μθ + μ∂ E + θ † ) on Y † which is the conjugate of Y . The analogy can be enhanced to the more precise correspondence by the Nahm transforms between wild harmonic bundles on P1 and periodic monopoles, which will be explained elsewhere. 

2.7 Twistor Families of Mini-Complex Structures on R × C and (R/T Z) × C

65

2.7.4 Family of Mini-Complex Structures Corresponding to the complex structure of Xλ , we obtain the mini-complex structures on M and M. (See Sect. 2.7.1.) The three-dimensional manifolds with mini-complex structure are denoted by M λ and Mλ . In the case λ = 0, we use the mini-complex coordinate system (t, w) = (Im(z), w) on M 0 . It induces local mini-complex coordinate systems on M0 . We shall introduce two mini-complex coordinate systems (ti , βi ) (i = 0, 1) on M λ , which induce mini-complex local coordinate systems on Mλ . For that purpose, we shall introduce two complex coordinate systems (αi , βi ) (i = 0, 1) on Xλ from which (ti , βi ) are induced. Remark 2.7.4 The coordinate systems (ti , βi ) depend on λ, but we omit to denote the dependence to simplify the description. 

2.7.5 The Mini-Complex Coordinate System (t0 , β0 ) Let (α0 , β0 ) be the complex coordinate system of Xλ given by the following relation: (ξ, η) = α0 (1, −λ) + β0 (λ, 1) = (α0 + λβ0 , β0 − λα0 ). The inverse transform is described as follows: α0 =

ξ − λη , 1 + |λ|2

β0 =

η + λξ . 1 + |λ|2

It is easy to check dα0 dα 0 + dβ0 dβ 0 = dz dz + dw dw. The R-subspace L and the Z-action κ are described as follows in terms of (α0 , β0 ): L = {(s, 0) | s ∈ R}, κn (α0 , β0 ) = (α0 , β0 ) + nT ·

 1 − |λ|2 √ 1 + |λ|2

√ 2λ −1  . −1, 1 + |λ|2

(2.9) (2.10)

We set t0 := Im(α0 ). Because L is described as (2.9), (t0 , β0 ) is a mini-complex coordinate system of M λ . The induced Z-action κ on M λ is described as follows in terms of (t0 , β0 ):  1 − |λ|2 2λ√−1  , . κn (t0 , β0 ) = (t0 , β0 ) + nT · 1 + |λ|2 1 + |λ|2

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Clearly, κn is given along the integral curve of the real vector field: ∂t =

1 − |λ|2 2λ √ 2λ √ ∂ + −1∂ − −1∂β 0 . t β 0 0 1 + |λ|2 1 + |λ|2 1 + |λ|2

(2.11)

Remark 2.7.5 If (t0 , β0 ) is another mini-complex coordinate system of M λ such  that dt0 dt0 + dβ0 dβ 0 , we obtain (t0 , β0 ) = (t0 , aβ0 ), where a is a complex number such that |a| = 1.  Remark 2.7.6 For the twistor parameter μ ∈ Cμ ⊂ P1 , we obtain (α0† , β0† ) from (ξ † , μ† ) as above: (α0† , β0† ) =

1 (ξ † − μη† , η† + μξ † ) 1 + |λ|2

Because the orientations of L induced by z and z† = −z are mutually reversed, the induced mini-complex coordinate system of M †μ is (− Im(α0† ), β0† ). If μ = λ−1 = 

0, we have (α0† , β0† ) = − α0 , (λ−1 λ)β0 , and hence (t0† , β0† ) = (− Im(α0† ), β0† ) =

 t0 , (λ−1 λ)β0 . 

2.7.6 The Mini-Complex Coordinate System (t1 , β1 ) Let (α1 , β1 ) be the complex coordinate system of Xλ determined by the relation (ξ, η) = α1 (1, −λ) + β1 (0, 1). The transformations are described as follows: 

ξ = α1 , η = β1 − λα1 ,



α1 = ξ, β1 = η + λξ.

The R-subspace L and the Z-action κ are described as follows in terms of (α1 , β1 ): L = {(s, 0) | s ∈ R},

(2.12)

√ √ κn (α1 , β1 ) = (α1 , β1 ) + nT ( −1, 2λ −1).

(2.13)

We set t1 := Im α1 . Because L is described as (2.12), we obtain the mini-complex coordinate system (t1 , β1 ) on M λ . The induced Z-action κ on M λ is described as follows: √ κn (t1 , β1 ) = (t1 , β1 ) + nT (1, 2λ −1). Remark 2.7.7 For the twistor parameter μ ∈ Cμ ⊂ P1 , we obtain the complex coordinate system (α1† , β1† ) = (ξ † , η† + μξ † ) of X† μ , and the mini-complex

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67

coordinate system (t1† , β1† ) = (− Im(α1 )† , β1† ) of M † μ . If μ = λ−1 , we have (α1† , β1† ) = (λ−1 η, λ−2 (η + λξ )), and hence (t1† , β1† ) = (t1 − Im(λ−1 β1 ), λ−2 β1 ). We obtain the splittings of M λ = M †μ induced by (t1 , β1 ) and (t1† , β1† ) as in Remark 2.7.2, and they are different. Therefore, we obtain two difference modules depending on the choice of (ξ, η) or (ξ † , η† ) as mentioned in Remark 2.7.3. There are essentially only these two choices. See Sect. 2.9.7.  Remark 2.7.8 If λ = 0, we have (t0 , β0 ) = (t1 , β1 ) = (t, w).



2.7.7 Coordinate Change We have the following relation: 



α1 = α0 + λβ0 β1 = (1 + |λ|2 )β0 ,

α0 = α1 − (1 + |λ|2 )λβ1 β0 = (1 + |λ|2 )−1 β1 .

Hence, we have the following relation: 

t1 = t0 + Im(λβ0 ) β1 = (1 + |λ|2 )β0 ,



t0 = t1 − (1 + |λ|2 )−1 Im(λβ1 ) β0 = (1 + |λ|2 )−1 β1 .

We obtain the following relation of vector fields: ∂t1 = ∂t0 ,

∂β 1 =

λ 1 1 √ ∂t + ∂ . 1 + |λ|2 2 −1 0 1 + |λ|2 β 0

(2.14)

2.7.8 Compactification λ

Set M := Rt1 × P1β1 , which is equipped with the natural mini-complex structure. √ λ We have the Z-action κ on M given by κn (t1 , β1 ) = (t1 + T n, β1 + 2 −1λT n). λ The quotient space is denoted by M . It is a compactification of Mλ , and equipped λ cov := M λ \ M λ and with the naturally induced mini-complex structure. We set H∞ λ := Mλ \ Mλ . Let  λ : M λ −→ Mλ denote the projection. For λ = λ , we H∞ 1 2 λ λ have M 1 = M 2 though Mλ1 = Mλ2 as C ∞ -manifolds. Remark 2.7.9 If λ = μ−1 = 0, we may obtain another natural compactification induced by (t1† , β1† ). The compactifications depend on the splittings induced by  (t1 , β1 ) and (t1† , β1† ) (see also Remark 2.7.2).

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2.7.9 Mini-Holomorphic Bundles Associated with Monopoles Let ϕ : U −→ M be a local diffeomorphism. We regard U as the Riemannian manifold whose metric is obtained as the pull back of dt dt + dw dw. For λ ∈ C, let U λ denote the three-dimensional manifold with the mini-complex structure obtained as the pull back of the mini-complex structure of Mλ . For any monopole (E, h, ∇, φ) on U , we obtain the mini-holomorphic bundle on U λ underlying (E, h, ∇, φ) for each λ (see Sect. 2.4.2), which we denote by (E λ , ∂ E λ ).

2.7.9.1 Compatibility with the Dimensional Reduction from 4D to 3D  := Rs × U on which we consider the Riemannian Let U be as above. We set U 0 denote the complex manifold whose complex metric ds ds +dt dt +dw dw. Let U √ structure is given by local coordinate systems (z, w) = (s + −1t, w). For any λ denote the complex manifold whose complex structure is given by λ ∈ C, let U the local complex coordinate systems (ξ, η) as in Sect. 2.7.3.   Let (E, h, ∇, φ) be a monopole on U . We obtain the induced instanton (E, h, ∇)  as in Sect. 2.5.1. Let (E λ , ∂ Eλ ) denote the holomorphic bundle on U λ on U underlying the instanton. The following lemma is clear by the constructions, or can be checked by a direct computation. λ , ∂ Eλ ) is equal to the holomorphic bundle induced by the miniLemma 2.7.10 (E holomorphic bundle (E λ , ∂ E λ ) as in Sect. 2.5.2. 

2.8 OMλ -Modules and λ-Connections 2.8.1 Dimensional Reduction from OMλ -Modules to λ-Flat Bundles 2.8.1.1 Setting Let Ψ : M = ST1 × Cw −→ Cw denote the projection. Let Uw −→ Cw be a local diffeomorphism. Let U denote the fiber product of Uw and M over Cw . The induced morphism U −→ Uw is also denoted by Ψ . There exists a natural isomorphism U ST1 × Uw which is equipped with the naturally defined ST1 -action. We obtain the naturally induced local diffeomorphism U −→ M. We regard U as a Riemannian manifold whose metric is obtained as the pull back of dt dt + dw dw. For λ ∈ C, let U λ denote the three-dimensional manifold equipped with the minicomplex structure obtained as the pull back of the mini-complex structure of Mλ .

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69

We shall explain that λ-flat bundles on Uw are the dimensional reduction of miniholomorphic bundles on U λ (Corollary 2.8.2), as the special case of a more general equivalence including the non-integrable case (Lemma 2.8.1).

2.8.1.2 Some Vector Fields and Forms We have the local complex coordinates on Uw induced by w, which induce the differential forms dw and dw, and the vector fields ∂w and ∂w globally defined on Uw . We have the local mini-complex coordinate systems on U λ induced by (t1 , β1 ), and we obtain the globally defined differential forms dt1 , dβ1 and dβ 1 , and the globally defined complex vector fields ∂t1 , ∂β1 and ∂β 1 on U . Note that Ψ is √ described as Ψ (t1 , β1 ) = (1 + |λ|2 )−1 (β1 − 2 −1λt1 ) in terms of the coordinate systems (t1 , β1 ) and w. The tangent map of Ψ is described as follows: √ √ −2 −1λ 2 −1 · λ ∂w + ∂w , T Ψ (∂t1 ) = 1 + |λ|2 1 + |λ|2

T Ψ (∂β 1 ) =

1 ∂w . 1 + |λ|2

(2.15)

Similarly, we have the local mini-complex coordinate systems on U λ induced by (t0 , β0 ), the globally defined differential forms dt0 , dβ0 and dβ 0 , and the globally defined complex vector fields ∂t0 , ∂β0 and ∂β 0 on U . We also have the local mini-complex coordinate systems on U 0 induced by (t, w), the globally defined differential forms dt, dw and dw, and the globally defined complex vector fields ∂t , ∂w and ∂w on U .

2.8.1.3 A General Equivalence Let V be a C ∞ -vector bundle on Uw . Recall that a λ-connection of V in the C ∞ sense is a linear differential operator Dλ : C ∞ (Uw , V ) −→ C ∞ (Uw , V ⊗ ΩU1 w ) such that Dλ (gs) = (λ∂ + ∂)g · s + f Dλ (s) for any g ∈ C ∞ (Uw ) and s ∈ C ∞ (Uw , V ). (See [63, §2.2].) It is called flat if Dλ ◦ Dλ = 0. Let ∂V ,w (resp. Dλw ) denote the differential operator of V induced by ∂w (resp. ∂w ) and Dλ . The flatness of Dλ is equivalent to [∂V ,w , Dλw ] = 0.  = Ψ −1 (V ). We say that a linear differential operator We set V ) −→ C ∞ (U λ , V  ⊗ Ω 0,1λ ) ∂ V : C ∞ (U λ , V U satisfies a mini-complex Leibniz rule if ∂ V(f u) = ∂ U λ (f )u + f ∂ V(u) for any ). f ∈ C ∞ (U ) and u ∈ C ∞ (U, V λ Let D be a λ-connection of V in the C ∞ -sense. Noting the relation (2.15), we  such that the following obtain the linear differential operators ∂V,β 1 and ∂V,t1 of V

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2 Preliminaries

holds for any s ∈ C ∞ (Uw , V ) and f ∈ C ∞ (U ): ∂V,β 1 (f Ψ −1 (s)) = ∂β 1 f · Ψ −1 (s) +

1 f Ψ −1 (∂V ,w s). 1 + |λ|2

1 f Ψ −1 1 + |λ|2 √ √

 × − 2 −1Dλw s + 2 −1 · λ∂V ,w s .

(2.16)

∂V,t1 (f Ψ −1 (s)) = ∂t1 f · Ψ −1 (s) +

(2.17)

) −→ C ∞ (U, V  ⊗ Ω 0,1λ ) by ∂ V(u) = ∂  (u)dβ 1 + We define ∂ V : C ∞ (U, V V ,β 1 U ). By the relation (2.15), ∂ V satisfies the mini∂V,t1 (u) dt1 for any u ∈ C ∞ (U, V , ∂ V) is S 1 -equivariant. By the construction, the complex Leibniz rule. Note that (V T following holds: 

∂V,β 1 , ∂V,t1



√  −2 −1 −1  = ∂w , Dλw . Ψ 2 2 (1 + |λ| )

(2.18)

Lemma 2.8.1 The above procedure induces an equivalence between the following objects. • Vector bundles V on Uw equipped with a λ-connection in the C ∞ -sense.  on U equipped with an S 1 -equivariant linear • ST1 -equivariant vector bundles V T ∞ ) −→ C ∞ (U λ , V  ⊗ Ω 0,1λ ) satisfying the differential operator ∂ V : C (U λ , V U mini-complex Leibniz rule. , ∂ V ) be as in the statement of Proof Let us indicate the inverse construction. Let (V 1 Lemma 2.8.1. Let ∂V,β 1 (resp. ∂V,t1 ) denote the ST -equivariant differential operator  induced by ∂  and ∂ (resp. ∂t1 ). There exists a C ∞ -bundle V on Uw with on V V β1 . We obtain the differential operators an ST1 -equivariant isomorphism Ψ −1 (V ) V λ ∂V ,w and Dw of V by the following condition for any s ∈ C ∞ (Uw , V ): Ψ −1 (∂V ,w (s)) = (1 + |λ|2 )∂V,β 1 Ψ −1 (s), Ψ −1 (Dλw s) =

√ −1 (1 + |λ|2 )∂V,t1 Ψ −1 (s) + (1 + |λ|2 )λ∂V,β 1 Ψ −1 (s). 2

Then, by the relation (2.15), we have ∂V ,w (gs) = ∂w (g)s +g∂V ,w (s) and Dλw (gs) = λ∂w (g)s + gDλw (s) for any g ∈ C ∞ (Uw ) and s ∈ C ∞ (Uw , V ). Hence, ∂V ,w and Dλw induces a λ-connection of V . Two constructions are mutually inverse.  2.8.1.4 Mini-Holomorphic Bundles and Flat λ-Connections As a consequence of Lemma 2.8.1 and (2.18), we obtain the following.

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71

Corollary 2.8.2 The construction in Sect. 2.8.1.3 induces an equivalence between λ-flat bundles on Uw and ST1 -equivariant mini-holomorphic vector bundles on U .  2.8.1.5 λ-Flat Bundles of Infinite Rank Let E be a C ∞ -bundle on U λ equipped with a differential operator ∂ E : C ∞ (U λ , E) −→ C ∞ (U λ , E ⊗ ΩU0,1λ ) satisfying the mini-complex Leibniz rule. Let EC ∞ denote the sheaf of C ∞ -sections of E. Let CU∞w denote the sheaf of C ∞ functions on Uw . We obtain the CU∞w -module V1 := Ψ! (EC ∞ ), where Ψ! denotes the proper push-forward with respect to Ψ . We define ∂V1 ,w and DλV1 ,w on the sheaf V1 by ∂V1 ,w (u) = (1 + |λ|2 )∂E,β 1 (u), DλV1 ,w (u)

√ −1 (1 + |λ|2 )∂E,t1 (u) + λ(1 + |λ|2 )∂E,β 1 (u) = 2

for sections u of V1 . We can easily observe that ∂V1 ,w and DλV1 ,w determines a λconnection DλV1 of V1 in the C ∞ -sense. If (E, ∂ E ) is a mini-holomorphic bundle, then DλV1 is a flat λ-connection. 2.8.1.6 Remark This analogy between mini-holomorphic bundles on U λ and λ-flat bundles on Uw is one of the motivations of this study, as explained in Sect. 1.5. (See also Sect. 2.9.5.2.) It is also useful when we study the property of the mini-holomorphic bundles underlying monopoles induced by harmonic bundles. (See Corollary 2.8.4 and the computations in Sects. 5.2 and 5.3 below.) We shall generalize this analogy to the case where Uw is a ramified covering of a neighbourhood of ∞ ∈ P1 (Corollary 2.8.8 and Corollary 4.5.10). We study the formal and ramified version in Sect. 3.6.3. Note that in the formal version we can remove the ST1 -equivariance condition (see Proposition 3.6.8). We obtain the filtered version (Proposition 3.6.15 and Proposition 4.5.2) as a refinement. It is also useful to study the non-integrable case (Lemma 2.8.1, Proposition 2.8.3, Lemma 2.8.7 and Lemma 4.5.9). We recall that in the study of harmonic bundles we understand the asymptotic behaviour of harmonic bundles through the filtered extension of the underlying λflat bundles, pioneered by Simpson [80], and further studied in [62, 64]. Similarly, it is our viewpoint to understand the asymptotic behaviour of monopoles through the filtered extension of the underlying mini-holomorphic bundles, studied in Sect. 7. Moreover, we would like to apply the results for filtered extension of λ-flat bundles underlying harmonic bundles to study the filtered extension of the mini-holomorphic

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bundles. That is the main reason to pursue this analogy. (See Sect. 1.8.7 for an outline of the argument.)

2.8.2 Comparison of Some Induced Operators Let Uw and U be as in Sect. 2.8.1.1. Let V be a C ∞ -bundle on Uw with a Hermitian metric h. Let ∂ V be a holomorphic structure of V , and let θ ∈ C ∞ (Uw , End(V ) ⊗ Ω 1,0) which does not necessarily satisfy ∂ V θ = 0. Let h be a Hermitian metric of V . We obtain the Chern connection ∂ V + ∂V , and the adjoint θ † ∈ C ∞ (Uw , End(V ) ⊗ Ω 0,1) of θ . Let ∂V ,w (resp. ∂V ,w ) denote the differential operator of V induced by ∂ V and ∂w (resp. ∂V and ∂w ). Let f and f † be the endomorphisms of V determined  = Ψ −1 (V ) on U with the by θ = f dw and θ † = f † dw, respectively. We set V −1  metric h = Ψ (h). On one hand, we obtain the unitary connection ∇ and an anti-Hermitian  satisfying the following condition for s ∈ C ∞ (Uw , V ) as endomorphism φ of V in (2.6): ∇w (Ψ −1 (s)) = Ψ −1 (∂V ,w s),

∇w (Ψ −1 (s)) = Ψ −1 (∂V ,w s),

√  ∇t (Ψ −1 (s)) = Ψ −1 − −1(f + f † )(s) ,

 φ(Ψ −1 (s)) = Ψ −1 (f − f † )s .

 with ∇ and φ, we obtain a linear differential operator ∂  : From V V ) −→ C ∞ (U λ , V  ⊗ Ω 0,1λ ) as in Sect. 2.4.2 which satisfies the miniC ∞ (U λ , V U complex Leibniz rule. On the other hand, we obtain the λ-connection Dλ = ∂ V + λθ † + λ∂V + θ of V in (2) ) −→ C ∞ (U λ , V ⊗Ω 0,1λ ) the C ∞ -sense. From (V , Dλ ), we obtain ∂ V : C ∞ (U λ , V U as in Sect. 2.8.1. (1)

(1)

(2)

Proposition 2.8.3 We have ∂ V = ∂ V . Proof Note that √ √ 1 − |λ|2 2λ −1 2λ −1 ∇t0 = ∇t − ∇w + ∇w , 1 + |λ|2 1 + |λ|2 1 + |λ|2 ∇β 0

√ −1λ λ2 1 = ∇t + ∇w + ∇w . 2 2 1 + |λ| 1 + |λ| 1 + |λ|2

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73

We also obtain the following: 1 λ 1 (1) ∂ (1) √ ∂V,t + 2 0 1 + |λ| 2 −1 1 + |λ|2 V,β 0 √ 1 1 λ (φ + = ∇ − −1∇t0 ), β 1 + |λ|2 0 1 + |λ|2 2

(1) ∂V = ,β 1

(1)

(1)

∂V,t = ∂V,t = ∇t0 − 1

0

(2.19)

√ −1φ.

(2.20)

Hence, we obtain the following: (1) ∂V,β 1

(1)

∂V,t = 1

 λ√−1 1 λ  ∇ φ , = + ∇ − t w 1 + |λ|2 2 2

(2.21)

√ √ √ 1 − |λ|2 2λ −1 2λ −1 ∇ − ∇ + ∇w − −1φ. t w 2 2 2 1 + |λ| 1 + |λ| 1 + |λ|

(2.22)

By the construction of ∇ and φ, we obtain the following: (1) Ψ −1 (s) = ∂V ,β 1

(1) ∂V,t Ψ −1 (s) 1

=Ψ

−1

 1 Ψ −1 (∂V ,w + λf † )s , 1 + |λ|2

√  2 −1 (∂ + λf )s − (λ∂ + f )s . V ,w V ,w 1 + |λ|2 1 + |λ|2

 2λ√−1

†

(2) This is equal to ∂V constructed from Dλ = ∂ V + λθ † + λ∂V + θ in Sect. 2.8.1. 

2.8.2.1 Comparison of Mini-Holomorphic Bundles Induced by Harmonic Bundles ,  h, ∇, φ) Suppose that (V , ∂ V , θ, h) is a harmonic bundle on Uw . On one hand, (V , ∂ (1) is a monopole on U associated with (V , ∂ V , θ, h) as in Sect. 2.6.1, and (V  ) V ,  is the mini-holomorphic bundle on U λ underlying (V h, ∇, φ). On the other hand, , ∂ (2) Dλ is a flat λ-connection associated with (V , ∂ V , θ, h), and (V  ) is the miniV holomorphic bundle associated with (V , Dλ ) in Corollary 2.8.2. We obtain the following from Proposition 2.8.3. , ∂  ) , ∂  ) and (V Corollary 2.8.4 The associated mini-holomorphic bundles (V V V on U λ are the same.  (1)

(2)

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2.8.3 O

λ

M

-Modules and λ-Connections

2.8.3.1 Setting λ

λ 1 by Ψ λ (t1 , β1 ) = (1 + |λ|2 )−1 (β1 − Let √ Ψ : M −→ Pw be the map induced λ 2 −1λt1 ). Note that the restriction of Ψ to Mλ is equal to Ψ . Let Uw be any open subset of P1w such that ∞ ∈ Uw . We set U λ = λ (Ψ λ )−1 (Uw ) ⊂ M , which is equipped with the mini-complex structure as an open λ subset of M . We shall generalize Lemma 2.8.1 and Corollary 2.8.2 to this context.

Remark 2.8.5 We shall later generalize the construction to the case where Uw −→ P1 is a ramified covering around ∞. (See Sect. 4.5.5.) 

2.8.3.2 A General Equivalence Let (V , ∂ V ) be a holomorphic vector bundle on Uw . Let ∂V ,w denote the differential operator of V induced by ∂ V and ∂w . Let Dλw be a linear differential operator of V satisfying Dλw (f s) = λ∂w (f ) · s + f Dλw (s) for any f ∈ C ∞ (Uw ) and s ∈ C ∞ (Uw , V ). Note that Dλw is equivalent to a linear differential operator Dλ (1,0) : (2∞)) such that Dλ (1,0)(f s) = λ∂Uw (f )s + C ∞ (Uw , V ) −→ C ∞ (Uw , V ⊗ ΩU1,0 w λ (1,0) ∞ (s) for any f ∈ C (Uw ) and s ∈ C ∞ (Uw , V ). The restriction of ∂ V + fD λ(1,0) D to Uw \ {∞} is a λ-connection of V|Uw \{∞} in the C ∞ -sense. λ and  := (Ψ λ )−1 (V ) on U λ . We set U λ∗ := U λ \H∞ We obtain a vector bundle V ∗ 1   V := V|U λ∗ . We have already constructed the ST -equivariant differential operator ∗ ) −→ C ∞ (U λ∗ , V ∗ ⊗ Ω 0,1λ∗ ) satisfying ∂ V∗ (f u) = ∂ U λ∗ (f ) · ∂ V∗ : C ∞ (U λ∗ , V U ∗ ). u + f ∂ V∗ (u) for any f ∈ C ∞ (U λ∗ ) and u ∈ C ∞ (U λ∗ , V Lemma 2.8.6 The differential operator ∂ V∗ uniquely extends to a linear differential ) −→ C ∞ (U λ , V  ⊗ Ω 0,1λ ) satisfying the mini-complex operator ∂ V : C ∞ (U λ , V U Leibniz rule. Proof We may assume that Uw is an open disc around ∞. Let v be a holomorphic frame of (V , ∂ V ) on Uw . We have ∂V ,w v = 0. Let A be the matrix valued C ∞ function on Uw determined by Dλw v = vA. The pull back  v := (Ψ λ )−1 v is a C ∞ λ  frame of V on U . λ , we choose P λ cov such that  λ (P  ∈ H∞ ) = P . Then, For any P ∈ H∞ −1  (t1 , τ1 ) = (t1 , β1 ) around P induces a local mini-complex coordinate system on a λ , the operators ∂ neighbourhood U1λ of P . On U1λ \ H∞ ∗ ,t1 and ∂V ∗ ,τ 1 are described V as follows with respect to the frame  v: √ −2 −1 λ ∗ v = v (Ψ ) (A), ∂V∗ ,t1 1 + |λ|2

2

∂V∗ ,τ 1 v = −β 1 ∂V∗ ,β 1 v=0

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75

Note that A is√C ∞ with respect to w−1 and w−1 , and that (Ψ λ )∗ (w−1 ) = (1 + |λ|2 )τ1 (1 − 2 −1λt1 τ1 )−1 is C ∞ around P . Hence, ∂V∗ ,t1 and ∂V∗ ,τ 1 uniquely ). Then, the claim of the lemma is extend to operators ∂V,t1 and ∂V,τ 1 on C ∞ (U λ , V clear.  Lemma 2.8.7 The above construction induces an equivalence between the following objects. • Holomorphic vector bundles (V , ∂ V ) on Uw equipped with a linear differential operator Dλw on C ∞ (Uw , V ) such that Dλw (f s) = λ∂w (f )s + f Dλw (s) for any f ∈ C ∞ (Uw ) and s ∈ C ∞ (Uw , V ).  on U λ equipped with an S 1 -equivariant • ST1 -equivariant vector bundles V T ) −→ C ∞ (U λ , V  ⊗ Ω 0,1λ ) satisfying differential linear operator ∂ V : C ∞ (U λ , V U the mini-complex Leibniz rule. , ∂  ) be as in the statement of Proof We indicate the inverse construction. Let (V V ∞ Lemma 2.8.7. There exists a C -vector bundle V on Uw equipped with an ST1  (Ψ λ )−1 (V ). We set V ∗ := V|Uw \{∞} . We obtain a equivariant isomorphism V λ λ-connection DV ∗ corresponding to ∂ V|U λ∗ . There exists the map π λ : U λ −→ ST1 induced by (t1 , β1 ) −→ t1 . Fix t10 ∈ ST1 . The fiber C := (π λ )−1 (t10 ) is naturally equipped with a complex structure, and , ∂ )|C is naturally a holomorphic vector bundle on C. The induced morphism (V V ΨCλ : C −→ Uw is holomorphic, and the image is an open neighbourhood of |C (Ψ λ )−1 (V )|C . Note ∞. We have the isomorphism of C ∞ -vector bundles V λ −1 that (Ψ ) (V )|C\H∞ λ is equipped with the holomorphic structure induced by ∂ V ∗ , λ −1 (V )  and that V|C\H∞ λ (Ψ ) λ is holomorphic. It implies that ∂ V ∗ uniquely |C\H∞ extends to a holomorphic structure ∂ V of V . Let v be a holomorphic frame of (V , ∂ V ) on Uw . We obtain a frame  v = . There exists a matrix-valued C ∞ -function A on Uw determined (Ψ λ )−1 (v) of V by √ −2 −1 λ ∗ ∂V,t1 v = v· (Ψ ) (A). 1 + |λ|2 Let DλV ∗ ,w denote the differential operator of V ∗ induced by DλV ∗ and ∂w . By the construction of DλV ∗ , we have DλV ∗ ,w v |Uw \{∞} = v |Uw \{∞} · A. It implies that DλV ∗ ,w uniquely extends to a differential operator Dλw of V . , ∂ ). The two constructions are In this way, we obtain (V , ∂ V , Dλw ) from (V V mutually inverse.  2.8.3.3 Mini-Holomorphic Bundles and Meromorphic Flat λ-Connections The following corollary is an immediate consequence of Lemma 2.8.7 and (2.18).

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2 Preliminaries

Corollary 2.8.8 The construction in Sect. 2.8.3.2 induces an equivalence of the following objects. • Locally free OUw -modules V equipped with a meromorphic λ-connection Dλ : V −→ V ⊗ ΩU1 w (2∞).  • ST1 -equivariant locally free OU λ -modules V.  2.8.3.4 Another Description of the Construction Let us explain another more direct description of the equivalence in Corollary 2.8.8. λ For any open subset U ⊂ M , let K λ (U) denote the space of C ∞ -functions M

 λ , a choice P  ∈ ( λ )−1 (P ) = 0. For P ∈ U ∩ H∞ f on U such that ∂β 1 f|U \H∞ λ

induces a local mini-complex coordinate system (t1 , τ1 ) = (t1 , β1−1 ) around P . For any f ∈ K λ (U), we obtain ∂τ 1 f = 0. Because [∂t1 , ∂β 1 ] = 0, we obtain the M naturally defined action of ∂t1 on K λ (U), and f ∈ K λ (U) is mini-holomorphic M M if and only if ∂t1 f = 0. Thus, we obtain a sheaf K λ . We have the natural action ∂t1 on K λ , and the M

M

λ

kernel is O λ . For any open subset U ⊂ M , let KU denote the restriction of K λ M M to U. λ Let Uw be an open subset of P1w , and U λ = (Ψ λ )−1 (Uw ) ⊂ M . We have the naturally defined monomorphism of sheaves (Ψ λ )−1 (OUw ) −→ KU λ . Let V be a locally free OUw -module equipped with a meromorphic λ-connection Dλ : V −→ V ⊗ ΩU1 w (2∞). From OUw -module V, we obtain the locally free KU λ module  V ∞ := KU λ ⊗(Ψ λ )−1 (OUw ) (Ψ λ )−1 (V). ∞ determined by the following We obtain the differential operator ∂V∞ ,t1 on V condition for local sections f and s of KU and V as follows:

 ∂V∞ ,t1 f (Ψ λ )−1 (s) = ∂t1 (f )(Ψ λ )−1 (s) +

√ 1 f · (Ψ λ )−1 (−2 −1Dλw s). 2 1 + |λ|

 such that ∂  u = 0. Note that  V ∞ is equal to the sheaf of C ∞ -sections u of V V ,β 1 Hence, the kernel of ∂V∞ ,t1 is naturally isomorphic to the sheaf of mini-holomorphic . sections of V √ = (1+|λ|2)−1 (β1 −2 −1λt1 ), Remark 2.8.9 Although we use the map Ψ λ (t1 , β1 ) √ it is more natural to consider (t1 , β1 ) −→ β1 − 2 −1λt1 . We adopt Ψ λ for the consistency with the dimensional reduction from monopoles to harmonic bundles in Sect. 2.6.1. See Remark 3.6.7. 

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77

2.9 Curvatures of Mini-Holomorphic Bundles with Metric on Mλ 2.9.1 Contraction of Curvature and Analytic Degree Let U −→ Mλ be a local diffeomorphism. We obtain the induced metric and the induced mini-complex structure on U . We also obtain the complex vector fields ∂βi , ∂β i , and ∂ti (i = 0, 1) on U . Let (E, ∂ E ) be a mini-holomorphic bundle on U . We have the operators ∂E,β 0 and ∂E,t0 . Let h be a Hermitian metric of E, which induces the Higgs field φh and the Chern connection ∇h . Let F (h) denote the curvature of ∇h . We have the expression F (h) = F (h)β0 β 0 dβ0 dβ 0 + F (h)β0 ,t0 dβ0 dt0 + F (h)β 0 ,t0 dβ 0 dt0 . We set G(h) := 2F (h)β0 β 0 −

√ −1∇h,t0 φh .

(2.23)

Note that the Bogomolny equation is equivalent to G(h) = 0 (see Corollary 2.4.4). Definition 2.9.1 Suppose that Tr G(h) is described as a sum g1 + g2 , where g1 1 is  L on U and g2 is non-positive everywhere. Then, we set deg(E, ∂ E , h) := U Tr G(h) dvolU , where dvolU is the volume form induced by the Riemannian  metric. Note that deg(E, ∂ E , h) ∈ R ∪ {−∞}. Remark 2.9.2 Let Rs0 × U −→ Rs0 × Mλ be the induced local diffeomorphism. √ We obtain the complex structure on Rs0 × U such that (α0 , β0 ) = (s0 + −1t0 , β0 )  ∂ E) be the holomorphic vector induces local complex coordinate systems. Let (E,   denote bundle with the metric h induced by (E, ∂ E ) with h as in Sect. 2.5.2. Let F   the curvature of the Chern connection associated with (E, ∂ E, h). We have the = F α0 ,α 0 dα0 dα 0 + F    expression F α0 ,β 0 dα0 dβ 0 + Fβ0 ,α 0 dβ0 dα 0 + Fβ0 ,β 0 dβ0 dβ 0 . √  = 2(F α0 ,α0 + F  Then, −1ΛF β0 ,β 0 ) is equal to the pull back of G(h) by the projection Rs0 × U −→ U , where Λ is the adjoint of the multiplication of the Kähler form of Rs0 × U . (See [48].) Hence, deg(E, ∂ E , h) is an analogue of the analytic degree in [79].  Remark 2.9.3 In [67] and a previous version of this monograph, we set “G(h) = √ −1 F (h)β0 β 0 − 2 ∇h,t0 φh ”, which is the half of (2.23). There is no essential difference though constants in some formulas are changed. 

2.9.2 Chern-Weil Formula The standard Chern-Weil formula [79, 80] is translated as follows, which is implicitly contained in [11].

78

2 Preliminaries

Lemma 2.9.4 Let V be a mini-holomorphic subbundle of E. Let hV be the induced metric of V . Let pV denote the orthogonal projection of E onto V , which we regard as an endomorphism of E in a natural way. Then, we obtain the following formula: 2 1  2  

Tr G(hV ) = Tr G(h)pV − 2∂E,β 0 pV  − ∂E,t0 pV  . 2

(2.24)

Proof We use the notation in Remark 2.9.2. We obtain the induced holomorphic  of E  with the metric  subbundle V hV induced by V . We obtain the Chern connection  onto V , which is the pull back of pV . ∇  +∂E,h  . Let pV  be the projection of E h = ∂E  in a natural way. Let ι  and ι ⊥ denote the We regard pV as the endomorphism of E V V  and V ⊥ into E,  respectively. We set A := p  ◦ ∂  ◦ ι ⊥ . Because inclusions of V V E V pV ◦ F ( h) ◦ ιV = F ( hV) − A ◦ A† , we obtain √ √ √ −1 Tr ΛF ( hV) = −1 Tr(pV ◦ ΛF ( h) ◦ ιV) + −1Λ Tr(A ◦ A† ). √ √ Note that −1ΛF ( h) and −1ΛF ( hV) are the pull back of G(h) and G(hV ),  2 √ respectively. Because of (2.5), −1Λ Tr(A◦A†) is the pull back of −2∂E,β 0 pV  −  2 1  ∂E,t pV  . Then, we obtain the claim of the lemma. 2

0

As a direct consequence of the lemma, if |G(h)|h is L1 on U , then deg(V , hV ) makes sense in R ∪ {−∞} for any mini-holomorphic subbundle V of E.

2.9.3 Another Description of G(h) We have the differential operator ∂E,β 1 on C ∞ (U, E), which is given by the inner product of ∂β 1 and ∂ E . We have the differential operator ∂E,h,β1 on C ∞ (U, E) determined by the condition ∂β 1 h(u, v) = h(∂E,β 1 u, v) + h(u, ∂E,h,β1 v). Proposition 2.9.5 The following formula holds:   G(h) = 2(1 + |λ|2 )2 ∂E,h,β1 , ∂E,β 1  √  √ √ − −1 (1 − |λ|2 )∇h,t0 φ + 2λ −1∇h,β0 φ − 2λ −1∇h,β 0 φ   √ (2.25) = 2(1 + |λ|2 )2 ∂E,h,β1 , ∂E,β 1 − −1(1 + |λ|2 )∇h,t φ. Here, ∇h,t denote the inner product of ∇h and ∂t . (See (2.11).)

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79

Proof To simplify the description, we omit to denote the dependence on h, i.e., ∂E,h,β1 is denoted by ∂E,β1 , for example. By the relation (2.14), we obtain the following: 1 λ 1 ∂ √ ∂E,t0 + 2 1 + |λ| 2 −1 1 + |λ|2 E,β 0 √ 1 1 λ (φ + = ∇ − −1∇t0 ). β 1 + |λ|2 0 1 + |λ|2 2

∂E,β 1 =

(2.26)

We obtain the following: ∂E,β1 =

√ λ 1 1 (φ − ∇ − −1∇t0 ). β 0 1 + |λ|2 1 + |λ|2 2

We have the following: √  λ  1 2  ∇β0 , ∇β 0 − ∇β0 φ + −1[∇β0 , ∇t0 ] 2 1 + |λ| 2 2 √ √ √   |λ|  λ φ − −1∇t0 , φ + −1∇t0 . + ∇β 0 φ − −1[∇β 0 , ∇t0 ] + (2.27) 2 4 

  ∂E,β1 , ∂E,β 1 =

By Lemma 2.4.3, we obtain  λ λ√  − ∇β0 φ − −1 ∇β0 , ∇t0 = −λ∇β0 φ, 2 2 √  λ −1  ∇ φ− λ ∇β 0 , ∇t0 = λ∇β 0 φ. 2 β0 2 We also have √ √  |λ|2  |λ|2 √ φ − −1∇t0 , φ + −1∇t0 = − −1∇t0 φ. 4 2 Hence, we obtain the following: 

 ∂E,β1 , ∂E,β 1 =  =

1 1 + |λ|2



2 

1 1 + |λ|2

2   |λ|2 √ −1∇t0 φ Fβ0 ,β 0 − λ∇β0 φ + λ∇β 0 φ − 2

 1 1 − |λ|2 √ G(h) − λ∇β0 φ + λ∇β 0 φ + −1∇t0 φ . 2 2

Then, we obtain (2.25). We state some consequences.

(2.28) 

80

2 Preliminaries

Corollary 2.9.6  Suppose that U := Mλ \ Z, where Z is a finite subset of Mλ . Suppose that Tr ∂E,h,β1 , ∂E,β 1 and Tr ∇h,t φh are L1 on Mλ \ Z. Then, we obtain  deg(E, ∂ E , h) =



T

dt1

Cβ1

0

Tr



∂E,h,β1 , ∂E,β 1

√ −1dβ1 dβ 1 .

(2.29)

Proof The following formula holds: √ √ √ −1 −1 −1 2 −2 dvol = dβ0 dβ 0 dt0 = (1 + |λ| ) dβ1 dβ 1 dt1 = dw dw dt. 2 2 2 By using Fubini theorem, we obtain the following: √   

 −1 Tr ∇t φ dvol = dw dw ∂t Tr(φh ) dt = 0. 2 Mλ \Z Cw R/T Z Hence, by Proposition 2.9.5, we obtain the following. 

 Mλ \Z

Tr G(h) dvol =

Mλ



√ β1 , ∇  ] −1dβ1 dβ 1 dt1 Tr [∇ β1 

T

=

dt1 0

C η1

√

 β1 , ∇  Tr ∇ −1dβ1 dβ 1 . β1

(2.30) 

Thus, we obtain (2.29).

Note that for almost all 0 ≤ t1 ≤ T , the analytic degree of (E, ∂ E , h)|{t1 }×Cβ1 is defined as the integration of the first Chern form (see [79]): √   

−1 (2.31) deg (E, ∂ E , h)|{t1 }×Cβ1 = Tr [∂E,h,β1 , ∂E,β 1 ] dβ1 dβ 1 . 2π Cβ1 Then, (2.29) is rewritten as follows: 

T

deg(E, ∂ E , h) = 0

 2π deg (E, ∂ E , h)|{t1 }×Cβ1 dt1 .

(2.32)

We also obtain the following useful formula to relate the curvatures of the miniholomorphic bundles on Mλ and M0 , underlying a monopole. Corollary 2.9.7 Suppose that (E, ∂ E , h) is a monopole on U , i.e., G(h) = 0. Then, we obtain √   1 −1 1 ∂E,h,β1 , ∂E,β 1 = ∇h,t φ = Fw,w . 2 1 + |λ|2 1 + |λ|2 

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81

2.9.4 Change of Metrics Let h1 be another Hermitian metric of E. Let s be the automorphism of E determined by h1 = h · s, which is self-adjoint with respect to both h and h1 . The following is a variant of [79, Lemma 3.1]. Lemma 2.9.8 The following holds: G(h1 ) = G(h) √ √

  1  − 2∂E,β 0 s −1 ∂E,h,β0 s − ∇h,t0 − −1φh , s −1 ∇h,t0 + −1φh , s . 2 (2.33) Proof Because ∂E,h1 = ∂E,h + s −1 ∂E,h s, the following holds:   F (h1 )β0 ,β 0 = ∂E,h1 ,β0 , ∂E,β 0 = F (h)β0 ,β 0 − ∂E,β 0 (s −1 ∂E,h,β0 s). We also obtain the following: √   1 ∇h1 ,t0 = ∇h,t0 + s −1 ∇h,t0 + −1φh , s , 2 √ √  −1 −1  s ∇h,t0 + −1φh , s . φh1 = φh − 2 We obtain the following: √ √ √ √   1 − −1∇h1 ,t0 φh1 = − −1∇h,t φh − ∇h,t0 − −1φh , s −1 ∇h,t0 + −1φh , s . 2 Hence, we obtain (2.33).



We obtain the following direct consequence, which is also a variant of [79, Lemma 3.1]. Corollary 2.9.9 The following equality holds:  1  − ∂β 0 ∂β0 + ∂t20 Tr s 4 2 1  2   1   = Tr s G(h1 ) − G(h) − s −1/2 ∂E,h,β0 s h − s −1/2 ∂E,h,t s . 0 h 2 4

(2.34)

We also have the following inequality:     

 1  1  G(h) + G(h1 ) . − ∂β 0 ∂β0 + ∂t20 log Tr(s) ≤ h h1 4 2 

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2 Preliminaries

Corollary 2.9.10 If rank E = 1, we have G(h1 ) − G(h) = 2−1 Δ log s on U . Here, Δ denote the Laplacian of the Riemannian manifold Mλ . 

2.9.5 Relation with λ-Connections This subsection is a complement to Sect. 2.8. We use the notation there. Let us recall the condition for harmonic bundles given in terms of λ-connections [63, §2.2]. Let (V , Dλ ) be a λ-flat bundle on Uw with a Hermitian metric hV . Let Dλ = dV + dV denote the decomposition into the (0, 1)-part and the (1, 0)-part. We have the (1, 0)-operator δ  and the (0, 1)-operator δ  determined by the conditions ∂hV (u, v) = hV (dV u, v) + hV (u, δ  v),

λ∂hV (u, v) = hV (dV u, v) + hV (u, δ  v)

for u, v ∈ C ∞ (Uw , V ). We have the (0, 1)-operator ∂ V , the (1, 0)-operator ∂V , the section θ ∈ C ∞ (Uw , V ⊗ Ω 1,0 ) and the section θ † ∈ C ∞ (Uw , V ⊗ Ω 0,1 ) determined by dV = ∂ V + λθ † ,

dV = λ∂V + θ,

δ  = ∂V − λθ,

δ  = λ∂V − θ † .

Then, (V , Dλ , h) is called a harmonic bundle if (V , ∂ V , θ, h) is a harmonic bundle. If λ = 0, it is equivalent to ∂ V θ = 0. We set Dλ$ := δ  − δ  . Then, (V , Dλ , hV ) is a harmonic bundle if and only if [Dλ , Dλ$ ] = 0.  , Let dV ,w denote the inner product of dV and ∂w . We use the notation dV ,w , δw  and δ  − λd  are endomorphisms of etc., in similar meanings. Note that dw − λδw w w   2 V , which follow from d − λδ = (1 + |λ| )θ and δ  − λd  = −(1 + |λ|2 )θ † . , ∂ V) on As explained in Sect. 2.8, we obtain a mini-holomorphic bundle (V λ −1 λ λ U = Ψ (Uw ) ⊂ M from (V , D ). We also obtain the Hermitian metric  h := (Ψ λ )−1 (hV ). Let ∇V and φ denote the Chern connection and the Higgs field  , ∂ V ,  of (V h). Let ∂V be the differential operator induced by ∂V,t0 and h as in ,t0 Sect. 2.4.2. Lemma 2.9.11 We obtain the following formulas:     1 λ )−1 d  , δ   + λd  , δ   + λ[d  , δ   + |λ|2 d  , δ   , (Ψ ∇V,β ,∇V,β = w w w w w w w w 0 0 (1 + |λ|2 )2 

√





 2 −1   , δ  ] − λ[d  , δ  ] + λ[d  , δ  ] − λ2 [d  , δ  ] , ∇V,β , ∂  (Ψ λ )−1 [dw = w w w w w w w V ,t0 0 (1 + |λ|2 )2

√     −2 −1 λ )−1 δ  , d   − λ[δ  , d  ] + λ[δ  , d  ] − λ2 [δ  , d  ] , ∇V,β0 , ∂V,t0 = (Ψ w w w w w w w w (1 + |λ|2 )2 

 ∂V,t0 , ∂V ,t

 0

=

  4  , δ  ] − λ[d  , δ  ] − λ[d  , δ  ] + |λ|2 [d  , δ  ] . (Ψ λ )−1 [dw w w w w w w w 2 2 (1 + |λ| )

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83

We also have the following formula: φ=

 1   (Ψ λ )−1 (dw − λδw ) + (δw − λdw ) . 2 1 + |λ|

Proof By the construction, the following holds for any s ∈ C ∞ (Uw , V ): λ −1

∂V,t1 (Ψ )

√   −2 −1 λ −1    d (s) = (Ψ ) − λd w w s . 1 + |λ|2

We denote it as ∂V,t1 =

√  −2 −1 λ −1   dw − λdw . (Ψ ) 2 1 + |λ|

Similarly, we have   √ 2 −1 λ )−1 δ  − λδ  (Ψ w w 1+|λ|2

  1 λ −1 ∂V,β 1 = 1+|λ|2 (Ψ ) dw ⎪ ⎪ ⎩ 1 λ −1 (δ  ). ∂V,β1 = 1+|λ| 2 (Ψ ) w ⎧ ⎪ ∂ = ⎪ ⎨ V,t1

(2.35)

  Note that ∂V,t0 = ∂V,t1 and ∂V = ∂V . We obtain ,t ,t 0

1

λ ∇V,β 0 = (1 + |λ|2 )∂Vβ 1 − √ ∂V,t1 2 −1 =

1 λ (Ψ λ )−1 (dw ) + (Ψ λ )−1 (dw ). 1 + |λ|2 1 + |λ|2

(2.36)

Similarly, we obtain ∇V,β0 =

λ 1   (Ψ λ )−1 (δw )+ (Ψ λ )−1 (δw ). 2 1 + |λ| 1 + |λ|2

Then, we obtain the desired equalities by direct computations.



Lemma 2.9.12 We have the following formula: G( h) =

 √ 1 λ −1 λ λ$ ) −1Λ [D , D ] . (Ψ U w (1 + |λ|2 )

√ Here, ΛUw : ΩU1,1 −→ ΩU0,0 is determined by ΛUw (dw dw) = −2 −1. w w

(2.37)

84

2 Preliminaries

Proof By definition, we have √  1  . − −1∇V,t0 φ = − ∂V,t0 , ∂V ,t 0 2 By a direct computation, we obtain   √ 2 ∇V,β0 , ∇V,β 0 − −1∇t0 φ       2 λ −1        − d (Ψ ) + λd , δ + λδ − λd , δ − λδ − d = w w w w w w w w (1 + |λ|2 )2       2 λ −1   δ + δw , d w . (2.38) (Ψ ) , d = w w 1 + |λ|2 We also have    √   √       −1ΛUw Dλ , Dλ$ = −1ΛUw [d  , δ  ] − [d  , δ  ] = 2 δw , dw + δw , dw . 

Hence, we obtain (2.37).

, ∂ V,  h) is a monopole if and only if (V , Dλ , h) is a harmonic Corollary 2.9.13 (V bundle  h as inSect. 2.9.3. Lemma 2.9.14 Let ∂V, ,β 1 and  h,β1 be the operator induced by ∂V Then, we obtain the following equality:   ∂V,β 1 , ∂V, h,β1 =

λ −1     1 [dw , δh,w ] . Ψ 2 2 (1 + |λ| ) 

Proof It follows from (2.35). 2.9.5.1 λ-Flat Bundles of Infinite Rank with a Harmonic Metric

Let (E, ∂ E ) be a mini-holomorphic bundle on U λ . We obtain the Chern connection ∇ and the Higgs field φ. Let EC ∞ denote the sheaf of C ∞ -sections of E. We obtain the CU∞w -module V1 = Ψ! (EC ∞ ) equipped with the induced λ-connection DλV1 as explained in Sect. 2.8.1.5. Let h be a Hermitian metric of E. It induces a Hermitian metric h1 = Ψ! (h) of V1 . We obtain the operators  δw = (1 + |λ|2 )∂E,h,β1 ,

 δw =

√ − −1  (1 + |λ|2 )∂E,h,t + λ(1 + |λ|2 )∂E,h,β1 0 2

 (s) dw and δ  (s) = δ  (s) dw for any local section u of on V1 . We set δ  (s) = δw w λ$  V1 . We obtain DV1 = δ − δ  on V1 . We have the decomposition DλV1 = d  + d 

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85

into the (0, 1)-part and the (1, 0)-part. By the construction, we have ∂h1 (u1 , u2 ) =  local h1 (d  u1 , u2 )+h1 (u1 , δ  u2 ) and ∂h1 (u1 , u2 ) = h1 (d  u1 , u2 )+h √ 1 (u1 , δ u2λ) forλ$ sections ui of V1 . By similar computations, we obtain that −1ΛUw [DV1 , DV2 ] is the multiplication of (1+|λ|2 )G(h). Hence, [DλV1 , Dλ$ V1 ] = 0 if and only if (E, ∂ E , h) is a monopole. In this sense, we may regard a monopole on U λ as a λ-flat bundle with a harmonic metric of infinite rank.

2.9.5.2 Remark The analogy between monopoles on U λ and λ-flat bundles with a harmonic metric on Uw (Corollary 2.9.13 and Sect. 2.9.5.1) is one of the motivation of this study, as explained in Sect. 1.5. It is also useful for the construction of a Hermitian metric h1 of a given mini-holomorphic bundle such that G(h1 ) is small in Proposition 7.4.3. In the study of harmonic bundles, we study the asymptotic behaviour of a harmonic metric h of (V , Dλ ) by constructing a Hermitian metric h0 of V such that P∗h V = P∗h0 V , and that [Dλh0 , Dλ$ h0 ] is small. We can explicitly construct such h0 , and we can prove that h and h0 are mutually bounded. This kind of argument was pioneered by Simpson [80], and further applied in [62, 64]. Such a Hermitian metric is also useful in the proof of Kobayashi-Hitchin correspondence, i.e., the existence of globally defined harmonic metrics. By adopting a similar strategy, in Proposition 7.4.3 below, we shall construct a Hermitian metric h1 of a miniholomorphic bundle such that G(h1 ) is small. For the construction of such h1 , we use a metric h0 for λ-flat bundles as above through Lemma 2.9.11, Lemma 2.9.12 and Lemma 2.9.14.

2.9.6 Dimensional Reduction of Kronheimer Let us recall the dimensional reduction of Kronheimer [51]. Let ϕ : C2 −→ R × C  := ϕ −1 (U ). be the map (2.2). Let U be a neighbourhood of (0, 0) in R×C. We set U ∗ ∗   We put U := U \ {(0, 0)} and U := U \ {(0, 0)}. Let (E, ∂ E ) be a mini-holomorphic bundle on U ∗ with a Hermitian metric hE . We obtain the Chern connection ∇ and the Higgs field φ. Let F be the curvature of ∇.  := ϕ −1 (E) on U ∗ . It is equipped with the unitary connection ∇  := We put√E ∗ ∗ ϕ (∇) + −1ϕ (φ) ⊗ ξ , where ξ = −u1 du1 + u1 du1 − u2 du2 + u2 du2 .  is equal to ϕ ∗ (F ) + The curvature F

√

−1ϕ ∗ (∇φ) ∧ ξ .

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2 Preliminaries

Lemma 2.9.15 We have the following equality of currents on U :     dvolU = π Tr G(h) dvolU . ϕ∗ Tr ΛF

(2.39)

Here, ϕ∗ denote the push-forward of currents by the proper map ϕ. Proof Note (|u1 |2 + |u2 |2 )2 = ϕ ∗ (|w|2 + |t|2 ). We have ϕ ∗ dw = 2u1 du2 + u2 du1 , ϕ ∗ dw = 2u1 du2 + u2 du1 and ϕ ∗ dt = u1 du1 + u1 du1 − u2 du2 − u2 du2 . The forms ϕ ∗ dw, ϕ ∗ dw, ϕ ∗ dt and ξ are orthogonal. Moreover, we have |ϕ ∗ dw|2 = |ϕ ∗ dw|2 = 8(|u1 |2 + |u2 |2 ), |ϕ ∗ dt|2 = 4(|u1 |2 + |u2 |2 ) and |ξ |2 = 4(|u1|2 + |u2 |2 ). Hence, we have  ∗    ϕ (dw dw dt) ξ  = 8(|u1 |2 + |u2 |2 )2 du1 du1 du2 du2 . Let ψ(u1 ,u2 ) : R/2π −→ C2 be given by ψ(u1 ,u2 ) (e We have

√ −1θ )

√ 

√ = u1 e −1θ , u2 e− −1θ .

√ ∗ ξ = 2 −1(|u1 |2 + |u2 |2 )dθ. ψ(u 1 ,u2 ) Hence, we obtain   1 dw dw dt  · 4π(|w|2 + |t|2 )1/2 8(|w|2 + |t|2 )   π dw dw dt . = (2.40) 2(|w|2 + |t|2 )1/2



ϕ∗ |du1 du1 du2 du2 | =

We have |dw dw dt| = 2 dvolU and |du1 du1 du2 du2 | = 4 dvolU . Hence, we obtain the following: ϕ∗ dvolU =

π dvolU . 4(|w|2 + |t|2 )1/2

By direct computations, we have √ u1 ,u1 = 4u1 u2 ϕ ∗ Fwt − 4u1 u2 ϕ ∗ Fwt + 4|u2|2 Fww − 2|u1 |2 −1ϕ ∗ ∇t φ, F √ u2 u2 = −4u1 u2 ϕ ∗ Fwt + 4u1 u2 ϕ ∗ Fwt + 4|u1 |2 Fww − 2|u2 |2 −1ϕ ∗ ∇t φ. F Hence, we obtain √

 −1 2 2 ∗   ∇t φ Fu1 u1 + Fu2 u2 = 4(|u1 | + |u2 | )ϕ Fww − 2   ∗ 2 2 1/2 = ϕ 2(|w| + |t| ) G(h) . Thus, we obtain (2.39).

(2.41) 

2.9 Curvatures of Mini-Holomorphic Bundles with Metric on Mλ

87

2.9.7 Appendix: Ambiguity of the Choice of a Splitting We explain one of the main reasons why we use (t1 , β1 ) (or (t1† , β1† )). Let Z be a finite subset of M. Let (E, ∇, h, φ) be a monopole on M \ Z. Let F (∇) = F (∇)ww dw dw + F (∇)t w dt dw + F (∇)t w dt dw denote the curvature of ∇. Suppose the following conditions. Condition 2.9.16



• |F (∇)ww |h = O |w|−2 (log |w|)−2 as |w| → ∞. It implies |∇t φ|h = 

O |w|−2 (log |w|)−2 as |w| → ∞. • There exist positive numbers R, C,  > 0, a finite subset S ⊂ Q and an orthogonal decomposition E=



Eω

ω∈S



on ST1 × {|w| > R} such that F (∇)t w − ω∈S ωw−1 idEω = O |w|−1− as



|w| → ∞. Note that it implies that F (∇)t w + ω∈S ωw−1 idEω = O |w|−1− 

√ as |w| → ∞. It also implies that ∇w φ − ( −1ω)w−1 idEω = O |w|−1− 

√ and ∇w φ − ( −1ω)w−1 idEω = O |w|−1− as |w| → ∞. • We assume that S = {0}. Let p1 : M −→ M denote the projection. We set Z1 := p1−1 (Z). We set (E1 , h1 , ∇1 , φ1 ) := p1−1 (E, h, ∇, φ) on M \ Z1 . Let λ = 0. Let (E1λ , ∂ E λ ) denote 1

the holomorphic vector bundle on M λ \ Z1 underlying (E1 , h1 , ∇1 , φ1 ). Let (t2 , β2 ) be a mini-complex coordinate system of M λ . Let Fβt2 ,β dβ2 dβ 2 2 2 denote the curvature of the Chern connection of the holomorphic vector bundle (E1λ , ∂ E λ )|{t2 }×Cβ2 with the metric induced by h1 . Suppose the following conditions 1

Condition 2.9.17

 • |Fβt2 ,β |h = O |β2 |−2 (log |β2 |)−2 as |β2 | → ∞. It is equivalent to |Fβt2 ,β |h = 2 2 2 2 

O |w|−2 (log |w|)−2 as |w| → ∞. Proposition 2.9.18 Either one of the following holds. • There exist a positive number c1 and a non-zero complex number c2 such that (t2 , β2 ) = c1 (t1 , c2 β1 ). • There exist a positive number c1 a non-zero complex number c2 such that (t2 , β2 ) = c1 (t1† , c2 β1† ) = c1 (t1 − Im(λ−1 β1 ), c2 λ−2 β1 ). Proof After changing (t2 , β2 ) to c1 (t2 , β2 ) for some c1 > 0, we may assume that there uniquely exists a linear complex coordinate system (α2 , β2 ) of Xλ such that (i) the R-action is described as s • (α2 , β2 ) = (α2 + s, β2 ), (ii) (α2 , β2 ) induces  ∇, 1 , corresponding to the   (t2 , β2 ) on M λ . We have the instanton (E, h) on X \ Z

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2 Preliminaries

 denote the pull back of Z by the projection monopole (E1 , h1 , ∇1 , φ1 ), where Z λ , ∂ Eλ ) be the holomorphic vector bundle on Xλ \Z 1 underlying Xλ −→ M λ . Let (E λ, ∂ )   Note that (E λ , ∂ Eλ ) √ (E, h, ∇).

(E . Hence, according λ E1 |{t2 }×Cβ2 |{ −1t2 }×Cβ2 1 to Condition 2.9.17, for the expression   = F (∇)  α2 α2 dα2 dα 2 + F (∇) F (∇) α2 β 2 dα2 dβ 2  β2 α 2 dβ2 dα 2 + F (∇)  + F (∇) β2 β 2 dβ2 dβ 2 ,

(2.42)



−2 −2 as |β | → ∞. Note that for √  we have F (∇) 2 β2 β 2 |{ −1t2 }×Cβ2 = O |w| (log |w|) the expression  zw2 dz dw + F (∇)  wz dw dz + F (∇)  ww dw dw,  = F (∇)  zz dz dz + F (∇) F (∇) 

−2 −2 .  ww |  zz | |F (∇) Condition 2.9.16 implies that |F (∇) h =

h = O |w| (log |w|) = ω such that Moreover, there exists a decomposition E E  zw + |F (∇)



−1−  . C(ωw −1 ) idEω | h = O |w|

Because β2 is R-invariant, there exists a non-zero complex number a such that β2 = a(η + λξ ) = aβ1 . There exists a complex number b such that α2 = ξ + b(η + λξ ). We obtain z= w=

 1 α2 − a −1 bβ2 + |λ|2 α 2 − λa −1 β 2 (1 + λb) , 2 1 + |λ|

 1 − λα2 + a −1 β2 (1 + λb) + λα 2 − λa −1 bβ 2 . 2 1 + |λ|

√ Hence, the restriction of dz dw to { −1t2 } × Cβ2 is equal to b(1 + λb)

−1 |a|2 dβ2 dβ 2 . (1 + |λ|2 )

√ Similarly, the restriction of dw dz to { −1t2 } × Cβ2 is equal to b(1 + λb)

−1 |a|2 dβ2 dβ 2 . (1 + |λ|2 )

We obtain b(1+λb) = 0, i.e., b = 0 or b = −λ−1 . If b = 0, we obtain α2 = ξ = α1 , and hence t2 = t1 . If b = −λ−1 , we obtain α2 = −λ−1 η = −α1† , and hence t2 = t1† . Thus, we are done. 

λ )-Modules 2.10 Difference Modules and OMλ \Z (∗H∞

2.10 Difference Modules and O

λ

M \Z

89

λ )-Modules (∗H∞

2.10.1 Difference Modules with Parabolic Structure at Finite Place Let  ∈ C. Let Φ ∗ be an automorphism of the rational function field C(y) induced by Φ ∗ (y) = y + . Let V be a difference module as in Definition 1.6.1, i.e., V is a finite dimensional C(y)-module equipped with a C-linear automorphism Φ ∗ such that Φ ∗ (f s) = Φ ∗ (f ) · Φ ∗ (s) for f ∈ C(y) and s ∈ V . Let V be a lattice of V , i.e., a C[y]-free module V ⊂ V such that C(y) ⊗C[y] V = V . We set V  := V + (Φ ∗ )−1 (V ). We obtain the associated free OP1 (∗∞)module FV . We also obtain the associated free OP1 (∗∞)-modules F(Φ ∗ )−1 V and FV  . There exists the natural isomorphism FV Φ ∗ F(Φ ∗ )−1 (V ) . We may regard FV and F(Φ ∗ )−1 (V ) as OP1 (∗∞)-submodules of FV  , and FV  = FV + F(Φ ∗ )−1 (V ) holds. Definition 2.10.1 A parabolic structure of V at finite place is a tuple as follows. • A C[y]-free submodule V ⊂ V such that C(y)  ⊗C[y] V = V . • A function m : C −→ Z≥0 such that x∈C m(x) < ∞. We assume that FV (∗D) = F(Φ ∗ )−1 (V ) (∗D), where D := {x ∈ C | m(x) > 0}. (1)

(m(x))

< T for each x ∈ C. If • A sequence of real numbers 0 ≤ τx < · · · < τx m(x) = 0, the sequence is assumed to be empty. The sequence is denoted by τ x . • Lattices Lx,i ⊂ V ⊗ C((y − x)) for x ∈ C and i = 1, . . . , m(x) − 1. We formally set Lx,0 := V ⊗ C[[y − x]] and Lx,m(x) := (Φ ∗ )−1 V ⊗ C[[y − x]]. The tuple of lattices is denoted by Lx . Here, C[[y − x]] denotes the ring of formal power series with the variable y − x, and C((y − x)) denotes the ring of formal Laurent power series with the variable y − x.  Remark 2.10.2 The notion of parabolic structure of a difference module at finite place is apparently different from the ordinary notion of parabolic structure in the context of harmonic √ bundles. However, as we shall explain in Sect. 2.10.4, a parabolic structure of 2 −1λ-difference module at finite place is a reincarnation of an ordinary parabolic structure of the regular part of λ-flat bundle at {0, ∞} through the Mellin transform or the algebraic Nahm transform. 

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2 Preliminaries

2.10.2 Construction of Difference Modules from λ )-Modules O λ (∗H∞ M \Z

λ )-module of Let Z be a finite subset in Mλ . Let E be a locally free O λ (∗H∞ M \Z Dirac type. Let us observe that we obtain the √ associated difference module with parabolic structure at finite place with  = 2 −1λT . λ λ Let  λ : M −→ M denote the projection. Set Z cov := ( λ )−1 (Z). Let λ λ p1 : M −→ Rt1 and p2 : M −→ P1β1 be the projections. We obtain the λ

λ

λ cov)-module E cov := ( λ )∗ E. Let j : M \ Z −→ M denote OM λ \Z (∗H∞ λ cov)-module j E cov . For any a ∈ R, let the inclusion. We obtain the OM λ (∗H∞ ∗

cov ιa : p1−1 (a) −→ M . We obtain the locally free OP1 (∗∞)-modules ι−1 a j∗ E (a ∈ R). Because j∗ E cov is naturally Z-equivariant, we have the following isomorphism for any a ∈ R and n ∈ Z: λ

 

cov cov ) ι−1 ) . (Φ n )∗ ι−1 a j∗ (E a+T n j∗ (E

(2.43)

√ Here, Φ : P1 −→ P1 is given by Φ(β1 ) = β1 + 2  −1λT . 

We set Da,b := p2 Z cov ∩ {a ≤ t1 ≤ b}×P1β1 for any a ≤ b. By the scattering map, we obtain the isomorphism

  cov cov ι−1 ) (∗Da,b ) ι−1 ) (∗Da,b ). a j∗ (E b j∗ (E

(2.44)

For any a ∈ R, we set 

 cov ) . Va := H 0 P1 , ι−1 a j∗ (E For any subset S ⊂ C, let C[β1 ]S denote the localization of C[β1 ] with respect to β1 − x (x ∈ S). By (2.44), we obtain the isomorphism of C[β1 ]Da,b -modules: Va ⊗ C[β1 ]Da,b Vb ⊗ C[β1 ]Da,b . Hence, we obtain the isomorphism of C(β1 )-modules for any a ≤ b. Va ⊗ C(β1 ) Vb ⊗ C(β1 ). From (2.43), we have the C-linear isomorphism (Φ ∗ )n : Va+nT Va

(2.45)

λ )-Modules 2.10 Difference Modules and OMλ \Z (∗H∞

91

√ such that (Φ ∗ )n (β1 s) = (β1 + 2 −1λnT ) (Φ ∗ )n (s). We obtain the C-linear isomorphism (Φ ∗ )n : Va+nT ⊗ C(β1 ) Va ⊗ C(β1 )

(2.46)

√ ) · (Φ ∗ )n (s) for any such that (Φ ∗ )n (g(β1 )s) = g(β1 + 2 −1λnT

 g(β1 ) ∈ C(β1 ). cov We take a small  > 0 such that Z ∩ {− ≤ t1 < 0} × P1β1 = ∅. Set V (E) := V− ⊗ C(β1 ) which is a finite dimensional C(β1 )-vector space. It is identified with Vb ⊗ C(β1 ) for any b ∈ R by (2.45). It is equipped with C-linear automorphism Φ ∗ by (2.46). Set V (E) := V− for which we obtain C(β1 ) ⊗ V (E) = V (E). Let mZ : C −→ Z≥0 be the function such that     mZ (x) = p2−1 (x) ∩ p1−1 ({0 ≤ t1 < T }) ∩ Z cov . (1)

(m (x))

Z For each x ∈ C, the sequence 0 ≤ t1,Z,x < · · · < t1,Z,x

(2.47)

< T determined by

{(t1,Z,x , x) | i = 1, . . . , mZ (x)} = p2−1 (x) ∩ p1−1 ({0 ≤ t1 < T }) ∩ Z cov . (i)

(2.48)

We set (i)

(i)

τZ,x := t1,Z,x /T .

(2.49)

(i) The tuple τZ,x (i = 1, . . . , mZ (x)) is denoted by τ Z,x . For x ∈ C and for i = (i) (i+1) 1, . . . , mZ (x) − 1, we choose t1,Z,x < b(x, i) < t1,Z,x , and we set

LZ,x,i (E) := Vb(x,i) ⊗ C[[β1 − x]], which is independent of the choice of b(x, i). The tuple LZ,x,i (i = 1, . . . , mZ (x)− 1) is denoted by LZ,x . Thus, we obtain a parabolic structure at finite place

V (E), mZ , (τ Z,x , LZ,x (E))x∈C



of V (E).

2.10.3 Construction of O λ (∗H λ )-Modules from M \Z Difference Modules √ Let V be a (2 −1λT )-difference module with a parabolic at finite place  (i)structure   # (i)  (V , m, (τ x , Lx )x∈C ). We set Z1 := x∈C (T τx , x) τx ∈ τ x ⊂ Rt1 × Cβ1 .

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2 Preliminaries

We set Z :=  λ (Z1 ) ⊂ Mλ . Let us construct an O λ (∗H λ )-module M \Z E(V , V , m, (τ x , Lx )x∈C). λ We set U :=] − , T − /2[×P1β1 and H∞, :=] − , T − /2[×{∞}. We set (i)

(i)

D := {x ∈ C | m(x) > 0}. For x ∈ D, we set tx = T τx (i = 1, . . . , m(x)), (0) (m(x)+1) tx = −/2 and tx = T − . The following conditions determine a unique λ OU \Z1 (∗H∞, )-module E1 : • E1 (∗p2−1 (D)) is isomorphic to the pull back of FV (∗D) by the projection U −→ P1β1 . (i)

(i+1)

for i = 0, . . . , m(x). Take a small neighbourhood Ux of x • Take tx < ai < tx in C. Let ι{ai }×Ux denote the inclusion of {ai } × Ux to U \ Z1 . Then, ι∗{ai }×Ux E1 ⊂ FV (∗D)|Ux is equal to the OUx -submodule determined by Lx,i . Let κ1 : ] − , −/2[×P1 −→]T√− , T − /2[×P1 be the isomorphism defined by κ1 (t1 , β1 ) = (t1 + T , β1 + 2 −1λT ). By the construction, there exists the isomorphism   κ1∗ E1|]T −,T −/2[×P1 E1|]−,−/2[×P1 . λ )-module E(V , V , m, (τ , L ) Hence, we obtain a locally free O λ (∗H∞ x x x∈C ) of M \Z Dirac type. By the construction, the following is clear.

Lemma 2.10.3 The constructions in Sect. 2.10.2 and Sect. 2.10.3 are mutually inverse up to canonical isomorphisms.  λ )Remark 2.10.4 The relation between difference modules and O λ (∗H∞ M \Z modules of Dirac type is analogue to the relation between OC (∗D)-modules and filtered bundles over (C, D) for a compact Riemann surface C with a finite subset D ⊂ C. 

2.10.4 Appendix: Mellin Transform and Parabolic Structures at Finite Place Let us explain that the notion of parabolic structure at finite place of difference modules is related with the usual notion of parabolic structure of filtered λ-flat bundles through the algebraic Nahm transform, which is an analogue of Mellin transform.

λ )-Modules 2.10 Difference Modules and OMλ \Z (∗H∞

93

2.10.4.1 Mellin Transform Let λ be any complex number. Let M be a C[u, u−1 ]-module equipped with a Clinear endomorphism Dλu : M −→ M such that Dλu (f s) = f Dλu (s) + λ∂u (f √ )s. ∗ (β ) = β + 2 −1λ. We define the automorphism Φ ∗ of C-algebra C[β ] by Φ 1 1 1 √ We define C[β1 ]-action on M by β1 s = −2 −1uDλu (s) for any s ∈ M. We define ∗ of M by Φ ∗ (s) = us for any s ∈ M. Because the C-linear automorphism ΦM √ ∗ ∗ ∗ ΦM (β1 s) = Φ (β1 )ΦM (s), we obtain 2 −1λ-difference module, which is called the Mellin transform. We have the natural geometrization of the transformation of the Mellin transform. Let pi (i = 1, 2) denote the projections of P1u × Cβ1 onto the i-th component. Let V be an algebraic quasi-coherent OP1 (∗{0, ∞})-module with a λ-connection Dλ . We obtain the following complex C • (V, Dλ ) on P1u × Cβ1 :

where the first term sits in the degree 0. We have R i p2∗ (C • (V, Dλ )) = 0 unless i = 1. We obtain the algebraic OCβ1 -module M(V, Dλ ) := R 1 p2∗ (C • (V, Dλ )). Let fu : V −→ V be the automorphism defined as the multiplication of u. It induces the isomorphism (V, Dλ + λdu/u) (V, Dλ ). √ Let Φ : Cβ1 −→ Cβ1 be defined by Φ(β1 ) = β1 + 2 −1λ. We obtain the following morphisms:

Hence, we obtain the isomorphism Φ ∗ M(V, Dλ ) M(V, Dλ ). If M = H 0 (P1 , V), then H 0 (P1 , M(V, Dλ )) is the Mellin transform. Remark 2.10.5 The stationary phase formula for Mellin transform has been studied in [27] and [29].  2.10.4.2 Algebraic Nahm Transform for Filtered λ-Flat Bundles (Special Case) Let D ⊂ C∗ be a finite subset. We set D = D ∪ {0, ∞}. Let V be a locally free OP1 (∗D)-module of finite rank equipped with a λ-connection Dλ . Let (P∗ V, Dλ ) be a good filtered λ-flat bundle on (P1 , D). For simplicity, we assume the following. Condition 2.10.6 • D is non-empty. Moreover, at each point P of D, (P∗ V, Dλ ) does not have the regular part, i.e., in the Hukuhara-Levelt-Turrittin decomposition (1.5) of the

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2 Preliminaries

pull back by an appropriate ramified covering of a neighborhood of P , we have V0 = 0. P∗  • P∗ V is regular at 0 and ∞, i.e., Dλ is logarithmic with respect to Pa (V) for any a ∈ RD at 0 and ∞.  We shall construct an O

λ

M \Z

(∗H λ )-module from (P∗ V, Dλ ), where Z denotes

a finite subset explained below. We assume T = 1 in the construction of Mλ . Let V0 denote the stalk  of V at 0. We haveλthe filtration Pa (V0 ) (a ∈ R). We set GrP a (V0 ) = Pa (V0 ) P 0 such that {t1 − ≤ a ≤ t1 +}∩  = {t1 }. Suppose t1 ∈  π (Z). π (Z) For any t1 −  ≤ t1 ≤ t1 + , we have Pt1 −,−t1 − V ⊂ Pt1 ,−t1 V, and hence there  t1 −,−t1 − (P∗ V, Dλ ) −→ N  t  ,−t  (P∗ V, Dλ ). exists the following morphism N 1

t1 := Lemma 2.10.7 We set Z of C. Then, the induced morphism

 π −1 (t1 )

1

 which we naturally regard as a subset ∩ Z,

 t1 −,−t1 − (P∗ V, Dλ )(∗Z  t  ,−t  (P∗ V, Dλ )(∗Z t1 ) −→ N t1 ) N 1 1

(2.51)

is an isomorphism. √

−1 Proof If α is an eigenvalues of Res0 (Dλ ) on GrP t1 (V0 ), then 2 √α is contained in −1 t1 . If α is an eigenvalues of Res∞ (Dλ ) on GrP Z −t1 (V∞ ), then − 2 α is contained t1 . in Z t1 . There exists a neighbourhood U of β1 in C \ Z t1 . By the Let β1 ∈ C \ Z √ −1 consideration in the previous paragraph, Res0 (Dλ ) − 2 β1 (β1 ∈ U ) are invertible √

−1  P λ  on GrP t1 (V0 ), and Res∞ (D )+ 2 β1 (β1 ∈ U ) are invertible on Gr−t1 (V∞ ). Hence, we can easily check the claim of the lemma. 

By Lemma 2.10.7, we obtain the following isomorphism for any t1 −  ≤ t1 ≤ t1 + :  t  ,−t  (P∗ V, Dλ ) 1  .  t1 ,−t1 (P∗ V, Dλ ) 1  N N |P \Zt |P \Zt 1

1

1

(2.52)

1

By the isomorphisms (2.50) and (2.52), we obtain an O(R×P1 )\Z(∗(R × {∞})) t1 ,−t1 (P∗ V) 1  (t1 ∈ R). The isomorphism fu on  ∗ V) from N module N(P |P \Zt1 λ ) ∗ V, Dλ ). Hence, we obtain an O λ (∗H∞  ∗ V, Dλ ) N(P V induces κ ∗ N(P M \Z

1

module N(P∗ V).  There exist integers k1 ≥ . . . ≥ kr and a frame v1 , . . . , vr of Let (t10 , β10 ) ∈ Z. λ  N(P∗ V, D )(t 0 −,β1 ) such that the tuple (β1 − β10 )ki vi (i = 1, . . . , r) is a frame of 1  ∗ V, Dλ ) 0 N(P . We set

(t1 +,β1 ) √ α = 2−1 β10 .

We obtain the filtration W on the spaces Eα GrP (V ) t0 0 1

and E−α GrP (V ) obtained as the weight filtration of the nilpotent parts of −t 0 ∞ 1

P Res0 (Dλ ) and Res∞ (Dλ ), respectively. For k ≥ 0, let P GrW k Eα Grt 0 (V0 ) and P P GrW k E−α Gr−t 0 (V∞ ) denote the primitive parts. 1

1

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2 Preliminaries

P Proposition 2.10.8 dim P GrW k Eα Grt 0 (V0 ) is equal to the number of vi such that 1

P ki = k + 1, and dim P GrW k E−α Gr−t 0 (V∞ ) is equal to the number of vi such that 1

ki = −k − 1.

Proof It is enough to consider the case (t10 , β10 ) = (0, 0). Let c1 : P−,− V −→ P−, V and c2 : P−,− V −→ P,− V denote the natural inclusions. Let d1 : P−, V −→ P, V and d2 : P,− V −→ P, V denote the natural inclusions. We obtain the following exact sequence: c1 ⊕c2

d1 −d2

0 −→ P−,− V −→ P,− V ⊕ P−, V −→ P, V −→ 0. It induces the following exact sequence  −,− (P∗ V, Dλ ) −→ N  ,− (P∗ V, Dλ ) ⊕ N  −, (P∗ V, Dλ ) −→ 0 −→ N  , (P∗ V, Dλ ) −→ 0. N

(2.53)

On C, we consider the following complex: GrP 0 V0

OC

Res0 D

2

1

1

GrP 0 V0

OC

The quotient is isomorphic to the quotient of  −,− (P∗ V, Dλ ) −→ N  ,− (P∗ V, Dλ ). N We have a similar description of the quotient of  −, (P∗ V, Dλ ).  −,− (P∗ V, Dλ ) −→ N N Then, we can easily deduce the claim of the proposition from Lemma 2.10.9 below.  Lemma 2.10.9 Let V be an r-dimensional vector space with a base e1 , . . . , er . Let f be the endomorphism of V determined by f (ei ) = ei+1 (i = 1, . . . , r − 1) and f (er ) = 0. Then, the cokernel f − z idV : V ⊗ OCz −→ V ⊗ OCz is isomorphic to OC /zr OC .

r Ci=2 ei , and let Proof We set F = f − z idV . Let V  denote the vector space ρ : V −→ V  denote the projection. It  is easy to see that ρ ◦ F is an epimorphism, and the kernel of ρ ◦ F is generated by rj =1 zr−j ej . Then, the claim of the lemma easily follows.  Remark 2.10.10 We can consider this kind of transformation for stable good filtered λ-flat bundles of degree 0 which do not necessarily satisfy Condition 2.10.6. It

2.11 Filtered Prolongation of Acceptable Bundles

97

will be explained elsewhere. The transformation naturally appear as the algebraic counterpart of the Nahm transform between periodic monopoles and wild harmonic bundles on (P1 , D). Some special cases were studied in [13, 14, 30], and a more general case is studied in [69]. 

2.11 Filtered Prolongation of Acceptable Bundles 2.11.1 Filtered Bundles on a Neighbourhood of 0 in C Let Y be a neighbourhood of 0 in C. Let E be a locally free OY (∗0)-module of finite rank. A filtered bundle over E is an increasing sequence of locally free OY submodules Pa E ⊂ E (a ∈ R) satisfying the following conditions. • Pa E (a ∈ R) are lattices of E, i.e., Pa E(∗{0}) = E. • Pa+n E = Pa E(n{0}) for any a ∈ R and n ∈ Z. • For any a ∈ R, there exists  > 0 such that Pa+ E = Pa E. In that case, we also say that P∗ E is a filtered bundle on (Y, 0), for simplicity. For any a ∈ R, we set P0 , a finite subset I(N ) ⊂ b ∈ yp−1 C[yp−1 ]  degyp−1 b <  p and a decomposition of difference modules N ⊗C((y −1 )) C((yp−1 )) = q



Nb ,

(Nb = 0)

(3.1)

b∈I (N )

such that each Nb has a Φ ∗ -invariant lattice Lb satisfying (Φ ∗ −(1+b) idNb )Lb ⊂ −p yp Lb . The set I(N ) and the decomposition (3.1) are uniquely determined. Proof If  = 0, the assumption implies that the eigenvalues of the C((yq−1 ))automorphism Φ ∗ are invertible elements in C[[yp−1]] for some p ∈ qZ>0 . Hence, the claim is easily checked. If  = 0, the claim is known as a part of the classification of difference modules. See [12, Proposition 5]. (It is not difficult to prove it directly by a standard method in the study of formal connections.)  By following [12], we say that the level of a difference module N is 0 if there −q exists a Φ ∗ -invariant lattice L of N such that (Φ ∗ − id)(L) = yq L = y −1 L. In Proposition 3.2.1, each Nb is isomorphic to the tensor product of a difference module of the level 0 and a difference module C((yp−1 )) e with the difference operator Φ ∗ defined as Φ ∗ (e) = (1 + b)e.

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109

∗ Remark 3.2.2 In Proposition 3.2.1, defined as a C-linear

∗Φ − (1 + b)  id is well −p endomorphism of Nb . Note that Φ − (1 + b) id Lb ⊂ yp Lb holds if and only if

 −p there exists a frame v = (v1 , . . . , vr ) of Lb such that Φ ∗ − (1 + b) id vi ∈ yp Lb . Indeed, the “only if” part of the claim is clear. Suppose that there exists such a frame v of Lb . For any f ∈ C[[yp−1 ]], we obtain





∗ Φ − (1 + b) id (f vi ) = Φ ∗ (f ) Φ ∗ − (1 + b) id (vi ) + (1 + b)(Φ ∗ (f ) − f )vi . (3.2) −p−1

Because Φ ∗ (f ) − f ∈ yp

−p

C[[yp−1 ]], (3.2) is contained in yp Lb .



The following lemma is standard. Lemma 3.2.3 Let g : N1 −→ N2 be a morphism of C((yq−1 ))-difference modules of level ≤ 1. Choose p ∈ qZ≥0 such that both Ni ⊗ C((yp−1 )) have decompositions of C((yp−1 ))-difference modules as in (3.1) Ni ⊗ C((yp−1 )) =



Ni,b .

b∈yp−1 C[yp−1 ]



Then, we have g N1,b ⊂ N2,b , where we set Ni,b = 0 if b ∈ I(Ni ). Proof We indicate only an outline. We may assume p = q. It is enough to prove that g = 0 if N1 = Ni,bi with b1 = b2 . There exist Φ ∗ -invariant lattices Li ⊂ Ni −q such that (Φ ∗ − (1 + bi )Ni )Li ⊂ yq Li . Let v i be a frame of Li . Let A be the matrix representingg with respect to the frames v i , i.e., g(v 1 ) = v 2 A. There exists an expansion A = j ≥N Aj (yq−1 )j . Suppose A = 0. We may assume that AN = 0. Let Bi be the matrices determined by Φ ∗ (v i ) = v i Bi . We have the expansions  −j Bi = (1 + bi )Ir(i) + j ≥q Bi,j yq , where we set r(i) = rank Ni , and Ir(i) denote q−1 −j the r(i)-square identity matrices. We also have the expansions bi = j =1 bi,j yq . Let j (0) := min{j | b1,j − b2,j }. Because B2 Φ ∗ (A) = AB1 and 1 ≤ j0 , we obtain −N−j0 . It implies AN = 0, b2,j (0)AN = b1,j (0)AN by taking the coefficients of yq which contradicts with the assumption. Hence, we obtain A = 0.    For p ∈ qZ>0 , we set Galq,p := γ ∈ C | γ p/q = 1 which acts on C((yp−1 )) by γ • f (yp−1 ) = f (γ −1 yp−1 ). It is identified with the Galois group of the extension C((yp−1 )) over C((yq−1 )). Note that the Galq,p -action on C((yp−1 )) induces a Galq,p action on N ⊗C((yq−1)) C((yp−1 )). We obtain the following lemma from the uniqueness of the index set I(N ) and the decomposition (3.1).

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3 Formal Difference Modules and Good Parabolic Structure

Lemma 3.2.4 The Galq,p -action on C((yp−1 )) preserves I(N ) ⊂ C((yp−1 )). Moreover, the following holds. −p

• For b ∈ I(N ), let Lb be a lattice of Nb satisfying (Φ ∗ −(1+b) id)Lb ⊂ yp Lb . Then, for any γ ∈ Galq,p , γ • Lb is a lattice of Nγ •b satisfying (Φ ∗ − (1 + γ • −p  b) id)Lγ •b ⊂ yp Lγ •b . Remark 3.2.5 If  = 0, for a difference module N of level 0, there exist frame u = (u1 , . . . , ur ) and a matrix A ∈ Mr (C) such that Φ ∗ (u) = u(Ir + yq−1 A), where Ir ∈ Mr (C) denotes the identity matrix. (See [12].)  Remark 3.2.6 In Sect. 3.6, we shall explain that formal difference modules of level ≤ 1 on C((yq−1 )) are equivalent to formal λ-connections on C((wq−1 )) whose Poincaré rank is strictly smaller than q. 

3.2.2 Formal Difference Modules of Pure Slope Let N be a difference C((yq−1 ))-module. For integers m and , a C[[yq−1 ]]-lattice of N is called yqm (Φ ∗ ) -invariant if yqm (Φ ∗ ) (L) = L. Note that the C-linear automorphism of L|∞ = L/yq−1 L is induced by yqm (Φ ∗ ) , which is denoted by (yqm (Φ ∗ ) )|∞ .    For any ω ∈ Q, we set Z(q, ω) := p ∈ qZ>0  pω ∈ Z , i.e., Z(q, ω) = qZ>0 ∩ ω−1 Z if ω = 0, and Z(q, 0) = qZ>0 . Definition 3.2.7 We say that a difference C((yq−1 ))-module N has pure slope ω ∈ Q pω  if there exist p ∈ Z(q, ω) and a yq (Φ ∗ )p/q -invariant lattice of N . Lemma 3.2.8 • Suppose that a difference C((yq−1 ))-module N has a pure slope ω. Then, for any pω p ∈ Z(q, ω), there exists a yq (Φ ∗ )p/q -invariant lattice of N . • Take any s ∈ qZ>0 . Then, a difference C((yq−1 ))-module N has a pure slope ω if and only if the induced difference C((ys−1 ))-module C((ys−1 )) ⊗C((y −1 )) N has a q pure slope ω. • Take any s ∈ Z(q, ω). If a difference C((yq−1 ))-module N has a pure slope ω, there exists a Galq,s -invariant lattice L of the induced difference C((ys−1 ))module C((ys−1 )) ⊗C((yq−1)) N such that yssω Φ ∗ (L) = L. Proof Let us study the first claim. By the assumption, there exists p1 ∈ Z(q, ω) p ω such that there exists a yq 1 (Φ ∗ )p1 /q -invariant lattice L1 of N . Let p be any element pω of Z(q, ω). If p ∈ p1 Z>0 , it is easy to see that L1 is yq (Φ ∗ )p/q -invariant. Let us

3.2 Formal Difference Modules

111

consider the case where there exists  ∈ Z>0 such that p1 = p. We set L = −1 ipω · (Φ ∗ )ip/q (L1 ). Then, we obtain i=0 yq yq (Φ ∗ )p/q (L1 ) = pω

−1

(i+1)pω

yq

· (Φ ∗ )(i+1)p/q (L1 )

i=0

=

−1

yq (Φ ∗ )ip/q (L1 ) + yq ipω

(−1)pω

pω −pω

· yq yq

L1 = L.

i=1

(3.3) The first claim in the general case immediately follows. Let us study the second claim. The “only if” part of the claim is obvious. Let us study the “if” part of the claim. It is enough to study the case s ∈ Z(q, ω). Suppose that C((ys−1 )) ⊗C((yq−1)) N has a pure slope ω. Let L be a yssω Φ ∗ -invariant

lattice of C((ys−1 )) ⊗C((y −1)) N . It is also a (yssω )s/q (Φ ∗ )s/q -invariant lattice of q

C((ys−1 )) ⊗C((yq−1 )) N . Note that (yssω )s/q = yqsω . We set L1 := L ∩ N for the natural inclusion N −→ C((ys−1 )) ⊗C((y −1 )) N . Because both L and N are preserved by q

the action of yqsω (Φ ∗ )s/q on C((ys−1 )) ⊗C((yq−1)) N , we obtain that L1 is yqsω (Φ ∗ )s/q invariant. Hence, N has pure slope ω. As for the third claim, we have already proved that there  exists a lattice L1 of C((ys−1 )) ⊗C((y −1 )) N , such that yssω Φ ∗ (L1 ) = L1 . Then, L = γ ∈Galq,s γ • L1 has q the desired property.  Example 3.2.9 A difference C((yq−1 ))-module of level ≤ 1 has pure slope 0.



C∗ ,

Example 3.2.10 For any  ∈ Z and α ∈ let Lq (, α) denote the difference C((yq−1 ))-module Lq (, α) = C((yq−1 )) e with the difference operator Φ ∗ defined as Φ ∗ (e) = yq− α · e. 

Then, Lq (, α) has pure slope /q. C((yq−1 ))-modules

Lemma 3.2.11 Let Ni (i = 1, 2) be difference with pure slopes −1 ωi . Any morphism of difference C((yq ))-modules N1 −→ N2 is 0 unless the following condition is satisfied. • ω1 = ω2 . pω • For any p ∈ Z(q, ω), and for any yq 1 (Φ ∗ )p/q -invariant lattices Li of Ni , the induced automorphisms on Li|∞ have a common eigenvalue. Proof Let f : N1 −→ N2 be a morphism of difference C((yq−1 ))-modules. Take any pω p ∈ Z(q, ω1 )∩Z(q, ω2 ). There exist yq i (Φ ∗ )p/q -invariant lattices Li of Ni . Let v i be frames of Li . Let ri := rank Ni . We obtain the matrices Ai ∈ Mri (C[[yq−1 ]]) such  pω −j yq (Φ ∗ )p/q v i = v i Ai . For the expansion Ai = ∞ j =0 Ai,j yq , Ai,0 are invertible.

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3 Formal Difference Modules and Good Parabolic Structure

Let B be the matrix determined by f (v 1 ) = v 2 · B. If B = 0, there exists the  j expansion B = j ≤j0 Bj yq such that Bj0 = 0. Because f is compatible with the difference operators, we obtain −pω1

B(yq−1 ) · A1 · yq

 −pω −q = A2 · B (1 + (p/q)yq )−1/q · yq 2 .

(3.4)

By comparing the expansions of the both sides of (3.4), we obtain Bj0 = 0 unless ω1 = ω2 holds and Ai,0 have a common eigenvalue. Then, the claim of the lemma follows.  We can classify difference modules with pure slope by the following proposition and Lemma 3.2.8. Proposition 3.2.12 ([12, 23, 76, 86]) Let N be a difference C((yq−1 ))-module with pure slope ω. Suppose that qω ∈ Z. Then, there exists a unique decomposition of difference C((yq−1 ))-modules N =



Nα

(3.5)

α∈C∗

such that each Nα is isomorphic to the tensor product of Lq (qω, α) and a difference C((yq−1 ))-module of level ≤ 1. Proof Let L be a yq Φ ∗ -invariant lattice of N . We obtain the generalized eigen decomposition qω

L|∞ =



Eα (L|∞ )

(3.6)

α∈C∗

of (yq Φ ∗ )|∞ , i.e., the restriction of α −1 (yq Φ ∗ )|∞ to

Eα (L|∞ ) are−1unipotent. It is a standard fact that there exists a decomposition L = Lα of C[[yq ]]-lattice such qω = E (L ). (See [12, §3], for example.) Then, that (i) yq Φ ∗ (Lα ) = Lα , (ii) Lα|∞ α |∞

the induced decomposition N = C((yq−1 )) ⊗C[[y −1 ]] Lα has the desired property. q The uniqueness follows from Lemma 3.2.11.  qω

qω

with pure slope ω. Suppose Let Ni (i = 1, 2) be difference C((yq−1 ))-modules

qω ∈ Z. Each Ni has a decomposition Ni = α∈C∗ Ni,α as in (3.5). We obtain the following lemma from Lemma 3.2.11. Lemma 3.2.13 Any morphism F : N1 −→ N2 preserves the decompositions Ni =

 α∈C∗ Ni,α . Let N be a difference C((yq−1 ))-difference of pure slope ω ∈ Q. Take p ∈ qZ>0 such that pω ∈ Z. Then, there exists a decomposition C((yp−1 )) ⊗C((yq−1 )) N = The following lemma is easy to see.



Nα .

(3.7)

3.3 Good Filtered Bundles of Formal Difference Modules

113

Lemma 3.2.14 For any γ ∈ Galq,p , we obtain γ • Nα = Nαγ −pω . As a result, the following holds. • Let F be the automorphism of C((yp−1 ))⊗C((yq−1)) N obtained by the multiplication −pω

of αyp

on Nα in (3.7). Then, F is equivariant with respect to the Galq,p -action 

3.2.3 Slope Decomposition of Formal Difference Modules We can understand the structure of general formal difference modules by the slope decomposition in the following theorem. Theorem 3.2.15 (see [12, 23, 76, 86]) For any difference C((yq−1 ))-module N , there exists a unique decomposition of difference C((yq−1 ))-modules N =



Sω (N )

ω∈Q

such that each Sω (N ) has pure slope ω. It is called the slope decomposition.



Lemma 3.2.11 implies the following lemma. −1 Lemma 3.2.16 Any morphism of difference

C((yq ))-modules F : N1 −→ N2 preserves the slope decompositions Ni = Sω (Ni ). 

Corollary 3.2.17 Let N be any difference C((yq−1 ))-module. For any difference C((yq−1 ))-submodule N1 ⊂ N , we have Sω (N1 ) ⊂ Sω (N ). For any difference C((yq−1 ))-quotient module N2 of N , Sω (N2 ) is a quotient of Sω (N ) 

3.3 Good Filtered Bundles of Formal Difference Modules 3.3.1 Filtered Bundles over C((yq−1 ))-Modules Let E be any C((yq−1 ))-module of finite rank. A filtered bundle over E is defined to 

 be an increasing sequence P∗ E = Pa E  a ∈ R of C[[yq−1]]-lattices of E such that (i) for any a ∈ R, there exists  > 0 such that Pa E = Pa+ E, (ii) Pa+n E = yqn Pa E for any a ∈ R and n ∈ Z. This is the formal version of the notion of filtered bundle in Sect. 2.11.1. We set  GrP a (E) := Pa (E) P0 such that ϕq,p P ∗ N ). obtain the endomorphism Res(Φ ∗ ) on Gra (ϕq,p Lemma 3.3.21 Res(Φ ∗ ) is equivariant with respect to the natural Galq,p -action ∗ on GrP a (ϕq,p N ). Proof It follows from Lemma 3.2.4 and Lemma 3.2.14. Res(Φ ∗ )



is compatible with the canonical decomposiBy Lemma 3.3.21, P ∗ tion (3.9). By using the identification GrP a (N ) G0 Grpa/q (ϕq,p N ), we obtain an ∗ endomorphism on GrP a (N ), denoted by Res(Φ ). We also obtain the monodromy P weight filtration W on Gra (N ) with respect to the nilpotent part of Res(Φ ∗ ).

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3 Formal Difference Modules and Good Parabolic Structure

3.4 Geometrization of Formal Difference Modules 3.4.1 Ringed Spaces A ringed space is defined to be a topological space X equipped with a sheaf of algebras AX on X, called the structure sheaf. In this monograph, the structure sheaf of any ringed space is assumed to be commutative. A morphism of ringed spaces F : (X, AX ) −→ (Y, AY ) is a continuous map F : X −→ Y with a morphism of sheaves of algebras F −1 (AY ) −→ AX . For an AY -module F , we obtain an AX module F ∗ (F ) := AX ⊗F −1 AY F −1 (F ). If X is a subspace of Y and if F −1 (AY ) = AX , F ∗ (F ) is denoted as F|X . Even if F −1 (AY ) = AX , we shall often denote F ∗ (F ) as F|X to simplify the description if there is no risk of confusion. For an AX -module F  , we obtain an AY -module F∗ (F  ) as the standard pushforward.

3.4.2 Some Formal Spaces For any q ∈ Z≥1 , let ∞  y,q denote the ringed space which consists of the point −1 ∞y,q with the ring O∞  y,q is the completion of  y,q := C[[yq ]]. In other words, ∞ −1 the projective line P1yq along yq = ∞. We also set O∞  y,q (∗∞y,q ) := C((yq )). If p ∈ qZ≥1 , we obtain the naturally defined morphism of the ringed spaces ϕq,p : ∗ ∞  y,p −→ ∞  y,q induced by the inclusions ϕq,p : C[[yq−1]] −→ C[[yp−1]]. There ∗ also exists the natural ring homomorphism ϕq,p : O∞  y,q (∗∞) −→ O∞  y,p (∗∞). The space ∞  y,1 is also denoted by ∞  y. cov := R ×{∞ cov We set H∞,q cov (resp. OH cov (∗H∞,q )) denote the sheaf ∞,q ∞,q t y,q }. Let OH cov −1 −1 cov denote ∞,q of locally constant functions on H∞,q to C[[yq ]] (resp. C((yq ))). Let H cov the ringed space which consists of the topological space H∞,q with the sheaf of algebras OH∞,q cov . cov induced by κ (t) = Let T ∈ R>0 . Let κ denote the natural Z-action on H∞,q n ∗ ∗ t +nT (n ∈ Z). We obtain the isomorphism κn OH∞,q cov OH cov induced by κn (y) = ∞,q √ cov is equipped with a Z-action κ. ∞,q y + 2 −1nT λ. In this sense, the ringed space H cov The sheaf OH∞,q cov (∗H∞,q ) is also naturally Z-equivariant. cov by the action κ. We obtain the Let H∞,q denote the quotient space of H∞,q cov sheaves OH∞,q and OH∞,q (∗H∞,q ) as the descents of OH∞,q cov and OH cov (∗H∞,q ), ∞,q ∞,q denote the ringed space obtained as the topological space respectively. Let H H∞,q with the sheaf of algebras OH∞,q . cov −→ H cov −→ H ∞,q ∞,q and H∞,q The projections H ∞,q are denoted by q . cov cov     We also denote H∞,q , H∞,q , etc., by H∞,q,T , H∞,q,T , etc., respectively, when we emphasize the dependence on T .

3.4 Geometrization of Formal Difference Modules

121

cov are the formal completion of the spaces Mλ ∞ and H ∞ Remark 3.4.1 The space H λ λ and H λ cov . (See Sect. 2.7 for the notation.) and M along H∞  ∞

The ringed space R with the sheaf of C-valued locally constant functions is cov −→ R induces ∞,q also denoted by R. We set ST1 := R/T Z. The projection H ∞,q −→ S 1 . It also induces an isomorphism H∞,q S 1 . For a morphism πq : H T T 1  t ∈ ST , let H∞,q t denote the formal space obtained as the point set {t} ⊂ H∞,q ST1 with the ring OH∞,q ,t . For any OH∞,q (∗H∞,q )-module E, let E|H∞,q t  denote ∞,q . Note that if we ∞,q t −→ H the pull back of E by the natural morphism H 1 choose  t ∈ R which is mapped to t ∈ ST , there exists the induced isomorphism ∞,q t. Hence, we may regard E|H t  as C((yq−1 ))-module in a { t} × ∞  y,q H ∞,q way depending on the lift  t. cov cov (∗H∞,q )-module of finite rank. For any t ∈ R, let Let E cov be a locally free OH∞,q cov . For any t , t ∈ R, ∞,q denote the pull back of E cov by {t} × ∞  y,q −→ H E cov 1 2 |{t }×∞  y,q

cov cov there exists the natural isomorphism E|{t  y,q −→ E|{t2 }×∞  y,q . 1 }×∞

3.4.3 Difference Modules and OH∞,q (∗H∞,q )-Modules Let E be a locally free OH∞,q (∗H∞,q )-module of finite rank. We obtain E cov := cov . There exists the isomorphism ∞,q q−1 E on H cov cov ΠT ,0 : E|{0}× ∞  y,q −→ E|{T }×∞  y,q

induced by the parallel transport. There also exists the natural identification cov cov E|{T ∞,q 0 = E|{0}×∞ }×∞  y,q = E|H  y,q .

We define the difference operator Φ ∗ : E|H∞,q 0 −→ E|H∞,q 0 as the composite of the following morphisms: ΠT ,0

cov cov E|H∞,q 0 = E|{0}× ∞,q 0 . ∞  y,q −→ E|{T }×∞  y,q = E|H

cov cov  ,E It is also equivalent to the following constructions. Let H 0 H∞,q denote the

cov cov  cov cov ∗ 0 as the on H∞,q . We define Φ on H H∞,q , E space of the sections of E composite of the following maps:

cov cov  κ1∗

cov ∗ cov 

cov cov  H 0 H∞,q ,E −→ H 0 H∞,q , κ1 E ,E = H 0 H∞,q .

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3 Formal Difference Modules and Good Parabolic Structure

∗ cov ∗ Here, we obtain the second equality by the identifications

covκ1 Ecov  = (qcov◦ κ1 ) E = ∗ cov 0

E|{0}×∞ q E = E . Under the natural isomorphism H H∞,q , E  y,q = E|H∞,q 0 , the two constructions are the same. Note that the constructions are compatible with direct sums, tensor products and inner homomorphisms.

Proposition 3.4.2 The above construction induces an equivalence √ of the categories of locally free OH∞,q (∗H∞,q )-modules of finite rank and (2 −1λT )-difference C((yq−1 ))-modules.

√ Proof Let (N , Φ ∗ ) be a (2 −1λT )-difference C((yq−1 ))-module. Let p2 : −1 cov → ∞ cov ∞,q  y,q denote the projection. We obtain the OH∞,q cov (∗H∞,q )-module p H 2 N

 which is naturally Z-equivariant by the action κn∗ (p2−1 (s)) = p2−1 (Φ ∗ )n (s) . We obtain an OH∞,q (∗H∞,q )-module as the descent. This is a converse construction of the previous one.  The properties for difference C((yq−1 ))-modules are translated to the properties of OH∞,q (∗H∞,q )-modules. Definition 3.4.3 Let E be a locally free OH∞,q (∗H∞,q )-module. • The level of E is 0 (resp. less than 1) if the level of the difference C((yq−1 ))-module E|H∞,q 0 is 0 (resp. less than 1). (See Sect. 3.2.1.)

• E has pure slope ω ∈ Q if the difference C((yq−1 ))-module E|H∞,q 0 has pure slope ω. (See Sect. 3.2.2) • E is unramified if the difference C((yq−1 ))-module E|H∞,q 0 is unramified. (See Definition 3.3.13.) 

Any OH∞,q (∗H∞,q )-module E of finite rank has the unique slope decomposition E=



Sω E,

(3.15)

ω∈Q

where each Sω E has pure slope ω.

3.4.4 Lattices and the Induced Local Systems Let E be a locally free OH∞,q (∗H∞,q )-module of finite rank. A lattice of E means a locally free OH∞,q -submodule E0 ⊂ E such that OH∞,q (∗H∞,q )⊗E0 = E. Similarly, cov cov of finite rank, a lattice of E cov means a for a locally free OH∞,q cov (∗H∞,q )-module E cov cov cov cov cov . locally free OH∞,q cov -submodule E such that OH∞,q cov (∗H∞,q )⊗E 0 ⊂E 0 =E cov := Let E be a locally free OH∞,q (∗H∞,q )-module of finite rank. We set E q−1 (E). Any lattice E0 of E induces a lattice E0cov = q−1 (E0 ) of E cov . For any

3.5 Filtered Bundles in the Formal Case

123

cov . It induces k ∈ Z>0 , we obtain a Z-equivariant local system E0cov /yq−k E0cov on H∞,q a local system Lock (E0 ) on H∞,q . The following lemma is obvious.

Lemma 3.4.4 Let E be an OH∞,q (∗H∞,q )-module. • There exists a lattice E0 of E if and only if E has pure slope 0. • The level of E is less than 1 if and only if there exists a lattice E0 of E such that the monodromy of Loc1 (E0 ) is unipotent. • The level of E is 0 if and only if there exists a lattice E0 of E such that the monodromy of Locq (E0 ) is the identity. 

3.5 Filtered Bundles in the Formal Case 3.5.1 Pull Back and Descent of OH∞,p (∗H∞,p )-Modules ∞,p −→ For any p ∈ qZ≥1 , there exists the naturally induced morphism Rq,p : H ∞,q , whose underlying map Rq,p : H∞,p −→ H∞,q is the identity, and H the morphism of sheaves R∗q,p OH∞,q −→ OH∞,p is induced by the extension

C[[yq−1 ]] −→ C[[yp−1 ]]. The Galq,p -action on C[[yp−1 ]] induces a Galq,p -action on ∞,p , which we may regard as the Galois groups action for the ramified covering H ∞,p −→ H ∞,q . H For any OH∞,q -module E, we obtain the R∗q,p E as the pull back as in Sect. 3.4.1. If E is an OH∞,q (∗H∞,q )-module, then R∗q,p E is naturally an OH∞,p (∗H∞,p )module. Let E1 be an OH∞,p -module. We obtain the OH∞,q -module Rq,p∗ E1 obtained as the push-forward. If E1 is an OH∞,p (∗H∞,p )-module, then Rq,p∗ E1 is also an OH∞,q (∗H∞,q )-module. If E1 is a Galq,p -equivariant OH∞,p -module, Rq,p∗ E1 is equipped with the induced Galq,p -action. The invariant part of Rq,p∗ E1 is called the descent of E1 with respect to the Galq,p -action.

3.5.2 Filtered Bundles We consider the family version of filtered bundles in Sect. 3.3.1 in a straightforward way. Definition 3.5.1 A filtered bundle over a locally free OH∞,q (∗H∞,q )-module E of

finite rank is defined to be a family of filtered bundles P∗ (E|H∞,q t  ) (t ∈ ST1 ) over E|H∞,q t  . The family is also denoted as P∗ E, for simplicity. We also say that P∗ E is ∞,q , H∞,q ).  a filtered bundle on (H

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Remark 3.5.2 In Definition 3.5.1, we do not impose any condition between the filtrations P∗ (E|H∞,q t1  ) and P∗ (E|H∞,q t2  ) under the natural isomorphisms E|H∞,q t1  E|H∞,q t2  . We shall explain what condition should be imposed in Sects. 3.5.3–3.5.5, and we shall eventually introduce the goodness condition in Definition 3.5.15. 

3.5.2.1 Subbundles and Quotient Bundles ∞,q , H∞,q ). For any locally free O  (∗H∞,q )Let P∗ E be a filtered bundle on (H H∞,q  submodule E ⊂ E, we obtain the filtered bundle P∗ E  over E  by setting Pa (E|H t  ) := Pa (E|H∞,q t  ) ∩ E|H t  . We also obtain the filtered bundle over ∞,q

E  = E/E  by setting Pa (E|H

∞,q

∞,q

) as the image of Pa (E|H∞,q t  ) −→ E|H t 

∞,q t 

.

3.5.2.2 Basic Functoriality Let Ei (i = 1, 2) be locally free OH∞,q (∗H∞,q )-modules. Let  

P∗ (Ei ) = P∗ (Ei|H∞,q t  )  t ∈ ST1  be filtered bundles over Ei . The induced filtered bundles P∗ (E1|H∞,q t  ) ⊕    P∗ (E2|H∞,q t  ) t ∈ ST1 over E1 ⊕ E2 is denoted by P∗ (E1 ⊕ E2 ). The filtered bundle     P∗ (E2|H∞,q t  ) ⊗ P∗ (E2|H∞,q t  )  t ∈ ST1 over E1 ⊗ E2 is denoted by P∗ (E1 ⊗ E2 ). Let Hom(E1 , E2 ) denote the sheaf of OH∞,q (∗H∞,q )-homomorphisms from E1 to E2 . Similarly, we obtain a naturally defined filtered bundle overHom(E1 , E2 ), which

is denoted by Hom(P∗ E1 , P∗ E2 ). Let P∗ OH∞,q (∗H∞,q ) denote the filtered

 bundle over OH∞,q (∗H∞,q ) defined by Pa OH∞,q (∗H∞,q ) = yq[a] OH∞,q . We

 define P∗ E ∨ = Hom P∗ E, P∗ (OH∞,q (∗H∞,q )) . 3.5.2.3 Pull Back Let P∗ E be a filtered bundle over a locally free OH∞,q (∗H∞,q )-module E. Because

 y,p −→ ∞  y,q , R∗q,p (E)|H∞,p t  (t ∈ ST1 ) are the pull back of E|H∞,q t  by ϕq,p : ∞

∗ 

 ∗ we obtain the filtered bundles P∗ Rq,p (E)|H∞,p t  = ϕq,p P∗ (E|H∞,q t  ) as in 

 Sect. 3.3.1. The tuple P∗ (R∗q,p (E)|H∞,p t  )  t ∈ ST1 is denoted by R∗q,p (P∗ E), and called the pull back of P∗ E. We obtain the following lemma from Lemma 3.3.6.

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125

Lemma 3.5.3 Let E and P∗ E be as above. Let E  be a free OH∞,q (∗H∞,q )submodule of E. We put E  = E/E  . They are equipped with the induced filtered bundle P∗ E  and P∗ E  , respectively. Then, R∗q,p (P∗ E  ) (resp. R∗q,p (P∗ E  )) is equal to the filtered bundle over R∗q,p E  (resp. R∗q,p E  ) induced by R∗q,p (P∗ E) and the inclusion R∗q,p E  ⊂ R∗q,p E (the projection R∗q,p E −→ R∗q,p E  ).  We obtain the following lemma from Lemma 3.3.7. ∞,q , H∞,q ). Then, there Lemma 3.5.4 Let P∗ Ei be filtered bundles over (H exist natural isomorphisms R∗q,p (P∗ E1 ⊕ P∗ E2 ) R∗q,p (P∗ E1 ) ⊕ R∗q,p (P∗ E2 ), 

R∗q,p (P∗ E1 ⊗ P∗ E2 ) R∗q,p (P∗ E1 ) ⊗ R∗q,p (P∗ E2 ), and R∗q,p Hom P∗ E1 , P∗ E2

∗  Hom Rq,p (P∗ E1 ), R∗q,p (P∗ E2 ) .  3.5.2.4 Push-Forward Let P∗ E1 be a filtered bundle over a locally free OH∞,p (∗H∞,p )-module E1 . Because Rq,p∗ (E1 )|H∞,q t  = ϕq,p∗ (E1|H∞,p t  ), we obtain the induced filtered



 bundle P∗ Rq,p∗ (E1 )|H∞,q t  = ϕq,p∗ P∗ (E1|H∞,p t  ) . The tuple

  P∗ (Rq,p∗ (E1 )|H∞,q t  )  t ∈ ST1

is denoted by Rq,p∗ (P∗ E1 ), and called the push-forward of P∗ E1 . We obtain the following lemma from Lemma 3.3.8. Lemma 3.5.5 Let E1 and P∗ E1 be as above. Let E1 ⊂ E1 be a locally free OH∞,p (∗H∞,p )-submodule. We set E1 = E1 /E1 . They are equipped with the induced filtered bundles P∗ (E1 ) and P∗ (E1 ), respectively. Then, Rq,p∗ (P∗ E1 ) (resp. Rq,p∗ (P∗ E1 )) is equal to the filtered bundle induced by Rq,p∗ (P∗ E1 ) and the inclusion Rq,p∗ (E1 ) ⊂ Rq,p∗ (E1 ) (the projection Rq,p∗ (E1 ) −→ Rq,p∗ (E1 )).  ∞,q , H∞,q ). Let P∗ E1 be a filtered bundle on Let P∗ E be a filtered bundle on (H  (H∞,p , H∞,p ). We obtain the following lemma from Lemma 3.3.9. Lemma 3.5.6 The natural isomorphism Rq,p∗ (E1 ⊗ R∗q,p E) Rq,p∗ (E1 ) ⊗ E induces Rq,p∗ (P∗ E1 ⊗ R∗q,p P∗ E) Rq,p∗ (P∗ E1 ) ⊗ P∗ E. The natural isomorphism Rq,p∗ (Hom(R∗q,p E, E1 )) Hom(E, Rq,p∗ E1 ) induces Rq,p∗ (Hom(R∗q,p P∗ E, P∗ E1 )) Hom(P∗ E, Rq,p∗ P∗ E1 ). 

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3.5.2.5 Descent If P∗ E1 is Galq,p -equivariant, then we obtain the induced filtered

bundle P∗ E2 over the descent E2 of E1 by taking the Galq,p -invariant part of P∗ Rq,p∗ (E1 )|Hq,∞ t  (t ∈ ST1 ) as in Sect. 3.3.1. We obtain the following lemma from Lemma 3.3.10. Lemma 3.5.7 R∗q,p (P∗ E2 ) is naturally isomorphic to P∗ E1 .



We obtain the following lemma from Lemma 3.3.11. Lemma 3.5.8 Let E1 be a Galq,p -equivariant locally free OH∞,p (∗H∞,p )submodule of E1 . We set E1 := E1 /E1 . We obtain the locally free OH∞,q (∗H∞,q )submodule E2 ⊂ E2 as the descent of E1 . We also obtain the locally free OH∞,q (∗H∞,q )-quotient module E2 −→ E2 as the descent of E1 . Then, the decent P∗ E2 of P∗ E1 is equal to the filtered bundle over E2 induced by P∗ E2 with the inclusion E2 ⊂ E2 . Similarly, the decent P∗ E2 of P∗ E1 is equal to the filtered bundle over E2 induced by P∗ E2 with the projection E2 −→ E2 .  We obtain the following lemma from Lemma 3.3.12. ∞,q , H∞,q ), R∗q,p (P∗ E) is naturally Lemma 3.5.9 For a filtered bundle P∗ E on (H Galq,p -equivariant, and P∗ E is isomorphic to the descent of R∗q,p P∗ E. 

3.5.3 Basic Filtered Objects with Pure Slope Let q ∈ Z>0 ,  ∈ Z and α ∈ C∗ . We consider the Z-action on  Lλq cov (, α) := cov OH∞,q cov (∗H∞,q ) eq,,α given by κ1∗ (eq,,α )

  √ √ G(2 −1T λy −1 ) eq,,α . = α (1 + |λ|2 )/q yq− (1 + 2 −1λT y −1 )−/q exp q (3.16)

Here, G(x) = 1 − x −1 log(1 + x). We obtain the induced OH∞,q (∗H∞,q )-module, which is denoted by  Lλq (, α). It has pure slope /q. t denote the restriction of eq,,α to {t} × ∞  y,q . We define the Let a ∈ R. Let eq,,α filtrations   Lλq cov(, α)|{t }×∞ P∗(a)   y,q

by setting t degP (eq,,α )=a−

t . T

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127

Because the filtrations are preserved by the Z-action, we obtain a tuple of the  

Lλq (, α)|H∞,q t  )  t ∈ ST1 . filtrations P∗(a) Lλq (, α) = P∗(a)( Remark 3.5.10 The filtered bundles in this section naturally appear in the context of periodic monopoles. See Sect. 5.1. 

3.5.4 Good Filtered Bundles over OH∞,q (∗H∞,q )-Modules with Level ≤1 Let E be a locally free OH∞,q (∗H∞,q )-module of finite rank with level ≤ 1. (See Definition 3.4.3.) Note that any Φ ∗ -invariant lattice L of the difference C((yq−1 ))module N := E|H∞,q 0 induces a lattice Υ (L) of E. In particular, it induces lattices Υ (L)|H∞,q t  of E|H∞,q t  (t ∈ ST1 ).

Definition 3.5.11 Suppose that the level of E is less than 1. A filtered bundle P∗ (E|H∞,q t  | t ∈ ST1 ) is called good if there exists a filtered bundle P∗ (N ) over N which is good with respect to Φ ∗ such that P∗ (E|H∞,q t  ) = Υ (P∗ N )|H∞,q t  .  Lemma 3.5.12 Let P∗ E be a good filtered bundle over E. Let E  be a locally free OH∞,q (∗H∞,q )-submodule of E. We set E  = E/E  . They are equipped with the induced filtered bundles P∗ E  and P∗ E  , respectively. Then, P∗ E  and P∗ E  are also good in the sense of Definition 3.5.11. Proof We explain the proof for E  . The other case can be argued similarly. Note that the level of E  is also less than 1, which follows from Lemma 3.2.13 and Lemma 3.2.16. We set N  = E|H 0 , which is a C((yq−1 ))-difference submodule of ∞,q

N , which is equipped with the induced filtration P∗ N  . By Lemma 3.3.19, P∗ N  is good. It is easy to see that P∗ (E|H t  ) = Υ (P∗ N )|H∞,q t  . Hence, P∗ E  is good.  ∞,q

If moreover E is unramified, there exists a decomposition E=



Eb

(3.17)

b∈S(q)

such that for a lattice Eb,0 of Eb , the monodromy of Locq (Eb,0 ) is equal to the multiplication of 1 + b. The following lemma is easy to see. Lemma 3.5.13 Suppose that the level of E is less  than 1 and that E is unramified. Then, a filtered bundle P∗ (E) = P∗ (E|H∞,q t  )  t ∈ ST1 is good if and only if the following condition is satisfied. • For each a ∈ R, there exists a lattice Pa (E) of E such that Pa (E|H∞,q t  ) = Pa (E)|H∞,q t  .

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3 Formal Difference Modules and Good Parabolic Structure

• The lattice Pa (E) is compatible with the

decomposition (3.17), i.e., by setting Pa (Eb ) = Pa E ∩ Eb , we obtain Pa (E) = Pa Eb . • The monodromy of Locq (Pa Eb ) are equal to the multiplication of 1 + b. 

3.5.5 Good Filtered Bundles over OH∞,q (∗H∞,q )-Modules Let E be a locally free OH∞,q (∗H∞,q )-module of finite rank. Definition 3.5.14 We say that E is unramified modulo level ≤ 1 if there exists a decomposition E=



E,α

(3.18)

∈Z α∈C∗

such that each E,α is the tensor product of  Lλq (, α) and a locally free OH∞,q (∗H∞,q )-module of level ≤ 1.  Note that the decomposition (3.18) is uniquely determined. Definition 3.5.15 If E is unramified modulo level ≤1, a filtered bundle P∗ E is called good if the following conditions are satisfied.

• P∗ E is compatible with the decomposition (3.18), i.e., P∗ (E) = ,α P∗ (E,α ).

λ  Lλq (, α)−1 ⊗E,α • There exists a good filtered bundle P∗  Lq (, α)−1 ⊗E,α over  (Definition 3.5.11) such that

λ 

λ  P∗ (E,α|H∞,q t  ) = P∗(0)  Lq (, α)|H∞,q t  ⊗ P∗  Lq (, α)−1 ⊗ E,α |H∞,q t  .  Recall that for any locally free OH∞,q (∗H∞,q )-module E of finite rank there exists p ∈ qZ>0 such that R∗q,p E is unramified modulo level ≤ 1. (See Proposition 3.2.12 and Theorem 3.2.15.) Definition 3.5.16 For a locally free OH∞,q (∗H∞,q )-module E of finite rank, a filtered bundle P∗ E over E is good if there exist p ∈ qZ>0 such that (i) R∗q,p (E) is unramified modulo level ≤ 1, (ii) R∗q,p (P∗ E) is a good filtered bundle over R∗q,p (E) (Definition 3.5.15). 

3.5.5.1 An Equivalence For a good filtered bundle P∗ E over a locally free OH∞,q (∗H∞,q )-module E, ∞,q 0, we obtain a filtered bundle P∗ (E|H 0 ) by taking the restriction to H ∞,q

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129

over E|H∞,q 0 . By the construction, it is good for the difference C((yq−1 ))-module E|H∞,q 0 . The following proposition is obvious by the definitions. Proposition 3.5.17 By the restriction from P∗ E to P∗ (E|H∞,q 0 ), we obtain an equivalence between good filtered bundles on locally free OH∞,q (∗H∞,q )-modules  and good filtered difference C((yq−1 ))-modules. 3.5.5.2 Some Properties The following lemma is obvious. ∞,q , H∞,q ). Then, it is Lemma 3.5.18 Let P∗ E be a good filtered bundle on (H

compatible with the slope decomposition (3.15), i.e., P∗ E = P∗ Sω E.  ∞,q , H∞,q ). Let E  be a Lemma 3.5.19 Let P∗ E be a good filtered bundle on (H  locally free OH∞,q (∗H∞,q )-submodule of E. We set E = E/E  . Then, the induced filtered bundle P∗ E  (resp. P∗ E  ) over E  (resp. E  ) is good. Proof By Lemma 3.5.3 and Lemma 3.5.9, it is enough to study the case where E and E  are unramified modulo ≤ 1. Because P∗ E is compatible with the decomposition (3.18), it is enough to study the case where the level of E is less than 1, which we have already studied in Lemma 3.5.12.  Lemma 3.5.20 Let P∗ Ei (i = 1, 2) be good filtered bundles over locally free OH∞,q (∗H∞,q )-modules Ei . Then, the induced objects P∗ (E1 ) ⊕ P∗ (E2 ), P∗ (E1 ) ⊗ P∗ (E2 ), and Hom(P∗ E1 , P∗ E2 ) are also good filtered bundles. In particular, for a good filtered bundle P∗ E, its dual (P∗ E)∨ is also a good filtered bundle. Proof By Lemmas 3.5.4 and 3.5.9, it is enough to consider the case where Ei are unramified modulo level ≤1. Because the good filtered bundles P∗ Ei are compatible with the decompositions as in (3.18), we have only to consider the case where the levels of Ei are less than 1. Then, it follows from the claims in the case of C((yq−1 ))difference modules (Lemma 3.3.20). 

3.5.6 Global Lattices on the Covering Space Let P∗ E be a good filtered bundle over a locally free OH∞,q (∗H∞,q )-module

cov  We set E cov := q−1 (E). We obtain the induced filtrations P∗ E|{t (t }× ∞  y,q

cov cov H∞,q ). The slope decomposition E = Sω (E) induces a decomposition E

cov , we obtain the decomposition P (E cov Sω (E)cov . For each t ∈ H∞,q ∗ |{t }×∞  y,q )

). P∗ (Sω (E)cov |{t }×∞  y,q

E. ∈ = =

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3 Formal Difference Modules and Good Parabolic Structure

cov , the isomorphism S (E)cov Lemma 3.5.21 For any t1 , t2 ∈ H∞,q ω |{t1 }×∞  y,q cov induces the following isomorphisms for any a ∈ R: Sω (E)|{t2 }×∞  y,q cov Pa−t1 qω/T Sω (E)cov |{t1 }×∞  y,q Pa−t2 qω/T Sω (E)|{t2 }×∞  y,q .

(3.19)

Proof If E is unramified, the claim is clear by the definition of good filtrations. In general, there exists p ∈ qZ>0 such that R∗q,p (E) is unramified. We obtain the isomorphism ∗ cov Pa−t1 pω/T Sω (R∗q,p E)cov |{t1 }×∞  y,q Pa−t2 pω/T Sω (Rq,p E)|{t2 }×∞  y,q .

It is compatible with the natural Galq,p -action. By using the natural isomorphisms (3.10), we obtain (3.19).  By Lemma 3.5.21, for each a ∈ R, there uniquely exists a locally free OH∞,q cov submodule Pa (E cov ) of E cov satisfying Pa (E cov )|{t }×∞  y,q :=



cov Pa−t qω/T (Sω E|{t }×∞  y,q ).

ω∈Q

Thus, we obtain the global filtration P∗ of E cov . The following lemma is obvious by the construction. Lemma 3.5.22 Let (P∗ N , Φ ∗ ) be a good filtered difference C((yq−1 ))-module cov −→ ∞ ∞,q 0. Let p2 : H ∞,q obtained as the restriction of P∗ E to H  y,q denote −1 cov

p2 (N ), we obtain the projection. Then, under the natural isomorphism E P∗ (E cov ) = p2−1 (P∗ N ).   P We set Gra (E cov ) := Pa (E cov ) P