Test Configurations, Stabilities and Canonical Kähler Metrics: Complex Geometry by the Energy Method (SpringerBriefs in Mathematics) 9811604991, 9789811604997

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SPRINGER BRIEFS IN MATHEMATICS

Toshiki Mabuchi

Test Configurations, Stabilities and Canonical Kähler Metrics Complex Geometry by the Energy Method

SpringerBriefs in Mathematics Sereis Editors Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Palle Jorgensen, Iowa, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, Canada Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York, USA George Yin, Detroit, USA Ping Zhang, Kalamazoo, USA

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Toshiki Mabuchi

Test Configurations, Stabilities and Canonical Kähler Metrics Complex Geometry by the Energy Method

Toshiki Mabuchi Department of Mathematics Osaka University, Graduate School of Science Toyonaka, Osaka, Japan

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-981-16-0499-7 ISBN 978-981-16-0500-0 (eBook) https://doi.org/10.1007/978-981-16-0500-0 © The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dedicated to the Memory of Professor Shoshichi Kobayashi

Preface

This is a development of the lectures on complex geometry delivered at the Yau Mathematical Sciences Center, Tsinghua University during the period 2017–2020. The main purpose of this book is to discuss several topics in complex geometry related to the theory of canonical Kähler metrics for polarized algebraic manifolds. In 1977, S.-T. Yau solved the Calabi conjecture on the existence of Kähler– Einstein metrics both in the Ricci-flat case and in the Ricci-negative case, where for the Ricci-negative case, independent work was also done by T. Aubin. These allow us to obtain a number of deep consequences in algebraic geometry such as the Miyaoka–Yau inequality and a characterization of the quotients of the 2dimensional complex unit ball. For the Ricci-positive case, S.-T. Yau proposed that the existence of Kähler– Einstein metrics on Fano manifolds has some link to the stability in algebraic geometry. It is well-known that G. Tian and S. K. Donaldson introduced the concept of K-stability and K-polystability by slightly different formulations. Now the Yau– Tian–Donaldson Conjecture states that: A Fano manifold X admits a K ahler–Einstein ¨ metric if and only if (X, KX−1 ) is K–polystable. Recently, this conjecture was solved affirmatively by X. Chen, S. K. Donaldson, and S. Sun, and independently also by G. Tian. However, in constant scalar curvature Kähler metric cases, this conjecture is still open for general polarizations, or more generally in extremal Kähler cases. In this book, these unsolved cases of this conjecture will also be discussed. For a Fano manifold, a Kähler–Einstein metric is a typical canonical Kähler metric which plays a very important role in complex geometry. However, such a metric does not necessarily exist on the manifold. In this book, a canonical Kähler metric on a general Fano manifold will also be discussed. Especially, if the Futaki character of a Fano manifold is non-vanishing, the following candidates for canonical Kähler metrics are known: (i) Kähler–Ricci solitons, (ii) extremal vii

viii

Preface

Kähler metrics, (iii) generalized Kähler–Einstein metrics, where the last one is often referred to as M-solitons. In the last chapter, we explain how these three types of metrics differ. Since (i) and (ii) are well-known, we focus on the study of the last one. Finally, as an appendix, we include a lecture given at Tsinghua University in 2018 in which we discuss how the geometry of Lp -spaces allows us to obtain a natural compactification of the moduli space of graded algebras (such as canonical rings) of a certain type. In contrast to the GIT-limits in algebraic geometry (or to the Gromov–Hausdorff limit in Riemannian geometry), we have some straightforward compactification of the moduli space. Toyonaka, Japan January 2020

Toshiki Mabuchi

Contents

1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 The Deligne Pairings with Metrics . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Definition of the Chow Norm .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 The First and Second Variation Formulas for the Chow Norm.. . . . . . Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 3 5 6 8

2 The Donaldson–Futaki Invariant . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Test Configurations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Test Configurations Associated to One-Parameter Groups .. . . . . . . . . . 2.3 Definition of the Donaldson–Futaki Invariant.. . . .. . . . . . . . . . . . . . . . . . . . 2.4 Expression of DF1 as an Intersection Number .. . .. . . . . . . . . . . . . . . . . . . . 2.5 The Relationship Between the Chow Norm and DFi . . . . . . . . . . . . . . . . . Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

9 9 11 13 14 16 20

3 Canonical Kähler Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Canonical Kähler Metrics on Compact Complex Manifolds .. . . . . . . . 3.2 Conformal Changes of Metrics by Hamiltonian Functions.. . . . . . . . . . Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

21 21 22 23

4 Norms for Test Configurations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Norms for Test Configurations of a Fixed Exponent . . . . . . . . . . . . . . . . . 4.2 The Asymptotic 1 -norm of a Test Configuration . . . . . . . . . . . . . . . . . . . . 4.3 Relative Norms for Test Configurations .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 The Twisted Kodaira Embedding .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 The Donaldson–Futaki Invariant for Sequences . .. . . . . . . . . . . . . . . . . . . . Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25 25 26 27 29 30 33

5 Stabilities for Polarized Algebraic Manifolds . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 The Chow Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 The Hilbert Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

35 35 41

ix

x

Contents

5.3 K-stability .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Relative Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

44 45 52

6 The Yau–Tian–Donaldson Conjecture.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 The Calabi Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 The Yau–Tian–Donaldson Conjecture.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 The K-Energy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Extremal Kähler Versions of the Conjecture . . . . . .. . . . . . . . . . . . . . . . . . . . Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

53 53 54 54 56 57

7 Stability Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Strong K-Semistability of CSC Kähler Manifolds .. . . . . . . . . . . . . . . . . . . 7.2 Relative Balanced Metrics . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Strong Relative K-Semistability of Extremal Kähler Manifolds .. . . . 7.4 K-Polystability of Extremal Kähler Manifolds . . .. . . . . . . . . . . . . . . . . . . . 7.5 A Reformulation of the Definition of the Invariant F ({μj }) .. . . . . . . . Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

59 59 61 62 64 64 68

8 Existence Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 A Result of He on the Existence of Extremal Kähler Metrics.. . . . . . . 8.2 Some Observations on the Existence Problem .. . .. . . . . . . . . . . . . . . . . . . . Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

69 69 70 73

9 Canonical Kähler Metrics on Fano Manifolds . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Kähler Metrics in Anticanonical Class . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Extremal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 An Obstruction of Matsushima’s Type . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 An Invariant as an Obstruction to the Existence . .. . . . . . . . . . . . . . . . . . . . 9.5 Examples of Generalized Kähler–Einstein Metrics .. . . . . . . . . . . . . . . . . . 9.6 Extremal Metrics on Generalized Kähler–Einstein Manifolds .. . . . . . 9.7 The Product Formula for the Invariant γX . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8 Yao’s Result for Toric Fano Manifolds .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.9 Hisamoto’s Result on the Existence Problem . . . . .. . . . . . . . . . . . . . . . . . . . Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

75 75 76 78 80 81 88 92 94 97 98

A Geometry of Pseudo-Normed Graded Algebras .. . . . .. . . . . . . . . . . . . . . . . . . . A.1 Differential Geometric Viewpoints . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2 Lp -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.3 An Orthogonal Direct Sum of Lp -Spaces . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.4 A Boundedness Theorem for Lp -Spaces . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.5 The Moduli Space of Lp -Spaces . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.6 A Multiplicative System of Lp -Spaces .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.7 Degeneration Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

99 99 99 100 103 109 111 114

Solutions. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 117 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125

Chapter 1

Introduction

Abstract In this chapter, by fixing notation, we introduce various basic concepts to discuss background materials. • In Sect. 1.1, we introduce several basic concepts such as Kähler classes, polarized algebraic manifolds, CSC Kähler metrics, extremal Kähler metrics and special one-parameter groups. • In Sect. 1.2, we discuss Deligne pairings with metrics. • In Sect. 1.3, we give a precise definition of the Chow norm introduced by S. Zhang which plays a very important role in our study of canonical metrics. • In Sect. 1.4, we give the first and second variation formulas for the Chow norm, and in particular we can show the convexity of the Chow norm along special one-parameter groups. Keywords Polarized algebraic manifolds · Deligne pairings · The Chow norm

1.1 Preliminaries For a compact connected complex manifold X, we call K a Kähler class on X if K is the set of all Kähler forms ω in a fixed Dolbeault cohomology class on X. Here a smooth d-closed (1, 1)-form ω on X is called Kähler if ω is positive definite everywhere on X. By choosing a system (z1 , z2 , . . . , zn ) of holomorphic local coordinates on X, we can write a Kähler form ω on X as ω =

√ −1  gα β¯ dzα ∧ dzβ¯ , 2π

(1.1)

α,β

where dω = 0. The Ricci form Ric(ω) for ω is √ n −1  Ric(ω) := Rα β¯ dzα ∧ dzβ¯ = − dd c log ωn , 2π

(1.2)

α,β=1

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0_1

1

2

1 Introduction

where 2πdd c :=

√ ¯ The scalar curvature Sω for ω is −1∂ ∂. n 

Sω := Trω Ric(ω) =

¯

g βα Rα β¯ .

α,β=1 ¯

Here (g βα ) is the inverse matrix of (gα β¯ ). Put Zω := gradC ω Sω , where for each complex-valued smooth function ψ on X, we define a complex vector field 1  βα ¯ ∂ψ ∂ gradC g ω ψ = √ ∂zβ¯ ∂zα −1 α,β

(1.3)

of type (1, 0) on X. Then ω is called CSC Kähler if Sω is a constant function. More generally, if the vector field Zω is holomorphic, then ω is called extremal Kähler (cf. [8, 9]) and in such a case, Zω is called the extremal vector field for (X, ω). Definition 1.1 (X, L) is called a polarized algebraic manifold if L is an ample line bundle on a compact complex connected manifold X. However, unless otherwise stated, L is always assumed to be very ample (with the only exception for L = KX−1 ). For a Hermitian vector space V of complex dimension N, let σ : C∗ → GL(V ) be an algebraic group homomorphism. Then for a suitable choice of an orthonormal basis for V , we can diagonalize σ in the form ⎛ α1 t ⎜ t α2 σ (t) = ⎜ ⎝ ...

0

⎞

0 ⎟⎟

⎠,

t ∈ C∗ ,

t αN

where the integers α1 , α2 , . . . , αN independent of t are called the weights of the C∗ -action on V via σ . Let F be the finite cyclic subgroup of SL(V ) defined by F : = { ζ idV ; ζ ∈ C is an N-th root of unity }. By setting α0 := (α1 + α2 + · · · + αN )/N, we have rational numbers α¯ i := αi − α0 . Then by setting σ SL (t) := {det σ (t)}−1/N σ (t), we have an algebraic group homomorphism σ SL : C∗ → SL(V )/F , called the special linearization of σ , written as ⎛ α¯ 1 t ⎜ t α¯ 2 σ SL (t) = ⎜ ⎝ ...

0

⎞

0 ⎟⎟

⎠,

t α¯ N

t ∈ C∗ ,

1.2 The Deligne Pairings with Metrics

3

modulo F . Let R+ be the multiplicative group of positive real numbers. Then σ SL , when restricted to R+ , defines a homomorphism: R+ → SL(V ) such that the eigenvalues of σ SL (t), t ∈ R+ , are all positive real numbers. Then α¯ 1 , α¯ 2 , · · · , α¯ N are called the weights of the R+ -action on V via σ SL . Moreover for σ and σ SL , ⎛ ⎜ ⎜ ⎝

α1

0

α2 ...

⎞

⎛

⎠,

⎜ ⎜ ⎝

0 ⎟⎟ αN

⎞

α¯ 1

0

α¯ 2 ...

0 ⎟⎟ ⎠

α¯ N

in sl(V ) are called the fundamental generators of σ , σ SL , respectively. Let A1 := { z ∈ C } be a complex affine line on which C∗ acts by multiplication of complex numbers. Let π : E → A1 be an algebraic vector bundle over A1 such that C∗ acts on E covering the C∗ -action on A1 and inducing linear maps between fibers of E. Let ρ be a Hermitian metric for the fiber E1 over 1 ∈ A1 . Since the origin in A1 is fixed by the C∗ -action, the fiber E0 over the origin is preserved by the C∗ -action, so that the C∗ -action on E0 is given by a representation σ : C∗ → GL(E0 ). Then the affirmative solution [56] of the equivariant Serre conjecture for abelian groups allows us to trivialize E equivariantly in the form E ∼ = E 0 × A1 , where C∗ acts on E0 × A1 by C∗ × (E0 × A1 ) → E0 × A1 ,

(t, (e, z)) → (σ (t)e, tz).

Then by Lemma 2 in Donaldson [21], a trivialization as above can be chosen isometrically in such a way that the Hermitian metric ρ on E1 induces a Hermitian metric for the fiber E0 on which S 1 in C∗ acts isometrically.

1.2 The Deligne Pairings with Metrics Let π : Y → T be a flat projective morphism of irreducible complex varieties of relative dimension n := dim Y − dim T ≥ 0 such that T is smooth. Here a complex variety means a reduced (possibly reducible) algebraic variety defined over C. Let Li = OY (Di ),

i = 0, 1, . . . , n,

4

1 Introduction

be line bundles over Y , where Di is a Cartier divisor on Y . Then the intersection δ := D0 · D1 · · · Dn is a q-dimensional algebraic cycle on Y with q = dim Y − (n + 1) = dim T − 1. Since T is smooth, the pushforward π∗ δ defines a line bundle on T ,

L0 , L1 , . . . , Ln Y/T , called the Deligne pairing of L0 , L1 , . . . , Ln . Let i , i = 0, 1, . . . , n, be local (with respect to T ) sections for Li such that the corresponding divisors have an empty intersection. Then the symbol

0 , 1 , . . . , n  represents a local basis for L := L0 , L1 , . . . , Ln Y/T . For a Hermitian metric hi for the line bundle Li , we can write hi = e−φi ,

i = 0, 1, . . . , n,

by using the associated Kähler potential φi := − log hi . Then the Deligne pairing L has a Hermitian metric e−φ with a continuous Kähler potential φ = φ0 , φ1 , . . . , φn Y/T , smooth over the smooth locus of π, which is multilinear and characterized by the following properties: dd φ0 , φ1 , . . . , φn Y/T =

n

c

dd c φi ,

Y/T i=0

(1.4)

φ0 , φ1 , . . . , φn Y/T − φ0 , φ1 , . . . , φn Y/T =

Y/T

(φ0 − φ0 )

n

dd c φi ,

i=1

(1.5)  where Y/T denotes the fiber integration for the fiber space Y sitting over T . For more details, see [5, 60, 68, 93]. Remark 1.1 (cf. [68, 93]) For D0 := zero(0 ), by choosing D0 as a difference of irreducible reduced divisors, it is possible to define φ = φ0 , φ1 , . . . , φn Y/T by induction on n by the formula





φ() = φ ( ) + Y/T

φ0 (0 )

n

i=1

dd c φi ,

1.3 Definition of the Chow Norm

5

where for φ = φ0 , φ1 , . . . , φn Y/T , φ = φ1 , . . . , φn D0 /T ,  = 0 , 1 , . . . , n ,

 = 1 , . . . , n , hφ = e−φ , hφ = e−φ , h0 = e−φ0 , we put

φ() = − log hφ (, ), φ ( ) = − log hφ ( ,  ), φ0 (0 ) = − log h0 (0 , 0 ). Note also that, for n = 0, L0 Y/T is nothing but the norm of L0 with respect to the finite flat morphism π : Y → T , and in particular for  = 0 , 

{φ()}(t) =

{φ0 (0 )}(P ),

t ∈ T.

P ∈π −1 (t )

Remark 1.2 (cf. [68, 93]) For bases 0 , 1 , · · · , n ,  0 ,  1 , · · · ,  n  over an open subset of T , the transition function is defined as follows: It suffices to consider the case where, for some k, we have i =  i for all i = k. For a point t in the open set, let ⎧ ⎫ ⎨ ⎬  zero(i ) nα (t)Pα (t) Yt ∩ = ⎩ ⎭ i =k

α

be a zero cycle on Yt := π −1 (t), where for each t, nα (t) is an integer, and {Pα (t)} is a finite set of points in Yt . Then by setting f :=  k /k , we have 

 0 ,  1 , · · ·

,  n 

=



 f (Pα (t))

nα (t )

0 , 1 , · · · , n .

α

1.3 Definition of the Chow Norm For V = CN with the standard Euclidean norm ρ, we consider its dual V ∗ = CN = {z = (z1 , . . . , zN )}. Then the group SL(V ) = SL(N, C) acts on V ∗ by the contragradient representation. For y = (y1 , . . . , yN ) ∈ V and z = (z1 , . . . , zN ) ∈ V ∗ , we put |y| 2 :=

N 

|yi |2

and

|z|2 =

i=1

N  i=1

Furthermore, for y = (y (0) , y (1), · · · , y (n) ) ∈ V n+1 , we put y2 :=

n

j =0

|y (j ) |2 ,

|zi |2 .

6

1 Introduction

and let ωFS ([y (j )]) be the Fubini–Study form dd c log |y (j ) |2 for the projective space P(V ) = {[y (j )] ; 0 = y (j ) ∈ CN }, j = 0, 1, . . . , n. Put W := (Symd V ∗ )⊗n+1 for a positive integer d. Then for every w ∈ W , we define its Chow norm wCH(ρ) ≥ 0 by wCH(ρ)

⎧ ⎨

⎫   n |w(y)| N−1 (j ) ⎬ log := exp ωFS ([y ]) , ⎩ P(V )n+1 ⎭ yd j =0

where w = w(y) is viewed as a homogeneous polynomial in (y (0), y (1), · · · , y (n) ) of multi-degree (d, d, · · · , d). Obviously,  CH(ρ) defines a Finsler norm on W . For more details, see [70, 93].

1.4 The First and Second Variation Formulas for the Chow Norm Let X ⊂ P(V ∗ ) be an n-dimensional projective variety for V ∗ = {z = (z0 , . . . , zN )} as in the preceding section, and we consider the Fubini–Study form ωFS = dd c log |z|2 = dd c log

N 

|zi |2

i=0

on P(V ∗ ). Put d := degP(V ∗ ) X and W := (Symd V ∗ )⊗n+1 . Let 0 = Xˆ ∈ W be the ˆ in P(W ) is the Chow point Chow form for X, so that the corresponding point [X] ∗ for X viewed as an algebraic cycle on P(V ). Let σ : C∗ → SL(V ) be an algebraic group homomorphism, where SL(V ) acts on V ∗ by contragradient representation. (More generally, the arguments down below are valid also for a Lie group homomorphism σ from R+ to SL(V ) with rational weights −b1 , · · · , −bN .) Since OP(V ∗ ) (−1) \ {zero section} = V ∗ \ {0}, the mapping V ∗ \ {0}  z → |σ (t)z|2 ∈ R defines a Hermitian metric on the line bundle OP(V ∗ ) (−1) for each fixed t ∈ C∗ . Then by taking its dual, we can view φt = log |σ (t)z|2

(1.6)

1.4 The First and Second Variation Formulas for the Chow Norm

7

as a Kähler potential for the polarization class c1 (OP(V ∗ ) (1)) on P(V ∗ ). Note that σ (t)∗ ωFS = dd c φt . Put φ˙ t := ∂φt /∂t. Then (see for instance [70]): d ˆ CH(ρ) = n + 1 log σ (t) · X dt 2



φ˙ t (dd c φt )n ,

(1.7)

X

where SL(V ) acts on W = (Symd V ∗ )⊗n+1 induced by the contragradient representation of SL(V ) on V ∗ . For a suitable basis for V , we can diagonalize σ (t) as ⎛ −b1 t ⎜ t −b2 σ (t) = ⎜ ⎝ ...

0

⎞

0 ⎟⎟

⎠,

t ∈ C∗ ,

t −bN

where −bα ∈ Z, α = 1, 2, · · · , N, are the weights of the C∗ -action on V via σ . Then by the contragradient representation, we can write σ (t)z = (t b1 z1 , t b2 z2 , . . . , t bN zN ),

z ∈ V ∗.

ˆ CH(ρ) , where t = exp s for s ∈ R. Then by (1.6), Put f (s) := log σ (t) · X φt = log(e2sb1 |z1 |2 + · · · + e2sbN |zN |2 ). Note that ωFS = dd c log(|z1 |2 + · · · + |zN |2 ). Put f˙(s) = ∂f/∂s. Then by (1.7), we have the first variation formula for the Chow norm: f˙(0) = (n + 1)

X

b1 |z1 |2 + · · · + bN |zN |2 n ωFS . |z1 |2 + · · · + |zN |2

(1.8)

Consider the rectangle Ra,b = {ζ ∈ C ; a ≤ Re ζ ≤ b, 0 ≤ Im ζ ≤ 2π}, where real numbers a, b are such that a < b. Let η : X × Ra,b → P(V ∗ ) be the map defined by η(x, ζ ) := σ (exp ζ ) · x. Then we have the following second variation formula for the Chow norm: f˙(b) − f˙(a) =

X×Ra,b

n+1 η∗ ωFS ≥ 0.

(1.9)

In particular, f (s) is a convex function of s, i.e., f¨(s) ≥ 0 for all s, where f¨(s) := ∂ 2 f/∂s 2 . For more details, see for instance [48, 70, 93].

8

1 Introduction

Problems 1.1 Show that, if f¨(0) = 0 for f (s) above, then the one-parameter group σ : C∗ → SL(V ) preserves the subvariety X in P(V ∗ ). 1.2 For a smooth irreducible projective variety X with positive first Chern class, let ω be a CSC Kähler metric in the polarization class c1 (X). Show that Ric(ω) = ω.

Chapter 2

The Donaldson–Futaki Invariant

Abstract In the study of the Calabi conjecture for Fano manifolds, G. Tian introduced the concept of special degenerations to characterize the stability related to the existence of Kähler–Einstein metrics. Later, S.K. Donaldson reformulated this concept to obtain a purely algebraic concept of test configurations. • In Sects. 2.1 and 2.2, we define the concept of test configurations and study its elementary properties. • In Sect. 2.3, we define Donaldson–Futaki invariants DFi for test configurations. • In Sect. 2.4, the Donaldson–Futaki invariant DF1 for a test configuration is characterized as an intersection number. • In Sect. 2.5, the Chow weight (the derivative of the Chow norm at infinity) along a special one-parameter group for a test configuration is expressed in terms of the Donaldson–Futaki invariants. Keywords Test configurations · The Donaldson–Futaki invariant

2.1 Test Configurations Let (X, L) be a polarized algebraic manifold. Let C∗ act on the complex affine line A1 := {z ∈ C} by multiplication of complex numbers: C∗ × A1  (t, x) → tx ∈ A1 . A pair (X , L ) is called a test configuration ([20]; see also [79]) for (X, L) if C∗ acts on X such that there exists a C∗ -equivariant projective morphism π : X → A1 of complex varieties satisfying the following conditions: • L is a relatively very ample line bundle on the fiber space X over A1 such that the C∗ -action on X lifts to an action on L inducing linear maps between fibers. © The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0_2

9

10

2 The Donaldson–Futaki Invariant

• For some positive integer γ , there exists an isomorphism (X1 , L1 ) ∼ = (X, L⊗γ ), 1 where (Xz , Lz ) denotes the fiber of (X , L ) over each z ∈ A . Here γ is called the exponent of the test configuration (X , L ) for (X, L). The action of each t ∈ C∗ on X (resp. L ) is written as g(t) ∈ Aut(X )

(resp. g(t) ˜ ∈ Aut(L )),

where Aut(X ) (resp. Aut(L )) is the set of all holomorphic automorphisms of X (resp. L ) viewed just as a complex variety. We now put XA1 \{0} := X × (A1 \ {0}). Then by identifying X1 with X, we have the C∗ -equivariant isomorphism XA1 \{0} = X1 × (A1 \ {0}) ∼ = X \ X0 (x, t) ←→

(2.1)

g(t) · x,

where C∗ acts on XA1 \{0} = X × (A1 \ {0}) by multiplication of complex numbers just on the second factor (A1 \ {0}). Now by attaching an end X × {∞} to XA1 \{0} , we have a C∗ -equivariant closure XP1 \{0} := X × (P1 \ {0}) of XA1 \{0} . Then by gluing X and XP1 \{0} together, we have the C∗ -equivariant compactification X¯ := XP1 \{0} ∪ X of X via the identification (2.1) such that the C∗ -action fixes the infinity fiber X¯∞ . Similarly, via the C∗ -equivariant identification L \ L0

⊗γ ∼ = LA1 \{0} := L1 × (A1 \ {0})

g(t) ˜ ·  ←→ (, t), ⊗γ

we can glue L and LP1 \{0} := L1 × (P1 \ {0}) together to obtain ⊗γ L¯ := LP1 \{0} ∪ L .

on which the group C∗ acts fixing the infinity fiber L¯∞ . Obviously, L¯ is relatively very ample on the fiber space X¯ over P1 . Thus we obtain the compactified family (X¯ , L¯ ) =



(X¯z , L¯z ),

z∈P1 ⊗γ where the restriction of L¯ over P1 \ {0} is trivialized as the bundle LP1 \{0} .

• A test configuration (X , L ) for (X, L) is called a product configuration if X over A1 is isomorphic to the product X × A1 just as a complex variety.

2.2 Test Configurations Associated to One-Parameter Groups

11

• A test configuration (X , L ) for (X, L) is called trivial if (X , L ) is a product configuration and furthermore C∗ acts trivially on the first factor of X ∼ = X×A1 . A test configuration

z

1

0

=X

π

z

0

1

A1

2.2 Test Configurations Associated to One-Parameter Groups For a polarized algebraic manifold (X, L), we put Vγ := H 0 (X, L⊗γ ) for each positive integer γ . Fix a Hermitian metric h for L such that ω := Ric(h) = −dd c log h is Kähler. Let ργ be the Hermitian structure for Vγ induced by h and ω such that

v1 , v2 ργ :=

(v1 , v2 )h ωn ,

v1 , v2 ∈ Vγ ,

(2.2)

X

where ( , )h is the pointwise Hermitian pairing for L by the Hermitian metric h. Let S 1 be the maximal compact subgroup of C∗ . An algebraic group homomorphism σ : C∗ → GL(Vγ )

12

2 The Donaldson–Futaki Invariant

is called a special one-parameter group if σ (S 1 ) acts isometrically on (Vγ , ργ ). In view of Sect. 1.1, we may choose another approach by assuming that σ is an algebraic group homomorphism: σ : C∗ → SL(Vγ )/F, where F is the finite cyclic subgroup of SL(Vγ ) as in Sect. 1.1. Then by taking ˜ ∗ → SL(Vγ ) of ˜ ∗ of C∗ , we have a lift σ˜ : C a finite cyclic unramified cover C ∗ ˜ σ , where C is also an algebraic torus. In this approach also, σ is called a special one-parameter group if σ˜ (S 1 ) acts isometrically on (Vγ , ργ ). For both of these approaches, we can consider a C∗ -invariant closed subset Xσ of A1 × P(Vγ∗ ) obtained as the closure of the set 

{t} × σ (t)X,

t ∈C∗

where X is viewed as a subvariety of P(Vγ∗ ) by the Kodaira embedding associated to the complete linear system |L⊗γ | on X, and C∗ acts on A1 × P(Vγ∗ ) by C∗ × A1 × P(Vγ∗ )  (t, (z, x)) → (tz, σ (t)x). Here the GL(Vγ )-action on P(Vγ∗ ) is induced by the contragradient representation. Let Lσ be the restriction to Xσ of the pullback pr∗2 OP(Vγ∗ ) (1) by the projection pr2 : A1 × P(Vγ∗ ) → P(Vγ∗ ). Clearly, the natural C∗ -action on Xσ lifts to a natural C∗ -action on Lσ . For the fiber over 1 ∈ A1 , we have the isomorphism ((Xσ )1 , (Lσ )1 ) ∼ = (X, L⊗γ ). • Hence (Xσ , Lσ ) is a test configuration of exponent γ for (X, L), and is called the test configuration associated to the special one-parameter group σ . • (Xσ , Lσ ) is called trivial if σ SL is a trivial homomorphism. Let (X , L ) be a test configuration, of exponent γ , for (X, L) as in Sect. 2.1. Then the direct image sheaf E := π∗ L ⊗m is viewed as a vector bundle over A1 on which C∗ acts. In view of Sect. 1.1, the vector bundle E can be trivialized C∗ -equivariantly as E ∼ = E 0 × A1 ,

2.3 Definition of the Donaldson–Futaki Invariant

13

where for each positive integer m, the algebraic torus C∗ acts on the fiber E0 of E over the origin via a representation σm : C∗ → GL(E0 ). Put γm := mγ . By the identification of E0 with E1 = H 0 (X, L⊗γm ) = Vγm , the compact subgroup S 1 in C∗ acts isometrically on Vγm with the Hermitian structure ργm . In particular, σm above is a special one-parameter group, and we can write (X , L ⊗m ) = (Xσm , Lσm ),

m = 1, 2, · · · .

2.3 Definition of the Donaldson–Futaki Invariant Let (X, L) be a polarized algebraic manifold. We then consider a test configuration (X , L ) for (X, L) of exponent γ . For a sufficiently large integer m  1, we put Nm := dim H 0 (X, L⊗mγ ) = dim H 0 (X0 , L0⊗m ), wm := weight of the C∗ -action on det H 0 (X0 , L0⊗m ). Then by the Riemann–Roch theorem and also by the equivariant Riemann–Roch theorem, we can write Nm and wm as polynomials with rational coefficients, Nm = a0 mn + a1 mn−1 + · · · + an−1 m + an , wm = b0 mn+1 + b1 mn + · · · + bn m + bn+1 , where a0 = γ n c1 (L)n [X]/n! is positive. Hence we have the asymptotic expansion b0 mn+1 + b1 mn + · · · wm = mNm a0 mn+1 + a1 mn + · · · =

∞ 

=

b0 a0 b1 − a1 b0 −1 + m + ··· a0 a02

DFi (X , L )m−i .

i=0

From this expansion, we immediately obtain DF0 (X , L ) =

b0 , a0

DF1 (X , L ) =

a0 b1 − a1 b0 . a02

(2.3)

Definition 2.1 DF1 (X , L ) is called the Donaldson–Futaki invariant (see [20]) of the test configuration (X , L ).

14

2 The Donaldson–Futaki Invariant

Remark 2.1 If we replace L by L ⊗k for some positive integer k, then the terms a0 , a1 , b0 , b1 are replaced by a0 k n , a1 k n−1 , b0 k n+1 , b1 k n , respectively. Hence by (2.3), we see that DF1 (X , L ⊗k ) = DF1 (X , L ).

2.4 Expression of DF1 as an Intersection Number For a test configuration (X , L ) of exponent γ for (X, L), we assume that X is a normal complex variety. Let us consider the relative canonical divisor KX¯ /P1 := KX¯ − π¯ ∗ KP1 , where KX¯ /P1 is a Weil divisor on X¯ , and π¯ : X¯ → P1 is the natural projection. The purpose of this section is to prove the following formula ([65, 82]; see also [4]): Theorem 2.1   1 S¯ n n+1 ¯ ¯ (KX¯ /P1 · L ) + (L ) , DF1 (X , L ) = − 2vol n+1 where vol = γ n c1 (L)n [X] and S¯ := n vol−1 c1 (X) γ n−1 c1 (L)n−1 [X], and the intersection numbers (KX¯ /P1 · L¯ n ), (L¯ n+1 ) are taken on X¯ . Proof Let m  1. Note that the direct image sheaf E (m) := π¯ ∗ L¯ ⊗m over P1 is locally free of rank Nm . We then consider the line bundle N := det E (m)

(2.4)

on P1 . By viewing N just as a complex variety, we consider the group Aut (N ) of all holomorphic automorphisms of N . Then the C∗ -action on N induced by the C∗ -action on L is expressible as C∗ × N  (t, ν) → g(t)ν ∈ N , for some group homomorphism g : C∗ → Aut (N ). Actually, g(t), t ∈ C∗ , define bundle automorphisms of N covering the C∗ -action on P1 . We now choose an element 0 = ν1 ∈ N1 sitting over 1 ∈ A1 . Then σ (t) := g(t)ν1 ,

t ∈ C∗ (= A1 \ {0}),

2.4 Expression of DF1 as an Intersection Number

15

extends to a rational section of N over P1 , written also as σ = σ (z),

z ∈ P1 ,

which is holomorphic and nowhere vanishing when restricted to P1 \ {0}. Then the order μ := ordz=0 σ (z) of σ at the origin is nothing but deg N . Hence ν0 := lim z−μ σ (z) z→0

is a nonzero element of N0 . Then for every t ∈ C∗ ,   g(t)ν0 = g(t) lim z−μ σ (z) = lim z−μ {g(t)σ (z)} = lim z−μ (g(t){g(z)ν1 }) z→0

z→0

z→0

= lim z−μ g(tz) ν1 = t μ lim (tz)−μ g(tz) ν1 = t μ lim (tz)−μ σ (tz) = t μ ν0 . z→0

z→0

z→0

Hence μ is the weight wm of the C∗ -action on N0 . As in [4], it then follows that deg N

= μ = wm .

Since E (m) is a vector bundle over P1 of rank Nm , by a theorem of Birkhoff– Grothendieck, we can write E (m) as a direct sum E (m) =

Nm 

OP1 (αi )

(2.5)

i=1

for some integers αi . Then by the Riemann–Roch theorem for OP1 (αi ), we obtain χ(P1 , OP1 (αi )) = deg OP1 (αi ) + 1 − g = αi + 1, since the genus g of P1 is 0. Hence by (2.4) and (2.5), χ(P1 , E (m) ) =

Nm  i=1

χ(P1 , OP1 (αi )) = Nm +

Nm 

αi = Nm + deg N = Nm + wm .

i=1

(2.6) Since L¯ is relatively very ample, by m  1, we have R i π¯ ∗ L¯ ⊗m = 0 for i > 0. Hence χ(P1 , E (m) ) = χ(P1 , π¯ ∗ L¯ ⊗m ) = χ(X¯ , L¯ ⊗m ) =

mn mn+1 (L¯ n+1 ) − (K ¯ · L¯ n ) + O(mn−1 ), (n + 1)! 2 · n! X

(2.7)

16

2 The Donaldson–Futaki Invariant

where in the last equality, we used the Riemann–Roch theorem for normal projective ¯ varieties [4, Appendix A]. On the other hand, in view of the definition of vol and S, the Riemann–Roch theorem for the line bundle L shows that mn n mn−1 γ c1 (L)n [X] + γ n−1 c1 (X)c1 (L)n−1 [X] + O(mn−2 ) n! 2 · (n − 1)!   S¯ 1 1 mn 1 + · + O( 2 ) . = vol (2.8) n! 2 m m

Nm =

In view of (2.6) together with (2.7) and (2.8), we obtain wm = χ(P1 , E (m) ) − Nm =

mn+1 mn mn (L¯ n+1 ) − (KX¯ · L¯ n ) − 2 · γ n c1 (L)n [X] + O(mn−1 ), (n + 1)! 2 · n! 2 · n!

where by (KX¯ /P1 · L¯ n ) = (KX¯ · L¯ n ) + 2 · γ n c1 (L)n [X], we rewrite wm as wm =

mn+1 mn (L¯ n+1 ) − (K ¯ 1 · L¯ n ) + O(mn−1 ). (n + 1)! 2 · n! X /P

(2.9)

By (2.8) and (2.9), the terms ai , bi , i = 0, 1, in the formula (2.3) for DF1 (X , L ) are given by the following: a0 =

(KX¯ /P1 · L¯ n ) vol vol ·S¯ (L¯ n+1 ) , a1 = , b0 = , b1 = − . n! 2 · n! (n + 1)! 2 · n!

Hence it follows that DF1 (X , L ) =

  a0 b1 − a1 b0 1 S¯ n n+1 ¯ ¯ ( L = − · L ) + ) , (K 1 X¯ /P 2vol n+1 a02  

as required.

2.5 The Relationship Between the Chow Norm and DFi Let (X , L ) be a test configuration, of exponent γ , for (X, L). For each positive integer m, we consider the direct image sheaf E := π∗ L ⊗m ,

m = 1, 2, . . . ,

2.5 The Relationship Between the Chow Norm and DFi

17

on A1 . Let m  1. Then by the affirmative solution of the equivariant Serre conjecture (see Sect. 1.1), E is C∗ -equivariantly isomorphic to E0 × A1 , where Ez ∼ = H 0 (Xz , Lz⊗m ),

z ∈ A1 ,

denotes the fiber of E over z. Note that C∗ acts on E0 × A1 by C∗ × (E0 × A1 )  (t, (e, z)) → (σm (t)e, tz) ∈ E0 × A1 for an algebraic group homomorphism σm : C∗ → GL(E0 ) naturally induced by the C∗ -action on (X , L ). Put γm := mγ . Then X = X1 ⊂ P(E1∗ ) = P(Vγ∗m ), where Vγm = H 0 (X, L⊗γm ). For the degree d of X in the projective space P(Vγ∗m ), we set W := (Symd Vγ∗m )⊗n+1 , and let 0 = Xˆ ∈ W be the Chow form for X. Let ργm be the Hermitian structure for Vγm as in (2.2). By setting t = exp s for s ∈ R, we consider the real-valued function fm on R defined by ˆ CH(ργ ) , fm (s) := log σmSL (t)X m

s ∈ R,

where  CH(ργm ) denotes the Chow norm for W induced by ργm as in Sect. 1.3. Note that the C∗ -equivariant isomorphism E ∼ = E0 × A1 can be chosen in such a way that, by the natural identification E0 × {1} ∼ = Vγm , the Hermitian structure ργm on Vγm induces a Hermitian structure on E0 which is preserved by the action of S 1 in C∗ . Let a0 be as in Sect. 2.3. Then by setting f˙m (s) = ∂fm /∂s, we obtain (see for instance [51]): Theorem 2.2

lim f˙m (s) = (n + 1)! a0

s→−∞

∞ 

DFi (X , L )mn+1−i .

i=1

Proof For k = 1, 2, · · · , we put Ek := H 0 (X0 , L0⊗km ) and Vk := H 0 (X, L⊗kmγ ). Let dk be the degree of X in the projective space P(Vk∗ ). Then by Mumford [62, Proposition 2.11], the weight μk of the R+ -action on det Ek induced by σmSL is μk = −

λm k n+1 + O(k n ), (n + 1)!

k  1,

(2.10)

where λm is the Chow weight of X ⊂ P(V1∗ ), i.e., the weight of the R+ -action on the line Xˆ0 in W1 := (Symd1 E1∗ )⊗n+1 via σmSL . Note here that Xˆ0 is the Chow form for the algebraic cycle X0 on P(E1∗ ) (= P(V1∗ )). We can write the Chow form Xˆ = 0 for the irreducible reduced algebraic cycle X ⊂ P(V1∗ ) as a sum Xˆ =

r  α=1

vα ,

18

2 The Donaldson–Futaki Invariant

where 0 = vα ∈ W1 is such that there is an increasing sequence of rational numbers e1 < e2 < · · · < er satisfying σmSL (t)vα = t eα vα , Hence Xˆt = σmSL (t)Xˆ1 = σmSL (t)Xˆ = limt →0 [Xˆt ] = [Xˆ0 ] in P(W1 ), and since  lim [Xˆt ] = lim

t→0

t→0



t e1 v1 +

r 

t ∈ R+ . r

t eα vα for all t ∈ C∗ . Since

α=1

 t eα −e1 vα

 = lim

t→0

α=2

v1 +

r 

 t eα −e1 vα

= [v1 ],

α=2

we may assume that Xˆ0 = v1 . Since λm is the weight of the R+ -action on the line RXˆ0 via σmSL , we have e1 = λm . It then follows that ˆ CH(ργ¯ ) = log t e1 (v1 + fm (s) = log σmSL (t)X

r 

t eα −e1 vα )CH(ργ¯ )

α=2

 = log exp(sλm ) v1 +

r 



t eα −e1 vα CH(ργ¯ )

α=2

= sλm + log v1 +

r 

t eα −e1 vα CH(ργ¯ ) .

α=2

Hence by setting ϕ(s) := log v1 +

r α=2

t eα −e1 vα CH(ργ¯ ) , we obtain

˙ f˙m (s) = λm + ϕ(s),

s ∈ R.

Let s1 ∈ R be such that s1  −1. By the mean value theorem, there exists an s˜1 ∈ R satisfying −1 + s1 < s˜1 < s1 such that ϕ(˜ ˙ s1 ) =

ϕ(s1 ) − ϕ(s1 − 1) = ϕ(s1 ) − ϕ(s1 − 1) = ϕ|t =es1 − ϕ|t =es1 −1 . s1 − (s1 − 1)

From the continuity of the Chow norm (cf. [60]), it follows that lim

s1 →−∞

ϕ(˜ ˙ s1 ) =

lim {ϕ|t =es1 − ϕ|t =es1 −1 } = ϕ|t =0 − ϕ|t =0 = 0.

s1 →−∞

(2.11)

2.5 The Relationship Between the Chow Norm and DFi

19

Since the function fm (s) is convex ([93]; see also Sect. 1.4), we observe that f˙m (s) (and hence ϕ(s)) ˙ is non-decreasing in s. Then by (2.11), lims→−∞ ϕ(s) ˙ = 0. Hence lim f˙m (s) = λm +

s→−∞

lim ϕ(s) ˙ = λm .

(2.12)

s→−∞

Recall that Nm = dim E1 . Put Nkm := dim Ek . By σmSL (t) = {det σm (t)}−1/Nm σm (t), the weight μk of the R+ -action on det Ek (induced by the R+ -action on E1 = E0 via the one-parameter group σmSL ) is expressible as μk = wkm − k(wm /Nm )Nkm ,

(2.13)

where wm and wkm are the weights of the C∗ -action on E1 and Ek , respectively, induced by the C∗ -action via σm . Since the natural map of S k (E1 ) to Ek is surjective (cf. [61, 62]) by m  1, as far as the R+ -action on Ek is concerned, wkm is the weight induced from the R+ -action on E0 via σm , while −k(wm /Nm )Nkm is the weight induced from the R+ -action on E0 via the scalar multiplication by {det σm (t)}−1/Nm . In view of (2.10) and (2.13), for k  1, we obtain   wkm wm λm n+1 n k + O(k ) = μk = kmNkm − − (n + 1)! (km)Nkm mNm ∞  ∞   −i −i DFi (X , L )(km) − DFi (X , L )m = kmNkm  = kmNkm

i=0 ∞ 

i=0

DFi (X , L )(km)

−i

−

i=1

= −kmNkm

∞ 

∞ 

 DFi (X , L )m

i=1

DFi (X , L )m

−i

+ O(k

−1

−i

 ) .

i=1

Since Nkm = a0 (km)n {1 + O(k −1 )}, it then follows that ∞

 λm = mn+1 a0 DFi (X , L )m−i . (n + 1)! i=1

From this together with (2.12), we now conclude that lim f˙m (s) = λm = (n + 1)!a0

s→−∞

as required.

∞ 

DFi (X , L )mn+1−i ,

i=1

 

20

2 The Donaldson–Futaki Invariant

Problems 2.1 (cf. Li and Xu [37]) For P3 (C) = { x = (x0 : x1 : x2 : x3 )} and A1 = Spec C[z], we consider the subscheme X in P3 (C) × A1 = {(x, z)} defined by the ideal I = (z2 (x0 + x3 )x3 − x22 , zx0 (x0 + x3 ) − x1 x2 , x0 x2 − zx1 x3 , x12 x3 − x02 (x0 + x3 )). Let pr2 be the restriction to X of the projection P3 (C) × A1 → A1 to the second factor. Note that X0 sitting in P3 (C) is defined by the ideal I0 = (x22 , x1 x2 , x0 x2 , x12 x3 − x02 (x0 + x3 )),

(2.14)

1 where Xz := pr−1 2 (z), z ∈ A , are the scheme-theoretic fibers. Put L := ∗ pr1 OP3 (1), where pr1 is the restriction to X of the projection P3 (C)×A1 → P3 (C) to the first factor. For C4 = Spec C[x0 , x1 , x2 , x3 ], the group C∗ = {t ∈ C∗ } acts on C4 × A1 by

(x0 , x1 , x2 , x3 , z) → (x0 , x1 , tx2 , x3 , tz),

t ∈ C∗ ,

which naturally induces C∗ -actions on L and X , since P3 (C) = (C4 \ {0})/C∗ , and since we can identify OP3 (−1) with the blow-up of C4 at the origin. Then it is easy to check that (X , L ) is a test configuration, of exponent γ = 1, for (X, L) = (P1 (C), OP1 (3)). To see this, show first of all that X1 ∼ = P1 (C). 2.2 Compute the Donaldson–Futaki invariant DF1 (X , L ) of the test configuration (X , L ) in Problem 2.1 above. 2.3 In Problem 2.1, consider the normalization ν : X˜ → X and the pullback L˜ := ν ∗ L . Show that the test configuration (X˜ , L˜ ) induced from (X , L ) is trivial.

Chapter 3

Canonical Kähler Metrics

Abstract In this chapter, we shall introduce various special metrics for compact complex manifolds such as Kähler–Einstein metrics, CSC Kähler metrics, extremal Kähler metrics, Kähler–Ricci solitons and generalized Kähler–Einstein metrics. • In Sect. 3.1, we give definitions of these special metrics. Here CSC Kähler metrics and extremal Kähler metrics are defined by using the scalar curvature Sω , while Kähler–Einstein metrics, Kähler–Ricci solitons and generalized Kähler– Einstein metrics are defined by using the Ricci potential fω . • In Sect. 3.2, we shall show that Kähler–Ricci solitons and generalized Kähler– Einstein metrics are Kähler–Einstein analogues in Bakry–Emery geometry by conformal changes via Hamiltonian functions of holomorphic vector fields. Keywords Kähler–Einstein metrics · CSC Kähler metrics · Extremal Kähler metrics · Kähler–Ricci solitons · Generalized Kähler–Einstein metrics

3.1 Canonical Kähler Metrics on Compact Complex Manifolds For a Kähler class K on a compact complex connected manifold X, we choose a Kähler form ω in K . Put n := dim X. By writing ω in terms of the notation in (1.1), we consider the associated Laplacian Δω :=

 α,β

¯

g βα

∂2 . ∂zα ∂zβ¯

 For the scalar curvature Sω := Trω Ric(ω), we have X (Sω − S0 ) ωn = 0 by taking the average S0 . Then we have a real-valued smooth function fω ∈ C ∞ (X)R such that Δω fω = Sω − S0 ,

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0_3

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22

3 Canonical Kähler Metrics

  where S0 = ( X ωn )−1 X n Ric(ω)ωn is a constant depending only on the Kähler class and independent of the choice of ω in K . Then the Futaki character [9, 23] F (y) :=

(yfω )ωn ,

y ∈ H 0 (X, O(T X)),

X

is independent of the choice of ω in K , and is a Lie algebra character known as an obstruction to the existence of CSC Kähler metrics. Recall that (cf. [8, 9]): • ω is called CSC Kähler if Sω is constant, • ω is called extremal Kähler if gradC ω Sω is holomorphic. If ω sits in the Kähler class c1 (X), then up to an additive constant, fω coincides with the Ricci potential for ω in Sect. 9.1, and we define (cf. [24, 35, 45]): • ω is called Kähler–Einstein if fω is constant, • ω is called Kähler–Ricci soliton if gradC ω fω is holomorphic, fω is holomorphic. • ω is called generalized Kähler–Einstein if gradC ω e However, if the condition [ω] = c1 (X) is dropped, we don’t know much about Kähler–Ricci solitons and generalized Kähler–Einstein metrics in such a broad sense.

Extremal Kähler CSC Kähler

Kähler–Einstein

3.2 Conformal Changes of Metrics by Hamiltonian Functions We now consider the case where c1 (X) > 0 and [ω] = c1 (X). Fix a holomorphic vector field y = 0 on X. Actually for Kähler–Ricci solitons and generalized Kähler– Einstein metrics, we choose holomorphic vector fields y associated to fω and efω ,

3.2 Conformal Changes of Metrics by Hamiltonian Functions

23

respectively. More generally, a holomorphic vector field y on X written in the form y = gradC ω ψ, for some ψ ∈ C ∞(X)R is called Hamiltonian, where we often impose a normalization condition X ψωn = 0 on ψ. Here ψ is called a Hamiltonian function ¯ the real vector field associated to y. In view of the identity iy ω = ∂(ψ/2π), yR := y + y¯ on X satisfies iyR ω = d(ψ/2π). Hence we have a moment map for y, ψ : X → R, where by the normalization condition, both a := minX ψ and b := maxX ψ are constants depending only on y, and independent of the choice of ω in the class c1 (X). Next we consider a smooth function ζ = ζ(s) : [a, b] → R, where a normalization condition on the Hamiltonian function ψ is not necessarily imposed. Then we put ζ (s) = s for Kähler–Ricci solitons, whereas we put ζ(s) = log s for generalized Kähler–Einstein metrics. In view of the counterpart (cf. [24, 47]) in complex geometry of the Bakry–Emery theory [2, 67, 84], by setting ω˜ := exp{ζ (ψ)/n} ω, we can write the corresponding volume form as ω˜ n = eζ(ψ) ωn , so that we define Ric(ω) ˜ := − dd c log ω˜ n = Ric(ω) − dd c ζ(ψ) = ω + dd c {fω − ζ(ψ)}. Then the Einstein condition Ric(ω) ˜ = ω is the constancy of fω − ζ(ψ). Since the Ricci potential fω for ω is unique up to an additive constant, for a suitable choice of an additive constant, we can express the Einstein condition as fω = ζ(ψ) for some ψ. Hence ω satisfying the Einstein condition Ric(ω) ˜ = ω is a Kähler–Ricci soliton or a generalized Kähler–Einstein metric, if ζ(s) = s or ζ(s) = log s, respectively.

Problems 3.1 Let y ∈ k := Lie(K) be a real vector field on a compact connected smooth manifold X with an effective action of a compact real Lie group K. By choosing a K-invariant real symplectic form ω on X, assume that some f ∈ C ∞ (X)R satisfies df = iy ω.

24

3 Canonical Kähler Metrics

Show that the value maxX f − minX f is independent of the choice of ω as long as ω defines the same de Rham cohomology class [ω]. √ 3.2 For ω˜ = exp{ζ(ψ)/n} ω as above, let Δω˜ := Δω − −1ζ˙ (ψ) y¯ be the ˙ associated Then for all ξ, η ∈ C ∞ (X)R , show  Laplacian, nwhere ζ (s)  := dζ(s)/ds. n ¯ ∂η) ¯ ω ω˜ that − X (∂ξ, = ˜ , where ( , )ω denotes the pointwise X (Δω˜ ξ )ηω Hermitian pairing by ω for 1-forms on X. 3.3 For a constant scalar curvature Kähler metric ω, let y be a Hamiltonian ∞ holomorphic vector field, i.e., y = gradC ω ψ for some ψ ∈ C (X)R . Show that F (y) = 0 for the Futaki character F .

Chapter 4

Norms for Test Configurations

Abstract In this chapter, we introduce the concept of norms for test configurations. Then by using such norms, we can define the Donaldson–Futaki invariant for sequences of test configurations. • In Sect. 4.1, we define norms of test configurations of a fixed exponent. • In Sect. 4.2, given a test configuration (X , L ), we obtain an asymptotic 1 norm from the sequence of test configurations (X , L ⊗m ), m = 1, 2, · · · . • In Sect. 4.3, we consider the relative version of norms of test configurations. • In Sect. 4.4, we define twisted Kodaira embeddings. • Finally, in Sect. 4.5, we define the Donaldson–Futaki invariant for sequences of test configurations. Keywords Norms for test configurations · Twisted Kodaira embeddings · The Donaldson–Futaki invariant for sequences of test configurations

4.1 Norms for Test Configurations of a Fixed Exponent For a polarized algebraic manifold (X, L), we fix a Hermitian metric h for L such that ω := Ric(h) is Kähler. Let μσ = (Xσ , Lσ ) be a test configuration, of exponent γ , associated to a special one-parameter group σ : C∗ → GL(Vγ ). Here for the subgroup S 1 of C∗ , σ (S 1 ) acts isometrically on the vector space Vγ := H 0 (X, L⊗γ ) endowed with the Hermitian metric ργ in (2.2). Put Nγ := dim Vγ . As in Sect. 1.1, for σ , we consider its special linearization σ SL such that, when restricted to R+ , we obtain a Lie group homomorphism σ SL : R+ → SL(Vγ ).

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0_4

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26

4 Norms for Test Configurations

Let u ∈ sl(Vγ ) be the fundamental generator (cf. Sect. 1.1) for σ SL . Then for a suitable choice of an orthonormal basis for Vγ , u is written in the form ⎛ −β1 ⎜ −β2 u = ⎜ ⎝ ...

0

0

⎞ ⎟ ⎟, ⎠

−βNγ

where −βi , i = 1, 2, . . . , Nγ , are the weights of the R+ -action on Vγ via σ SL , so that σ (es ) = exp(su) holds for all s ∈ R. We now put ⎞ ⎛ Nγ  |u|1 := γ −1 Nγ−1 ⎝ |βi |⎠ ,

|u|∞ := γ −1 max |βi |. 1≤i≤Nγ

i=1

(4.1)

Then |u|1 = 0 if and only if u = 0, i.e., (Xσ , Lσ ) is trivial. The same thing is true also for |u|∞ . By abuse of terminology, we define the following: Definition 4.1 The 1 -norm μσ 1 of the test configuration μσ = (Xσ , Lσ ) is |u|1 , while the ∞ -norm μσ ∞ of the test configuration μσ = (Xσ , Lσ ) is |u|∞ .

4.2 The Asymptotic 1 -norm of a Test Configuration Let μ = (X , L ) be a test configuration, of exponent γ , for a polarized algebraic manifold (X, L). As in Sect. 2.3, the algebraic torus C∗ acts on H 0 (X0 , L0⊗m ). Then as in Sect. 2.5, this action is given by a representation σm : C∗ → GL(H 0 (X0 , L0⊗m )),

m  1,

associated to the test configuration (X , L ⊗m ). By Sect. 1.1, we have its special linearization σmSL , and when restricted to R+ , we obtain a Lie group homomorphism σmSL : R+ → SL(H 0 (X0 , L0⊗m )). Put γm := mγ and Nm := dim H 0 (X0 , L0⊗m ). Let −βi , i = 1, 2, · · · , Nm , be the weights of the R+ -action on H 0 (X0 , L0⊗m ). Then the fundamental generator um for the special linearization σmSL is written as ⎛ ⎜ um = ⎜ ⎝

−β1

0

−β2 ...

0 −βNm

⎞ ⎟ ⎟ ⎠

4.3 Relative Norms for Test Configurations

27

by choosing a suitable basis for H 0 (X0 , L0⊗m ). Then a result of Hisamoto [30] shows that the following limit exists as an asymptotic 1 -norm μasymp of μ:  μasymp := lim |um |1 = lim m→∞

m→∞

γm−1 Nm−1

Nm 

 |βi | .

i=1

Remark 4.1 By definition, (X , L ⊗k )asymp = (X , L )asymp for all positive integers k. It is known that, as long as X is a normal complex variety, μasymp = 0 if and only if the test configuration μ is trivial ([4]; see also [18, 30]).

4.3 Relative Norms for Test Configurations As a reference for this section, see [76] (cf. also [54]). For a polarized algebraic manifold (X, L), we consider a possibly trivial algebraic torus T sitting in the identity component Aut0 (X) of Aut(X). Replacing L by its suitable power, we may assume that the T -action on X lifts to an action on L inducing linear maps between fibers. To study extremal Kähler metrics on X, we consider Tγ⊥ below in place of Gγ := SL(Vγ ). Now for gγ := Lie(Gγ ) = sl(Vγ ), we define a bilinear form , γ by

u, vγ := γ −n−2 Tr(uv),

u, v ∈ gγ .

(4.2)

Let tγ be the Lie algebra t := Lie(T ) viewed as a Lie subalgebra of gγ . Let zγ := {u ∈ gγ ; [u, tγ ] = 0} be its centralizer in gγ . Then for tγ and zγ , the corresponding connected linear algebraic subgroups in Gγ will be denoted by Tγ and Zγ , respectively. Let t⊥ γ be the orthogonal complement of tγ in zγ defined by t⊥ γ := { u ∈ zγ ; u, tγ γ = 0 }. ⊥ Note that, if T is trivial, then t⊥ γ = zγ = gγ . To see another expression of tγ , by considering the infinitesimal tγ -action on Vγ , we write Vγ as a direct sum

Vγ =

nγ 

Vγ ,i ,

i=1

where Vγ ,i = {v ∈ Vγ ; θ v = χγ ,i (θ )v for all θ ∈ tγ } for distinct characters χγ ,i ∈ t∗γ , i = 1, · · · , nγ . By choosing a basis for each Vγ ,i , we identify Gγ with SL(Nγ , C). Put Sγ :=

nγ

i=1

SL(Vγ ,i ) ⊂ SL(Nγ , C).

28

4 Norms for Test Configurations

The centralizer Hγ of Sγ in Gγ consists of all diagonal matrices in Gγ which act on each Vγ ,i , i = 1, . . . , nγ , by constant scalar multiplications. By setting hγ := Lie(Hγ ), we consider the orthogonal complement of tγ in hγ , t γ := { u ∈ hγ ; u, tγ γ = 0 }, and the associated algebraic torus in Hγ will be denoted by Tγ . Then Tγ⊥ := Tγ · Sγ is a reductive algebraic subgroup of Gγ whose Lie algebra is t⊥ γ . Let Tc be the maximal compact subgroup of T . Let ω = Ric(h) be a Tc -invariant Kähler form on X, so that we can choose h as a Tc -invariant Hermitian metric for L. Definition 4.2 For T above, we denote by 1PS(Tγ⊥ ) the set of all nontrivial special one-parameter groups σ : C∗ → SL(Vγ ) satisfying σ (C∗ ) ⊂ Tγ⊥ . Remark 4.2 If the algebraic torus T is trivial, then 1PS(Tγ⊥ ) is nothing but the set 1PS(Gγ ) of all nontrivial special one-parameter groups σ : C∗ → Gγ := SL(Vγ ). For each σ ∈ 1PS(Tγ⊥ ), let u ∈ t⊥ γ be its fundamental generator. By Sect. 2.2, we have the test configuration μσ = (Xσ , Lσ ) associated to the special one-parameter group σ . As in Sect. 4.2, for each positive integer m, the one-parameter group σ induces um ∈ sl(H 0 ((Xσ )0 , (Lσ⊗m )0 )). As in Sect. 2.5, H 0 ((Xσ )0 , (Lσ⊗m )0 ) is viewed as H 0 ((Xσ )1 , (Lσ⊗m )1 ) = Vγm with γm :=√mγ , so that we regard um as an element of sl(Vγm ) with real eigenvalues. Put tR := −1 tc , where tc is the maximal compact real Lie subalgebra of t. Then for each v ∈ tR , the element in sl(Vγm ) induced by v will be denoted as vm . Since u commutes with tγ , we see that um and vm are simultaneously diagonalized. Hence by choosing a suitable basis for Vγm , we can write ⎛ ⎜ um = ⎜ ⎝

⎞

−β1

0

−β2 ...

0 ⎟⎟

⎠,

−βNm

⎛ ⎜ vm = ⎜ ⎝

⎞

α1

0

α2 ...

0 ⎟⎟

⎠,

αNm

(4.3)

4.4 The Twisted Kodaira Embedding

29

where αi , βj , i, j ∈ {1, 2, · · · Nm }, are real constants. In [30], replacing the fundamental generator um by um + vm , we obtain μσ vasymp by setting  μσ vasymp

:= lim |um + vm |1 = lim m→∞

m→∞

γm−1 Nm−1

Nm 

 |αi − βi | .

i=1

Then a result in Hisamoto [31] shows that the asymptotic 1 -norm μσ Tasymp for μσ relative to T is defined by μσ Tasymp := inf μσ vasymp > 0. v∈tR

Remark 4.3 If T is trivial, then μTasymp coincides with μasymp in Sect. 4.2. Remark 4.4 For r := dim tR , we choose elements wi , i = 1, . . . , r, in tR such that the corresponding elements wi,m , i = 1, . . . , r, in tγm satisfy

wi,m , wj,m γm = δij ,

i, j ∈ {1, 2, · · · , r},

where δij is Kronecker’s delta, and , γm is the bilinear form on gγm as in (4.2). Then for um in (4.3), we define u m ∈ sl(Vγm ) with real eigenvalues by u m := um −

r 

um , wi,m γm wi,m .

i=1

In place of the above definition of μσ Tasymp, it is possible to use the formula μσ Tasymp := lim |u m |1 . m→∞

This new definition and the original definition for μσ Tasymp are known to give equivalent norms (see [31] for more details).

4.4 The Twisted Kodaira Embedding For a polarized algebraic manifold (X, L), let y be the extremal vector field for (X, L) as in Sect. 9.2. By the notation in Sect. 4.3, we choose an orthonormal basis {vi,α ; i = 1, 2, · · · , nγ , α = 1, 2, · · · , qi }

30

4 Norms for Test Configurations

for (Vγ , ργ ) such that each {vi,α ; α = 1, 2, · · · , qi }, i = 1, 2, · · · , nγ , is a basis for Vγ ,i . Let yγ be the element in tγ corresponding to y in t. Put √ # vi,α := {1 − γ −2 χγ ,i ( −1yγ )}1/2vi,α . # ; i = 1, 2, · · · , n , α = 1, 2, · · · , q } as {v # , v # , · · ·, v # }. Renumber the basis {vi,α γ i Nγ 1 2 # } is called an admissible basis for V . Then This renumbered basis {v1# , v2# , · · · , vN γ γ

the twisted Kodaira embedding Φγ# for L⊗γ is Φγ# : X → P(Vγ∗ ),

# x → (v1# (x) : v2# (x) : · · · : vN (x)), γ

where P(Vγ∗ ) is viewed as the complex projective space PNγ −1 (C) in terms of the √ admissible basis. Here by γ −2 (χγ ,i )∗ ( −1yγ ) = O(γ −1 ) ∈ R (cf. [50, 54]), we have # vi,α = {1 + O(γ −1 )} vi,α ,

and hence Φγ# above is well-defined for γ  1. If the extremal vector field y is trivial, then Φγ# coincides with the original Kodaira embedding Φγ : X → P(Vγ∗ ),

x → (v1 (x) : v2 (x) : · · · : vNγ (x)).

4.5 The Donaldson–Futaki Invariant for Sequences In this section, we give an example where the norms |u|1 and |u|∞ in (4.1) are needed. For a polarized algebraic manifold (X, L), we choose a possibly trivial algebraic torus T in Aut0 (X). Let MT denote the set of all sequences {μj } of test configurations μj = (Xj , Lj ), j = 1, 2, · · · , for (X, L) such that the exponent γj of μj satisfies γj → +∞,

as j → ∞,

where each μj is a test configuration associated to some σj ∈ 1PS(Tγ⊥j ). In the special case where T is trivial, Tγ⊥j is nothing but Gγj in terms of the notation in Remark 4.2, and MT will be written simply as M . In this section, we consider MT for a general T . Let 0 = uj ∈ tγj be the fundamental generator for σj . By setting

4.5 The Donaldson–Futaki Invariant for Sequences

31

t := exp(s/|uj |∞ ), s ∈ R, we define a real-valued function fj (s) on R by fj (s) := =

|uj |∞ −n γ log σj (t) · Xˆ j CH(ργ ) j |uj |1 j |uj |∞ −n γ log  exp(suj /|uj |∞ )Xˆ j CH(ργ ) , j |uj |1 j

(4.4)

where for the twisted Kodaira embedding Φγ#j : X → P(Vγ∗j ) for L⊗γj on X = (Xj )1 , we consider the Chow form Xˆ j for the irreducible reduced algebraic cycle Xj := Φγ#j (X) on P(Vγ∗j ). We now claim the following: Claim f˙j (0) ≤ C for some real constant C > 0 independent of j . Assuming this claim, we now define an invariant F1 ({μj }) ∈ R ∪ {−∞}, called the Donaldson–Futaki invariant for {μj }, as follows: For each s ≤ 0, we have lim f˙j (s) ≤ lim f˙j (0) ≤ C,

j →∞

j →∞

since limj →∞ f˙j (s) is a non-decreasing function of s by the convexity of fj (s). Then by letting s → −∞, we can define the following (cf. [52]): F1 ({μj }) :=

lim

lim f˙j (s) ≤ C.

s→−∞ j →∞

(4.5)

Proof of Claim Since σj is a special one-parameter group, by choosing an admissi# } and the associated orthonormal basis {v , · · · , v } for the ble basis {v1# , · · · , vN 1 Nj j Hermitian vector space (Vγj , ργj ), we have integers βk such that for all t ∈ C∗ , σj (t) · vk = t −βk vk ,

k = 1, · · · , Nj ,

where β1 +· · ·+βNj = 0. Let Bj (ω) := (n!/γjn ) Bergman kernel, and we further put Bj# (ω)

:=

(n!/γjn )

Nj

Nj 

2 k=1 |vk |h

be the γj -th asymptotic

|vk# |2h .

k=1

Then by the theorem of Tian–Yau–Zelditch [77, 92] and Lu [39], 1 Bj (ω) = 1 + Sω γj−1 + O(γj−2 ) = 1 + O(γj−1 ), 2

(4.6)

32

4 Norms for Test Configurations

so that we can write Bj# (ω) as Bj# (ω) = {1 + O(γj−1 )}2 Bj (ω) = 1 + O(γj−1 ).

(4.7)

Then by taking dd c log of both sides of this equality, we obtain ωFS − γj ω = O(γj−1 ), where ωFS = dd c log f˙j (0) =

Nj

# 2 k=1 |vk | .

(4.8)

Apply [48, Remark 4.6] to (1.8). Then by (4.8),

|uj |∞ −n d γ log σj (exp(s/|uj |∞ )Xˆ j CH(ργ ) j |uj |1 j ds |s=0

1 = γ −n (n + 1) |uj |1 j (n + 1)! −n = γ |uj |1 j

β |v # |2 + · · · + β |v # |2 1 1 h Nj Nj h X

# |2 |v1# |2h + · · · + |vN j h

Nj β |v |2 {1 + O(γ −1 )} k=1 k k h j Bj# (ω)

X

n ωFS

{ω + O(γj−2 )}n .

(4.9)

Since {v1 , · · · , vNj } is an orthonormal basis for (Vγj , ργj ), we have the identity  Nj Nj 2 n = k=1 βk |vk |h ω k=1 βk = 0, and hence by (4.7) and (4.9) together X with (4.1), (n + 1)! −n f˙j (0) = γ |uj |1 j

X

O(γj−1 )

Nj 

! |βk | · |vk |2h ωn

k=1 Nj

=

 (n + 1)! O(γj−n−1 ) |βk | = O(1). |uj |1

 

k=1

Remark 4.5 In (4.4) above, if we replace uj by its constant scalar multiple cj uj , the function fj (s) is still the same. Hence given μj = (Xσj , Lσj ), j = 1, 2, · · · , we can define F1 ({μj }) more generally even when each σj has rational weights, because by constant scalar multiplication, a rational matrix is changed to an integral matrix. Hence even in the case where each σj is a special one-parameter group from C∗ to GL(Vγj ), we can still define F1 ({μj }) by setting fj (s) :=

|uj |∞ −n γ log σjSL (t) · Xˆ j CH(ργ ) , j |uj |1 j

(4.10)

Problems

33

where uj is the fundamental generator for σjSL . In this case, σj is not necessarily an algebraic group homomorphism from C∗ to SL(Vγj ), but its lift ˜ ∗ → SL(Vγj ) σ˜ j : C ˜ ∗ of the algebraic torus C∗ is an algebraic group homomorto a suitable covering C phism. It is then easy to see that F1 ({μj }) thus obtained from the function fj (s) in (4.10) is nothing but F1 ({μσ˜ j }). Remark 4.6 In the definition of F1 ({μj }), all test configurations μj = (Xσj , Lσj ) are assumed to be nontrivial, i.e., nontriviality of the one-parameter groups σj : C∗ → Tγ⊥j is assumed. However, we can drop this assumption as follows: If μj is a trivial test configuration, we choose fj (s) as a constant function by setting fj (s) := Xˆ j CH(ργj ) ,

s ∈ R.

Then we can define F1 ({μj }) by the formula (4.5) even if the condition of nontriviality of μj is not assumed.

Problems 4.1 For the test configuration μ = (X , L ) for (X, L) = (P1 (C), OP1 (3)) in Problem 2.1, compute the asymptotic 1 -norm μasymp . 4.2 Let (X, L) = (P1 (C), OP1 (1)) for P1 (C) = {(x0 : x1 )}. Consider the product test configuration μ = (X , L ) of exponent 1 for (X, L), where by the representation C∗  t →



t 0 0 t −1

 ∈ SL(H 0 (X, L)),

the algebraic torus C∗ acts on H 0 (X, L) in terms of the basis {x0 , x1 } for H 0 (X, L). Compute the asymptotic 1 -norm μasymp for μ. 4.3 For {μj }j =1,2,··· ∈ MT , let {μjk }k=1,2,··· be its subsequence. Show that F1 ({μj }) ≤ F1 ({μjk }).

Chapter 5

Stabilities for Polarized Algebraic Manifolds

Abstract In this chapter, several stability concepts will be introduced from the viewpoints of the existence problem of canonical Kähler metrics. • Typical examples of such stabilities are the Chow stability and the Hilbert stability. We shall first study these classical stability concepts by showing that they are asymptotically equivalent. • Secondly, various kinds of K-stability will be discussed to study the existence problem of Kähler-Einstein metrics or more generally CSC Kähler metrics. • Thirdly, we introduce relative versions of stability concepts, which play a crucial role in the study of extremal Kähler metrics. • Finally, various relationships among the stability concepts will be discussed. Keywords The Chow stability · The Hilbert stability · K-stability · Relative stability

5.1 The Chow Stability Let X ⊂ P(V ∗ ) be an n-dimensional projective subvariety, where V is a finite dimensional complex vector space. As in Sect. 1.4, let 0 = Xˆ ∈ W be the Chow form for X, where W := (Symd V ∗ )⊗n+1 and d := degP(V ∗ ) X. For a reductive algebraic subgroup G of SL(V ), consider the induced G-action on P(V ∗ ). Definition 5.1 (1) X ⊂ P(V ∗ ) is called Chow polystable if the orbit G · Xˆ is closed in the vector space W . (2) Let X ⊂ P(V ∗ ) be Chow polystable. Then X ⊂ P(V ∗ ) is called Chow stable if in addition the isotropy subgroup of G at Xˆ is finite.

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0_5

35

36

5 Stabilities for Polarized Algebraic Manifolds

From now on in this section, consider the case where V = Vγ := H 0 (X, L⊗γ ) for a polarized algebraic manifold (X, L). Then we consider the Kodaira embedding X ⊂ P(Vγ∗ ),

x → hyperplane {v ∈ Vγ ; v(x) = 0} in Vγ ,

associated to the complete linear system |L⊗γ |, where this embedding is just an abstract one without assuming any orthonormality of the basis for the embedding. Note that the ordinary Kodaira embedding and the twisted Kodaira embedding coincide as an abstract Kodaira embedding. Assume that G = SL(V ). For a Kähler form ω in c1 (L), choose a Hermitian metric h for L such that Ric(h) = ω. Hence, in terms of the Hermitian structure ργ for V = Vγ as in (2.2), we can talk about a special one-parameter group. For each v ∈ V , we consider the nonnegative function |v|h on X defined by |v|h2 = (v, v)h . Definition 5.2 ω is called a balanced metric for L⊗γ if |v1 |2h + |v2 |2h + · · · + |vN |2h is a constant function on X for an orthonormal basis {v1 , v2 , · · · , vN } of (Vγ , ργ ). Theorem 5.1 ([93]; See Also [40]) X ⊂ P(Vγ∗ ) is Chow polystable if and only if there exists a balanced metric ω for L⊗γ . Proof We first use the following Hilbert–Mumford stability criterion (see for instance [48]): A G-orbit in W is closed if and only if for some point w in the orbit, σ (C∗ ) · w is closed in W for every special one-parameter group σ : C∗ → G. Then for the “if” part of the proof, it suffices to show that σ (C∗ ) · Xˆ is closed for every special one-parameter group σ : C∗ → G by assuming that a balanced metric ω for L⊗γ exists. Then by identifying V ∗ with CN = {(z1 , · · · , zN )} by a suitable choice of an orthonormal basis {v1 , . . . , vN } of V , we can diagonalize σ in the form σ (t) · z = (t b1 z1 , · · · , t bn zN ), where z = (z1 , · · · , zN ) ∈ V ∗ , and − bi  ∈ Z are the weights of the C∗ -action on V via σ . Since G = SL(V ), it follows that N i=1 bi = 0. Note also that |v1 |2h + · · · + |vN |2h = C

(5.1)

for some positive constant C. For the balanced metric ω, we have the Hermitian ˆ CH(ργ ) for s ∈ R. Then structure ργ for V = Vγ . Put f (s) := log σ (es ) · X by (1.8), f˙(0) = (n + 1)

X

b1 |v1 |2h + · · · + bN |vN |2h |v1 |2h + · · · + |vN |2h

n ωFS ,

(5.2)

 where ωFS := dd c log( ni=1 |vi |2 ). On the other hand, let dd c log operate on both sides of (5.1). It then follows that ωFS = −γ dd c log h = γ ω.

(5.3)

5.1 The Chow Stability

37

Now by (5.1)–(5.3), we obtain N N (n + 1)γ n  (n + 1)γ n  bi |vi |2h ωn = bi = 0. f˙(0) = C C X i=1

(5.4)

i=1

Moreover, by the second variation formula for the Chow norm, f¨(0) ≥ 0. On the other hand, for all t ∈ C∗ , we have ˆ CH(ργ ) = σ (|t|) · X ˆ CH(ργ ) , σ (t) · X

(5.5)

since for the subgroup S 1 of C∗ , its image σ (S 1 ) acts isometrically on (Vγ , ργ ) and hence on (W, CH(ργ )). Then the following cases are possible: Case 1: f¨(0) > 0. Note that the function f (s) is convex (see Sect. 1.4). We now see from (5.4) and the inequality f¨(0) > 0 that lim f (s) = +∞ =

s→−∞

lim f (s).

s→+∞

Hence by (5.5), in this case, σ (C∗ ) · Xˆ is closed in W . Case 2: f¨(0) = 0. Then by Problem 1.1, the one-parameter group σ : C∗ → G preserves the subvariety X in P(V ∗ ). Hence there exists an integer α such that ˆ σ (t) · Xˆ = t α X,

t ∈ C∗ .

Then by setting t = es with s ∈ R, we have 0 = f˙(0) = =

ˆ CH(ργ ) ) d(log σ (t) · X ds

|s=0

ˆ CH(ργ ) ) d(αs + log X ds

|s=0

=

ˆ CH(ργ ) ) d(log esα · X ds

|s=0

= α.

ˆ and in particular is closed. Hence σ (C∗ ) · Xˆ is a single point X, For the “only if” part of the proof, assume that X ⊂ P(Vγ∗ ) is Chow polystable, i.e., G · Xˆ is closed in W . For a Kähler metric ω = Ric(h), the associated Chow norm G · Xˆ  w → wCH(ργ ) ∈ R≥0

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5 Stabilities for Polarized Algebraic Manifolds

attains its minimum at some point g0 · Xˆ in the orbit (where g0 ∈ G). Put K := SU(Vγ , ργ ). By choosing an orthonormal basis {v1 , · · · , vN } for Vγ , we can identify G = SL(Vγ ) with SL(N, C). Then for some k, k1 ∈ K, we can write g0 = k Δ0 k1 , where Δ0 is a positive real diagonal matrix of order N whose α-th diagonal element is aα−1 . Since the Chow norm  CH(ργ ) is K-invariant, k in the above expression of g0 can be chosen to be arbitrary in K. Let 1 ≤ α1 < α2 ≤ N. Then by the ˆ we obtain minimality of the Chow norm at g0 · X, ˆ CH(ργ ) = 0, (d/ds)|s=0  exp(sv) · k Δ0 k1 · X

v ∈ g := sl(N, C).

(5.6)

ˆ so First, by rotating the Kodaira embedding by k1−1 , we may replace k1 · Xˆ by X, that we may assume k1 = 1 from the beginning. Next, let v be the diagonal matrix in g whose α-th diagonal element is δαα1 − δαα2 . Here δαβ is Kronecker’s delta, so that δαβ = 1 or 0, if α = β or α = β, respectively. Then by (1.8) and (5.6) applied to k = 1, we obtain

|zα 1 |2

n ωFS N 2 X α=1 |zα |

= X

|zα 2 |2 n ωFS , N 2 α=1 |zα |

(5.7)

 := dd c log( N |z |2 ). Let {v , · · · , v } be the basis where zα := aα zα and ωFS α=1 α N 1 ), i.e., v := a v , for Vγ associated to the system of coordinates z = t (z1 , · · · , zN α α α α = 1, 2, · · · , N. Then (5.7) is written as

|vα 1 |2

X

N

2 α=1 |vα |

ωFS

n



|vα 2 |2

= X

N

2 α=1 |vα |

ωFS

n

= C,

(5.8)

where C is a positive constant independent of the choice of α1 and α2 . Next, let k2 ∈ K be such that k2 z = t ((k2 z)1 , · · · , (k2 z)N ) for all z ∈ CN , where

(k2 z)α =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

√ (1/ 2)(zα1 − zα2 ), √ (1/ 2)(zα1 + zα2 ),

α = α1 , α = α2 ,

α1 = α = α2 .

zα ,

Moreover, let k3 ∈ K be such that k3 z = ((k3 z)1 , · · · , (k3 z)N ) for all z ∈ CN , where ⎧ √ √ ⎪ α = α1 , ⎪ ⎨ (1/√2)(z√α1 + −1 zα2 ), (k3 z)α = (1/ 2)( −1 zα1 + zα2 ), α = α2 , ⎪ ⎪ ⎩ α = α = α . z , α

1

2

5.1 The Chow Stability

39

Then by a straightforward computation, we obtain −{ |(k2z )α1 |2 − |(k2 z )α2 |2 } +

√ −1 { |(k3 z )α1 |2 − |(k3 z )α2 |2 } = 2zα 1 z¯ α 2 . (5.9)

 N N 2 = 2 c By k2 ∈ K, both N α=1 |(k2 z )α | α=1 |zα | and ωFS = dd log α=1 |(k2 z )α |2 hold, and the same thing is true also for k3 . As we obtain the equality (5.7) from (5.6) applied to k = 1, we obtain the following from (5.6) applied to k = k2 : X

|(k2 z )α1 |2 n ω = N 2 FS α=1 |zα |

X

|(k2 z )α2 |2 n ω . N 2 FS α=1 |zα |

(5.10)

Similarly, by applying (5.6) to k = k3 , we obtain the following: X

|(k3 z )α1 |2 n ω = N 2 FS α=1 |zα |

X

|(k3 z )α2 |2 n ω . N 2 FS α=1 |zα |

(5.11)

Then by (5.9)–(5.11), X

vα 1 v¯α 2 n ω = N |2 FS |v α=1 α

X

zα 1 z¯ α 2 n ω = 0. N |2 FS |z α=1 α

(5.12)

Let u be a local section for L⊗γ . Then |u|2h

FS

|u|2 := N 2 α=1 |vα |

defines a Hermitian metric for L⊗γ , so that h := (h FS )1/γ is a Hermitian metric for L. The corresponding Ricci form   n  ω 1 c 2 ω := Ric(h ) = −dd log h = dd log |vα | = FS γ γ





c

(5.13)

α=1

is a Kähler form on X. As in (2.2), let ρ be the Hermitian structure for Vγ induced by h and ω such that

v1 , v2 

ρ

(v1 , v2 )h ω , n

:= X

v1 , v2 ∈ Vγ .

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5 Stabilities for Polarized Algebraic Manifolds

Put uα := (γ n /C)1/2vα . In view of (5.13), we see from (5.8) and (5.12) that {uα ; α = 1, 2, . . . , N} is an orthonormal basis for (Vγ , ρ ). Then 2 |u1 |2h + · · · + |uN |2h = |u1 |2h + · · · + |uN |2h = (γ n /C)(|v1 |2h + · · · + |vN |h ) FS FS FS FS   2 2 |v | |v | = (γ n /C) N 1 + · · · + N N = γ n /C, 2 2 α=1 |vα | α=1 |vα |

and therefore ω is a balanced metric, as required.

 

C∗

→ G, we consider the real-valued For a special one-parameter group σ : ˆ CH(ργ ) , s ∈ R, where G = SL(V ) for V := Vγ . function fσ (s) := log σ (es ) · X Then by the same argument as in obtaining (2.12) in Theorem 2.2, λσ =

˙

lim f (s), s→−∞ σ

(5.14)

where the left-hand side λσ is the Chow weight for X ⊂ P(Vγ∗ ) for the C∗ -action via σ , i.e., the weight of the C∗ -action via σ on the complex line in W associated to ˆ For the special one-parameter group lim|t |→0 [σ (t)X]. σ −1 : C∗ → G defined by σ −1 (t) := σ (t)−1 = σ (t −1 ), t ∈ C∗ , we consider the real-valued ˆ CH(ργ ) on R. Then function fσ −1 (s) := log σ −1 (es ) · X λσ −1 =

lim f˙

−1 s→−∞ σ

(s),

where λσ −1 is the Chow weight for X ⊂ P(Vγ∗ ) for the C∗ -action via σ −1 . Since fσ −1 (s) = fσ (−s) for all s ∈ R, we obtain λσ −1 = − lim f˙σ (s).

(5.15)

s→+∞

Theorem 5.2 For a special one-parameter group σ : C∗ → G = SL(Vγ ), the orbit σ (C∗ ) · Xˆ is closed in W if and only if both λσ < 0 and λσ −1 < 0 hold. Proof By the second variation formula for the Chow norm (see Sect. 1.4), the realvalued function fσ (s) on R is convex. In view of (5.14) and (5.15), it is easy to check the following equivalence: σ (C∗ ) · Xˆ is closed ⇐⇒ ⇐⇒

lim fσ (s) = +∞ = lim fσ (s)

s→−∞

s→+∞

λσ < 0 and λσ −1 < 0.

 

5.2 The Hilbert Stability

41

Corollary 5.1 X ⊂ P(Vγ∗ ) is Chow polystable if λσ < 0 for every nontrivial special one-parameter group σ : C∗ → G for G = SL(Vγ ). Proof By the Hilbert–Mumford stability criterion as in the proof of Theorem 5.1, this corollary is straightforward from Theorem 5.2 above.   f

slope at −∞ is λσ

slope at +∞ is −λσ −1 f = fσ (s)

s

O

5.2 The Hilbert Stability For an n-dimensional polarized algebraic manifold (X, L), we set Vγ := H 0 (X, L⊗γ ) and Gγ := SL(Vγ ) for a positive integer γ . We further put Vkγ := H 0 (X, L⊗kγ ) for positive integers k. By choosing γ  1, we may assume that the natural maps Ψk,γ : S k (Vγ ) → Vkγ ,

k = 1, 2, . . . ,

are surjective (cf. [61]), where S k (Vγ ) denotes the k-th symmetric tensor product of Vγ . We now put Ik,γ := Ker Ψk,γ and nk,γ := dim Ik,γ . Then k,γ := det Ik,γ = ∧nk,γ Ik,γ is a complex line sitting in ∧nk,γ S k (Vγ ). Let 0 = pk,γ ∈ k,γ . We further consider the Kodaira embedding X ⊂ P(Vγ∗ ) associated to the complete linear system |L⊗γ |.

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5 Stabilities for Polarized Algebraic Manifolds

Definition 5.3 (1) X ⊂ P(Vγ∗ ) is called Hilbert polystable if the orbit Gγ · pk,γ is closed in the vector space ∧nk,γ S k (Vγ ) for k  1. (2) Let X ⊂ P(Vγ∗ ) be Hilbert polystable. Then X ⊂ P(Vγ∗ ) is called Hilbert stable if the isotropy subgroup of Gγ at pk,γ is finite for k  1. For the Hilbert stability and the Chow stability of polarized algebraic manifolds, we now consider their asymptotic versions. Definition 5.4 (1) (X, L) is called asymptotically Hilbert stable if the Kodaira embedding X ⊂ P(Vγ∗ ) is Hilbert stable for γ  1. (2) (X, L) is called asymptotically Chow stable if the Kodaira embedding X ⊂ P(Vγ∗ ) is Chow stable for γ  1. About the relationship between the Hilbert stability and the Chow stability, we have the following [49]: Theorem 5.3 A polarized algebraic manifold (X, L) is asymptotically Hilbert stable if and only if it is asymptotically Chow stable. Proof In the above theorem, the “if” part is a result of Fogarty. For more details, see the book by Mumford et al. [63, p. 215]. For the “only if” part of the proof, we assume that (X, L) is asymptotically Hilbert stable. For the Kodaira embedding Φγ : X → P(Vγ∗ ) associated to the complete linear system |L⊗γ | on X, we consider its image Xγ := Φγ (X) and the degree dγ := degP(Vγ∗ ) Xγ . Let 0 = Xˆ γ ∈ Wγ := (Symdγ Vγ∗ )⊗n+1 be the Chow form for the irreducible reduced algebraic cycle Xγ . Since it is a routine task to show the finiteness of the isotropy subgroup of Gγ := SL(Vγ ) at Xˆ γ for γ  1 from the finiteness of the isotropy subgroup of Gγ at pk,γ for k  1, the proof of the “only if” part is reduced to showing that Gγ · Xˆ γ is closed in Wγ for γ  1. Then by the Hilbert–Mumford stability criterion (as in the proof of Theorem 5.1), it suffices to show that the orbit σ (C∗ ) · Xˆ γ is closed in Wγ for every special one-parameter group σ : C∗ → Gγ , provided that γ  1. In order to show this, given a nontrivial special one-parameter group σ : C∗ → Gγ , we shall define a sequence of special one-parameter groups σα : C∗ → GL(Vγα ),

α = 1, 2, . . . ,

such that each γα is a multiple of γα−1 , where γ0 := γ  1. Let (Xσ , Lσ ) be the test configuration, of exponent γ , for (X, L) associated to the one-parameter group σ above, where π : Xσ → A1 is the natural projection. Put γα := kα γ for a

5.2 The Hilbert Stability

43

positive integer kα written in the form kα =

α

Ki  1,

α = 1, 2, . . . ,

i=1

where all Ki , i = 1, 2, . . . , α, are integers satisfying Ki  1. Then γα = Kα γα−1 , and hence γα is a multiple of γα−1 . For the direct image sheaves ⊗kα

E (α) := π∗ Lσ

,

α = 1, 2, · · · ,

the affirmative solution of the equivariant Serre conjecture for abelian groups gives (α) us a C∗ -equivariant isomorphism E (α) ∼ = E0 ×A1 , so that we have an identification (α) E0(α) ∼ = E1 = Vγα ,

(5.16) ⊗k

by an argument as in Sect. 1.1. Since Eα := H 0 ((Xσ )0 , (Lσ α )0 ) and E0(α) are C∗ -equivariantly identified, in view of (5.16), the natural C∗ -action on E0(α) induces a special one-parameter group σα : C∗ → GL(Eα ) (= GL(Vγα )), so that we can view (Xσ , Lσ⊗kα ) as the test configuration (Xσα , Lσα ) associated to σα . As in the proof of Theorem 2.2, by Mumford [62, Proposition 2.11], the weight wγα of the C∗ -action on det Eα via σα is wγα = −

λσ k n+1 + O(kαn ), (n + 1)! α

where λσ is the Chow weight of X ⊂ P(Vγ∗ ) for the C∗ -action via σ . Since kα tends to +∞ as α → ∞, and since we can write Nγα := dim Vγα as a polynomial of γα by the Riemann–Roch theorem, it follows from the equality γα = kα γ that λσ − (n+1)! kαn+1 + O(kαn ) wγα # $ lim = lim n α→∞ γα Nγα α→∞ kα γ c1 (L)n! [X] (kα γ )n + O((kα γ )n−1 )

= −

λσ n+1 (n + 1)γ c

1 (L)

n [X]

.

Hence by Corollary 5.1, it suffices to show the positivity of the limit on the left-hand side for every special one-parameter group σ : C∗ → G.

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5 Stabilities for Polarized Algebraic Manifolds

In order to show such positivity, by setting qα := wγα /(γα Nγα ), we claim that the sequence {qα } is monotone-increasing, i.e., qα−1 < qα ,

α = 1, 2, · · · .

(5.17)

From now on, Kα will be written simply as κ. Then we can write γα as κγα−1. Let ψα : S κ (Eα−1 ) → Eα be the natural map on the special fiber (over the origin). Then by γα−1  1, in view of [61] and [62], ψα is surjective. Moreover, by (5.16), Eα−1 and Eα are viewed as Vγα−1 and Vγα , respectively. Put ια := Ker ψα . Then the weight Hα , which we call the Hilbert weight, of the C∗ -action on det ια is (see [20]) Hα = {C∗ -weight on det S κ (Eα−1 )} − (C∗ -weight on det Eα ) =

κ wγα−1 Nγα − wγα . Nγα−1

In view of γα−1  1, the closeness of the C∗ -orbit σα (C∗ ) · pκ,γα−1 in Wγα implies that the Hilbert weight Hα is negative, i.e., 0 >

κ wγα−1 Nγα − wγα . Nγα−1

Since γα = κγα−1 , dividing this inequality by γα Nγα , we obtain (5.17). Note that q0 = wγ /(γ Nγ ) = 0, since we start from σ : C∗ → G = SL(Vγ ). Hence 0 = q0 < q1 < · · · < qα < · · · , and we obtain limα→∞ qα > 0, as required.

 

5.3 K-stability The concept of K-stability was introduced by Tian [79] in his study of Kähler– Einstein metrics on Fano manifolds. Later, by Donaldson [20], this concept was reformulated in a general setting by an algebraic geometric language. We consider here an n-dimensional polarized algebraic manifold (X, L). A test configuration (X , L ) for (X, L) is called normal if X is a normal complex variety. Definition 5.5 (1) (X, L) is called K-semistable if DF1 (X , L ) ≤ 0 for every normal test configuration (X , L ) for (X, L). (2) (X, L) is called K-polystable if (X, L) is K-semistable and furthermore, every normal test configuration (X , L ) satisfying DF1 (X , L ) = 0 is a product configuration.

5.4 Relative Stability

45

(3) (X, L) is called K-stable if (X, L) is K-semistable and furthermore, every normal test configuration (X , L ) satisfying DF1 (X , L ) = 0 is trivial. Definition 5.6 (1) (X, L) is called uniformly K-stable if every normal test configuration μ = (X , L ) for (X, L) satisfies DF1 (μ) ≤ −Cμasymp for some positive constant C independent of the choice of μ. (2) (X, L) is called strongly K-stable if F1 ({μj }) < 0 for all {μj } ∈ M , where F1 ({μj }) and M are as in Sect. 4.5.

5.4 Relative Stability As a reference for this section, see [76] (cf. also [55]). For a polarized algebraic manifold (X, L), if dim H 0 (X, O(T X)) = 0, then for the Kähler class c1 (L) on X to admit a special metric such as a CSC Kähler metric or an extremal Kähler metric, the stability concept should be modified suitably by choosing a smaller reductive algebraic group. In this section, we consider a possibly trivial algebraic torus T sitting in Aut0 (X). Then for every positive integer γ , we consider the reductive algebraic subgroup Tγ⊥ , as in Sect. 4.3, of the full special linear group Gγ := SL(Vγ ). As in Sect. 5.1, we consider the abstract Kodaira embedding Φγ : X → P(Vγ∗ ) associated to |L⊗γ |. Let Xγ := Φγ (X) be its image. As in the proof of Theorem 5.3, let 0 = Xˆ γ ∈ Wγ := (Symdγ Vγ∗ )⊗n+1 be the Chow form for the cycle Xγ on P∗ (Vγ ). Definition 5.7 (1) X ⊂ P(Vγ∗ ) is called Chow polystable relative to T if the orbit Tγ⊥ · Xˆ γ is closed in Wγ . (2) X ⊂ P(Vγ∗ ) is called Chow stable relative to T if X ⊂ P(Vγ∗ ) is Chow polystable relative to T and in addition, the isotropy subgroup of Tγ⊥ at Xˆ γ is finite. (3) (X, L) is called asymptotically Chow polystable relative to T if X ⊂ P(Vγ∗ ) is Chow polystable relative to T for γ  1. (4) (X, L) is called asymptotically Chow stable relative to T if X ⊂ P(Vγ∗ ) is Chow stable relative to T for γ  1. For each σ ∈ 1PS(Tγ⊥ ), let u ∈ t⊥ γ be its fundamental generator. As in Sect. 2.2, we have the test configuration μσ = (Xσ , Lσ ) associated to σ . For positive integers m, we put γm := mγ . As in Sect. 4.2, u induces um ∈ sl(H 0 ((Xσ )0 , (Lσ⊗m )0 )).

46

5 Stabilities for Polarized Algebraic Manifolds

Let vm be as in (4.3), where v ∈ tR is chosen arbitrarily. Then by Székelyhidi [76], the pairing , γm in (4.2) converges to a limit pairing , ∞ on zγ × t such that

um , vm γm → u, v∞ ,

as m → ∞.

Let wi , i = 1, . . . , r, be elements in t with real eigenvalues such that {w1 , · · · , wr } is an orthonormal basis for t (= tγ ) with respect to , ∞ . Then we put u := u −

r 

u, wi ∞ wi , i=1

where u is orthogonal to t in terms of the bilinear pairing , ∞ . We now define the relative Donaldson-Futaki invariant DFT1 (cf. [76]) for (Xσ , Lσ ) by DFT1 (Xσ , Lσ ) := DF1 (u ). Definition 5.8 (1) (X, L) is called K-semistable relative to T if for every positive integer γ , DFT1 (Xσ , Lσ ) ≤ 0 for all σ ∈ 1PS(Tγ⊥ ) such that Xσ is normal. (2) (X, L) is called K-stable relative to T if for every positive integer γ , the inequality DFT1 (Xσ , Lσ ) < 0 holds for all σ ∈ 1PS(Tγ⊥ ) as long as Xσ is normal. (3) (X, L) is called K-polystable relative to T if for every positive integer γ and for every σ ∈ 1PS(Tγ⊥ ), the inequality DFT1 (Xσ , Lσ ) ≤ 0 holds and, as long as Xσ is normal, the equality holds only when (Xσ , Lσ ) is a product configuration. Definition 5.9 (1) (X, L) is called uniformly K-stable relative to T if for all positive integers γ and all σ ∈ 1PS(Tγ⊥ ) such that Xσ is a normal complex variety, there exists a positive constant C independent of the choice of γ and σ such that the inequality DFT1 (μσ ) ≤ − Cμσ Tasymp holds for μσ = (Xσ , Lσ ). (2) (X, L) is strongly K-semistable relative to T if F1 ({μj }) ≤ 0 for all {μj } ∈ MT . (3) (X, L) is strongly K-stable relative to T if F1 ({μj }) < 0 for all {μj } ∈ MT . If the algebraic torus T is trivial, then all relative stabilities reduce to the ordinary stabilities without the term “relative to T ”. We now compare strong K-stability with uniform K-stability by assuming that T is trivial for simplicity, though a similar argument goes through also for a relative version. Theorem 5.4 If (X, L) is strongly K-stable, then (X, L) is uniformly K-stable.

5.4 Relative Stability

47

Proof By choosing a Hermitian metric h for L such that ω := Ric(h) is Kähler, we can talk about special one-parameter groups. Let (X, L) be strongly K-stable. Since Xσ is assumed to be normal for σ above, by the nontriviality of σ , we have the inequality μσ asymp > 0. In general, for every nontrivial normal test configuration μ = (X , L ) for (X, L), we put κ(μ) :=

DF1 (μ) . μasymp

Since the uniform stability means that κ(σ ) is bounded from above by −C, we assume for contradiction that there exist positive integers γj , j = 1, 2, · · · , and nontrivial special one-parameter groups σj : C∗ → SL(Vγj ) satisfying the following: (1) κ(μ1 ) ≤ · · · ≤ κ(μj ) ≤ κ(μj +1 ) ≤ · · · , (2) limj →∞ κ(μj ) ≥ 0, where μj is the test configuration (Xσj , Lσj ) associated to σj , and the limit in (2) can possibly be +∞. Then for the test configuration μj,m := (Xj , Lj⊗m ),

m  1,

the C∗ -action on the central fiber via σj induces σj,m : C∗ → GL(Ej,m ), where the space Ej,m := H 0 ((Xj )0 , (Lj )⊗m 0 )) is viewed as Vmγj . As in Sect. 2.5, we have SL σj,m : R+ → SL(Em ). SL . Then by setting t = Let uj,m ∈ sl(Vmγj ) be the fundamental generator for σj,m exp(s/|uj,m |∞ ), we define a real-valued function fj,m (s) by

fj,m (s) :=

|uj,m |∞ SL ˆ CH (ρmγ ) , (mγj )−n σj,m (t) · X j |uj,m |1

s ∈ R,

∗ ) by the where Xˆ is the Chow form for the algebraic cycle X sitting in P(Vmγ j Kodaira embedding via the complete linear system |L⊗mγj |. Then by Theorem 2.2, ∞

 (n + 1)! a0 lim f˙j,m (s) = (mγj )−n DFα (μj ) mn+1−α s→−∞ |uj,m |1 α=1

=

(n + 1)! a0 −n γj |uj,m |1

∞  α=1

DFα (μj ) m1−α ,

(5.18)

48

5 Stabilities for Polarized Algebraic Manifolds

where m  1 and a0 := γjn c1 (L)n [X]/n!. Let wj,m be the weight of the C∗ -action on det Ej,m via σˆ j,m . Put Nj,m := dim Ej,m . For m  1, wj,m and Nj,m are written as wj,m = b0 mn+1 + b1 mn + · · · + bn m + bn+1 , Nj,m =

a0 mn + a1 mn−1 + · · · + an ,

so that dividing wj,m by mNj,m , we obtain wj,m b0 mn+1 + b1 mn + · · · + bn m + bn+1 = n+1 n a0 m + a1 m + · · · + an m mNj,m =

∞ 

DFα (μj )m

−α

α=0

∞

b0  = + DFα (μj )m−α . a0

(5.19)

α=1

 α function of z in a neighborhood of By (5.19), ∞ α=1 DFα (μj )z is a holomorphic ∞ α−1 is holomorphic in z in a the origin with a zero at z = 0. Hence DF α (μj )z α=1 neighborhood of the origin. Hence, since μj asymp = limm→∞ |uj,m |1 , it follows from (5.18) above that  lim

m→∞

lims→−∞ f˙j,m (s) (n + 1)! a0

 =

DF1 (μj ) = κ(μj ). μj asymp

(5.20)

Take a sequence of positive real numbers εj , j = 1, 2, · · · , such that εj → 0 as j → ∞. Then by (5.20), we may choose an increasing sequence of positive integers mj , j = 1, 2, · · · , satisfying both mj  1 and limj →∞ mj = +∞ such that % % % % lims→−∞ f˙j,m (s) % % j − κ(μj )% ≤ εj , % % % (n + 1)! a0

j = 1, 2, · · · .

In view of (2) above, by setting κ∞ := limj →∞ κ(μj ) ∈ R≥0 ∪ {+∞}, we have lim f˙j,mj (s) → κ∞ (n + 1)! a0 ,

s→−∞

as j → ∞.

(5.21)

By convexity of the function fj,mj (s), its derivative f˙j,mj (s) is non-decreasing in s. Hence the following inequality holds for all s ∈ R: lim f˙j,mj (s) ≤ f˙j,mj (s).

s→−∞

(5.22)

5.4 Relative Stability

49

For special one-parameter groups σj,mj : C∗ → GL(Vmγj ), j = 1, 2, · · · , by identifying the spaces Ej,mj with Vmγj , we obtain test configurations μj,mj = (Xσj,mj , Lσj,mj ),

j = 1, 2, · · · .

SL Then the special linearization σj,m is not necessarily an algebraic group homomorj ∗ ˜ ∗, phism from C to SL(Vmγj ), but by choosing a suitable covering algebraic torus C the corresponding lift, denoted by σ˜ j , defines an algebraic group homomorphism

˜ ∗ → SL(Vmγj ). σ˜ j : C Then by setting μ˜ j := (Xσ˜ j , Lσ˜ j ), we see from Remark 4.5 that F1 ({μ˜ j }) =

lim

lim f˙j,mj (s).

s→−∞ j →∞

It then follows from (5.21) and (5.22) that  0 ≤ κ∞ (n + 1)! a0 = lim

j →∞

lim f˙j,mj (s)



s→−∞

≤ lim f˙j,mj (s), j →∞

s ∈ R.

Let s → −∞. Then 0 ≤ lims→−∞ limj →∞ f˙j,mj (s) = F1 ({μ˜ j }) in contradiction to the strong K-stability of (X, L), as required.   Theorem 5.5 (cf. [55]) If (X, L) is strongly K-stable relative to T , then (X, L) is asymptotically Chow stable relative to T . Proof For each positive integer γ , let Xˆ γ denote the Chow form for the image cycle of the twisted Kodaira embedding X ⊂ P(Vγ∗ ). Assume that (X, L) is strongly Kstable relative to T . By the notation in Sect. 4.3, it suffices to show the following: (1) (X, L) is asymptotically Chow polystable relative to T . (2) The isotropy subgroup of Tγ⊥ at Xˆ γ is finite for γ  1. In order to prove (1), we assume for contradiction that (X, L) is not asymptotically Chow polystable relative to T . Then there exists a sequence 1 ≤ γ1 < γ2 < · · · < γj < · · · of integers γj such that X ⊂ P(Vγ∗j ) is not Chow polystable relative to T . Then by Corollary 5.1, there exists a σj ∈ 1PS(Tγ⊥j ) for each j such that the Chow weight λσj := lim

r→−∞

d{log σj (er ) · Xˆ j CH(ργj ) } dr

(5.23)

50

5 Stabilities for Polarized Algebraic Manifolds

satisfies λσj ≥ 0. Put Xˆ j := Xˆ γj for simplicity. Consider the test configurations μj := (Xσj , Lσj ),

j = 1, 2, · · · .

Let uj = 0 be the fundamental generator for σj . Then for t := er , by setting s = |uj |∞ r, we obtain t = exp(s/|uj |∞ ). Then for fj in (4.4), we have f˙j (s) =

|uj |∞ −n d{log σj (t) · Xˆ j CH(ργj ) } dr · , γ |uj |1 j dr ds

(5.24)

and let s → −∞. Then r → −∞, and by (5.23) and (5.24), lim f˙j (s) =

s→−∞

|uj |∞ −n 1 dr = γj λσj γ −n λσj ≥ 0. |uj |1 ds |uj |1 j

By the convexity of the function fj (s), its derivative f˙j (s) is a non-decreasing function of s. Hence for each s ∈ R, 0 ≤

lim f˙j (s) ≤ f˙j (s).

s→−∞

In this inequality, let j → ∞. Then for each s ∈ R, 0 ≤ lim f˙j (s). j →∞

We further let s → −∞. Then we obtain F1 ({μj }) =

lim

lim f˙j (s) ≥ 0,

s→−∞ j →∞

in contradiction to the strong K-stability of the polarized algebraic manifold (X, L) relative to T . This finishes the proof of (1). In order to prove (2), we assume for contradiction that there exists an increasing sequence of positive integers 1  γ1 < γ2 < · · · < γj < · · · such that the isotropy subgroup Hj of Tγ⊥j at Xˆ j satisfies dim Hj > 0 for all j . By (1), the orbit Oj := Tγ⊥j · Xˆ j is closed in Wj := Wγj . Hence Oj is affine (and is a Stein space). Since Tγ⊥j is reductive, a theorem of Matsushima [59] asserts that Hj is reductive. By the positivity of dim Hj , the group Hj contains C∗ as a subgroup, i.e., G j (∼ = C∗ ) ⊂ H j .

5.4 Relative Stability

51

For a fixed maximal algebraic torus T in Aut0 (X) satisfying T ⊂ T , let Tc be the maximal compact subgroup of T . We choose a reference metric ω = Ric(h) (by which we endow Vγj with the Hermitian structure ργj ) in such a way that both h and ω are invariant under the action of Tc . Let P be the maximal connected linear algebraic subgroup of Aut0 (X), and let P be the identity component of the isotropy subgroup of SL(Vγj ) at [Xˆ j ] ∈ P(Wj ). Since 0 ∈ / Oj , the affine subset Oj ∩ CXˆ j ˆ of the affine line CXj is a finite set. Hence we have a natural isogeny ι : P → P. Let Tj be the identity component of ι−1 (T ). Then Tj is an algebraic torus containing Tγj as an algebraic subtorus. Note that Zγj is the centralizer of Tγj in SL(Vγj ). Clearly, both Gj and Tj sit in the identity component, denoted by Q, of Zγj ∩ P . Since Tj is a maximal algebraic torus in in P , it is also a maximal algebraic torus in Q. Hence there exists an element g0 in Q such that g0 Gj g0−1 ⊂ Tj .

(5.25)

Then by g0 [Xˆ j ] = [Xˆ j ], there exists a λ0 ∈ C∗ such that g0 Xˆ j = λ0 Xˆ j , i.e., λ−1 Xˆ j = g −1 Xˆ j . If g ∈ Gj , then g Xˆ j = Xˆ j , so that 0

0

−1 −1 ˆ ˆ ˆ ˆ g0 gg0−1 Xˆ j = g0 g(λ−1 0 Xj ) = λ0 (g0 g Xj ) = λ0 (g0 Xj ) = Xj .

Hence g0 Gj g0−1 fixes Xˆ j . Let 0 = u be a basis for the one-dimensional Lie algebra gj := Lie(Gj ). Then for all v ∈ tγj ,

g0 ug0−1 , vγj = γj−n−2 Tr((g0 ug0−1 )v) = γj−n−2 Tr(ug0−1 vg0 ). Now by g0 ∈ Q ⊂ Zγj and v ∈ tγj , we have g0−1 vg0 = v, so that the equality g0 ug0−1 v = g0 uvg0−1 holds. Hence

g0 ug0−1 , vγj = γj−n−2 Tr(uv) = 0,

(5.26)

where the last equality follows from u ∈ t⊥ γj . It then follows from (5.26) that g0 Gj g0−1 ⊂ Tγ⊥j . Since g0 Gj g0−1 fixes Xˆ j , we obtain g0 Gj g0−1 ⊂ Hj . From this together with (5.25), we obtain a special one-parameter group σj ∈ 1PS(Tγ⊥j ) such that σj (C∗ ) = g0 Gj g0−1 ⊂ Hj .

52

5 Stabilities for Polarized Algebraic Manifolds

It then also follows that σj−1 ∈ 1PS(Tγ⊥j ) and σj−1 (C∗ ) ⊂ Hj . We now consider the following sequences of test configurations: μj =

(Xσj , Lσj ),

νj = (Xσ −1 , Lσ −1 ), j

j

j = 1, 2, · · · , j = 1, 2, · · · .

In view of (5.25), it is easily seen that F1 ({μj }) = −F1 ({νj }).

(5.27)

On the other hand, since (X, L) is strongly K-stable relative to T , we have both F1 ({μj }) < 0

and

F1 ({νj }) < 0,  

in contradiction to (5.27), as required.

Problems 5.1 Let μ := (X , L ) be a nontrivial normal test configuration, of exponent γ , for a polarized algebraic manifold (X, L). Consider the test configurations μj = (X , L ⊗j ),

j = 1, 2, · · · ,

(5.28)

of exponent γj := j γ , for (X, L). Then by Sect. 2.2, each μj is a test configuration μσj associated to a special one-parameter group σj : C∗ → GL(Vγj ). Hence we have F1 ({μj }) ∈ R ∪ {−∞} as in Remark 4.5. Assume that the inequality F1 ({μj }) ≥

(n + 1)c1 (L)n [X] DF1 (μ) μasymp

(5.29)

always holds. Show that, if (X, L) is strongly K-stable, then (X, L) is K-stable. 5.2 Let μj , j = 1, 2, · · · , be a sequence of test configurations μσj of exponent γj for (X, L) associated to special one-parameter groups σj : C∗ → SL(Vγj ),

j = 1, 2, · · · .

such that F1 ({μj }) = 0 (cf. Remark 4.6). Assuming that (X, L) is strongly K-stable, show that there exists a positive integer N such that σj are trivial for all j ≥ N.

Chapter 6

The Yau–Tian–Donaldson Conjecture

Abstract In this chapter, we discuss the Yau–Tian–Donaldson conjecture from a historical point of view. • In Sect. 6.1, we briefly discuss the Calabi conjecture. The unsolved case of the Calabi conjecture motivates the Yau–Tian–Donaldson conjecture in Kähler– Einstein cases. • As mentioned in Sect. 6.2, the Yau–Tian–Donaldson conjecture in Kähler– Einstein cases was solved affirmatively by Chen, Donaldson and Sun and by Tian. • In Sect. 6.3, we define K-energy and modified K-energy for compact Kähler manifolds. This concept allows us to state the recent results of Chen and Cheng and of He on the existence of CSC Kähler metrics and extremal Kähler metrics. • Finally, in Sect. 6.4, various versions of the Yau–Tian–Donaldson conjecture will be considered in extremal Kähler cases. Keywords The Calabi conjecture · The Yau–Tian–Donaldson conjecture · The modified K-energy

6.1 The Calabi Conjecture For a compact complex connected manifold X, we view c1 (X) as a Dolbeault cohomology class on X. The following conjecture posed by Calabi [6, 7] in the 1950s on the existence of Kähler–Einstein metrics is the Calabi conjecture: 1. If c1 (X) < 0, then X admits a unique Kähler–Einstein form ω in the canonical class −c1 (X) such that Ric(ω) = −ω. 2. If c1 (X) = 0, then for every Kähler class K on X, there exists a unique Kähler– Einstein form ω ∈ K such that Ric(ω) = 0. 3. More generally, for every d-closed (1, 1)-form η in the class c1 (X), every Kähler class K on X admits a unique Kähler form ω such that Ric(ω) = η. It is well-known that the Calabi conjecture was solved affirmatively in the 1970s. Yau [86, 87] gave an affirmative answer to the whole conjecture by solving complex © The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0_6

53

54

6 The Yau–Tian–Donaldson Conjecture

Monge–Ampère equations using the continuity method, whereas (1) was proved independently also by Aubin [1]. These works represent a landmark in differential geometric studies of algebraic geometry. In the case where c1 (X) > 0, as advocated by Yau [88], new developments are made from the viewpoints of stability of the underlying manifolds. Related to the existence problem of Kähler–Einstein metrics on Fano manifolds, Tian [79] introduced the concept of K-stability, while Donaldson [20] reformulated the concept in a general setting by purely algebraic geometric languages.

6.2 The Yau–Tian–Donaldson Conjecture A compact complex connected manifold X is called Fano if its first Chern class c1 (X) is positive. As to the existence of Kähler–Einstein metrics on Fano manifolds, the following is known as the Yau–Tian–Donaldson conjecture: Conjecture 6.1 A Fano manifold X admits a Kähler–Einstein metric if and only if the polarized algebraic manifold (X, KX−1 ) is K-polystable. Recently, this conjecture was solved affirmatively by Chen et al. [13–15] and Tian [80] through the use of cone singularities applying the Cheeger–Colding–Tian theory. However, for CSC Kähler metrics and extremal Kähler metrics, the problem of characterizing the existence of such metrics by stability of underlying manifolds is still open in spite of various partial results.

6.3 The K-Energy Let K be a Kähler class on a compact complex connected manifold X. Fix a reference Kähler form ω0 ∈ K . Then ϕt , 0 ≤ t ≤ 1, is called a piecewise smooth path in C ∞ (X)R if the mapping [0, 1] × X → R,

(t, x) → ϕt (x),

is continuous and moreover there exists a partition 0 = a0 < a1 < · · · < ar = 1 of the unit interval [0, 1] such that the restrictions [ai−1 , ai ] × X → R,

(t, x) → ϕt (x),

 are C ∞ , i = 1, 2, · · · , r. Put V := X ωn . Every ω in K is expressible as ωϕ for some ϕ ∈ C ∞ (X)R , where ωϕ is the Kähler form in the class K defined by √ ωϕ := ω0 + dd ϕ = ω0 + c

−1 ¯ ∂ ∂ϕ. 2π

6.3 The K-Energy

55

Let ϕt , 0 ≤ t ≤ 1, be a piecewise smooth path in C ∞ (X)R satisfying ϕ0 = 0, ϕ1 = ϕ and ωϕt ∈ K . Put ϕ˙t = ∂ϕt /∂t. We then define the K-energy κ : K → R for K by 1 κ(ω) := − V

1 

0

 X

ϕ˙ t (S(ωϕt )

− S0 ) ωϕnt

(6.1)

dt,

where S(ωϕt ) is the scalar curvature of ωϕt , and   S0 is the average of the scalar curvature of ω0 defined by S0 := X S(ω0 )ω0n / X ω0n . Then (cf. [42]): Lemma 6.1 The right-hand side of (6.1) depends only on ω, and is independent of the choice of the path {ϕt }0≤t ≤1. Proof Define ψ = ψ(s, t) by setting ψ := sϕt for (s, t) ∈ [0, 1] × [0, 1]. Let Ψ = Ψ (s, t) be the 1-form on [0, 1] × [0, 1] defined by    n n ds − dt ψs (S(ωψ ) − S0 )ωψ ψt (S(ωψ ) − S0 )ωψ

 Ψ :=−

X

= −n X

X

n−1 (ψs ds + ψt dt){Ric(ωψ ) − λωψ } ∧ ωψ ,

∂ψ n c where ψs := ∂ψ ∂s , ψt := ∂t and λ := S0 /n. Then by Ric(ωψ ) = −dd log ωψ , c c we have (∂/∂s) Ric(ωψ ) = −dd (Δψ ψs ) and (∂/∂t) Ric(ωψ ) = −dd (Δψ ψt ). Hence & ' n−1 dΨ = −n (ψs ds + ψt dt) dd c {(Δψ + λ)ψs }ds + dd c {(Δψ + λ)ψt }dt ∧ ωψ X

−n

(ψs ds + ψt dt){Ric(ωψ ) − λωψ } ∧ θ, X

n−2 where θ := (n − 1)ωψ {ds(dd c ψs ) + dt (dd c ψt )}. The second integral on the right-hand side is obviously zero, while the first integral is

− ds ∧ dt X

(

) n ψs Δψ (Δψ + λ)ψt − ψt Δψ (Δψ + λ)ψs ωψ = 0.

Hence dΨ = 0. For the 2-chain  = [0, 1] × [0, 1] in the plane R2 = {(s, t)}, we consider its boundary ∂ = σ1 + σ2 − σ3 − σ4 , where σ1 = {(s, 0); 0 ≤ s ≤ 1}, σ2 = {(1, t); 0 ≤ t ≤ 1}, σ3 = {(s, 1); 0 ≤ s ≤ 1}, σ4 = {(0, t); 0 ≤ t ≤ 1}. It now follows from dΨ = 0 and σ4 Ψ = 0 that 0 =





dΨ =

Ψ = ∂

σ1 +σ2 −σ3 −σ4



Ψ =

Ψ− σ2

σ3 −σ1

Ψ.

(6.2)

56

6 The Yau–Tian–Donaldson Conjecture

Since ϕ0 = 0, we have −

1 V

1 

0

 σ1

Ψ = 0. Hence by (6.2),

 σ2

Ψ =

 σ3

Ψ , i.e.,

   1 1 n ds. ϕ˙ t (S(ωϕt ) − S0 )ωϕnt dt = − ϕ{S(ωsϕ ) − S0 }ωsϕ V 0 X X

Thus, (6.1) is independent of the choice of the path {ϕt }0≤t ≤1, as required.

 

For ω = ωϕ above, κ(ω) is written also as κ(ϕ). By setting ωt := ωϕt in (6.1), we now define the modified K-energy ([28, 73]; see also [46]) as follows: 1 κ(ϕ) ˆ := − V

1 

0

 ϕ˙t (S(ωt ) − S0 − X

Hωt ) ωtn

dt,

where the Hamiltonian function Hωt on (X, ωt ) associated to the extremal vector field yωt as in Sect. 9.2 is defined by gradC ωt

Hωt = yωt = yω0

and X

Hωt ωtn = 0.

Here the condition yωt = yω0 corresponds to the condition that ϕt is a function invariant under the S 1 -action generated by the real vector field yωR0 := yω0 + y¯ω0 . By recent works of Chen and Cheng [10–12] and He [29], the existence of CSC Kähler metrics and extremal Kähler metrics is characterized by the properness of the modified K-energy modulo a suitable subgroup of Aut0 (X) (see Sect. 8.1 in CSC Kähler cases; see also Theorem 9.7, Sect. 9.6, in extremal Kähler cases).

6.4 Extremal Kähler Versions of the Conjecture For a polarized algebraic manifold (X, L), by choosing a maximal compact subgroup K of Aut0 (X), we view its complexification K C as a connected reductive algebraic subgroup of Aut0 (X). Let Z be the central algebraic torus in K C , and we also consider a maximal algebraic torus Tmax in K C . Let T be either Tmax or Z. Existence Problem of Extremal Kähler Metrics Find a necessary and sufficient stability condition for X to admit an extremal Kähler metric in the class c1 (L). As an extremal Kähler version of the Yau–Tian–Donaldson conjecture, we expect that the required stability condition is one of the following: K-polystability, uniform K-stability, strong K-stability relative to either Tmax or Z. It could also occur that these stability conditions partially coincide. Existence and stability theorems for extremal Kähler metrics will be discussed in the next two chapters.

Problems

57

Problems 6.1 For a polarized algebraic manifold (X, L), let T and T be algebraic tori in Aut0 (X) such that T ⊂ T . Show that, if (X, L) is uniformly K-stable relative to T , then (X, L) is uniformly K-stable relative to T . 6.2 For (X, L) and Tmax in Sect. 6.4, let EEX be the set of all extremal Kähler metrics in the class c1 (L), and ECSC the set of all constant scalar curvature Kähler metrics in the class c1 (L). Consider the following conjectures: EEX = ∅ ⇐⇒ (X, L) is uniformly K-stable relative to Tmax ,

(1)

ECSC = ∅ ⇐⇒ F = 0 and (X, L) is uniformly K-stable relative to Tmax , (2) where F is the Futaki character as in Sect. 3.1. Show that, if Conjecture (1) is true, then Conjecture (2) is true.

Chapter 7

Stability Theorem

Abstract In this chapter, the existence of CSC Kähler metrics or extremal Kähler metrics on a polarized algebraic manifold implies various kinds of stability. • In Sect. 7.1, we shall show that the existence of a CSC Kähler metric implies strong K-semistability of the polarized algebraic manifolds. • In Sect. 7.2, we introduce the concept of relative balanced metrics. The existence of such a metric corresponds to relative Chow polystability. • In Sect. 7.3, we shall show that the existence of an extremal Kähler metric implies strong K-semistability of the polarized algebraic manifolds. • In Sect. 7.4, a result of Stoppa and Székelyhidi asserts that a polarized algebraic manifold (X, L) with an extremal Kähler metric in c1 (L) is K-polystable relative to a maximal algebraic torus Tmax in Aut0 (X). • Finally, in Sect. 7.5, we have an inequality by switching the order of the double limit in the definition of the invariant F1 ({μj }). Keywords Stability for extremal Kähler manifolds · Relative balanced metrics

7.1 Strong K-Semistability of CSC Kähler Manifolds For a polarized algebraic manifold (X, L), assume that the Kähler class c1 (L) admits a CSC Kähler form ω. Let h be a Hermitian metric on L such that Ric(h) = ω. We then consider a sequence {μj } of test configurations for (X, L) such that μj = (Xσj , Lσj ),

j = 1, 2, · · · ,

where σj : C∗ → SL(Vγj ), j = 1, 2, · · · , are nontrivial special one-parameter groups satisfying the condition γj → +∞,

as j → ∞.

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0_7

59

60

7 Stability Theorem

Let uj be the fundamental generator of σj . We then consider the functions |uj |∞ −n γ log σj (t) · Xˆ j CH(ργj ) , |uj |1 j

fj (s) =

j = 1, 2, · · · ,

as in (4.4), where t := exp(s/|uj |∞ ) for s ∈ R. As in the proof of Claim in Sect. 4.5, let −βk , k = 1, 2, · · · , Nj , be the weights of the C∗ -action on (Vγj , ργj ), and for its suitable orthonormal basis {v1 , · · · , vNj }, we can write σj (t) · vk = t −βk vk ,

k = 1, 2, · · · , Nj ,

such that β1 + · · · + βNj = 0. Here by (4.6), Bj (ω) = Cj + O(γj−2 ),

(7.1)

where for j  1, we have Cj := 1 + (1/2)Sω γj−1 > 0. In our case, since the extremal vector field y (= yγ ) is zero, Bj# (ω) in (4.7) is just Bj (ω), and we adapt the proof of Claim in Sect. 4.5 to our case by setting vk# = vk for all k. Then by (4.8) and (7.1), (n + 1)! −n f˙j (0) = γ |uj |1 j =

(n + 1)! −n γ Cj |uj |1 j

Nj

2 α=1 βk |vk |h {ω −2 X Cj + O(γj )

(

Nj 

X α=1

+ O(γj−2 )}n

βk |vk |2h ){ωn + O(γj−2 )},

where the right-hand side just above is ⎧ Nj ⎨

(n + 1)! −n γ Cj |uj |1 j ⎩

k=1

βk +

Nj 

|βk |O(γj−2 )

k=1

⎫ ⎬ ⎭

=

Nj

k=1 |βk |) n+2 γj |uj |1

O(

= O(γj−1 ).

Hence by limj →∞ γj−1 = 0, we have limj →∞ f˙j (0) = 0. Since limj →∞ f˙j (s) is a non-decreasing function of s, we finally obtain F1 ({μj }) = i.e., (X, L) is strongly K-semistable.

lim

lim f˙(s) ≤ 0,

s→−∞ j →∞

7.2 Relative Balanced Metrics

61

7.2 Relative Balanced Metrics For a polarized algebraic manifold (X, L), let T be an algebraic torus in Aut0 (X). Let γ be a positive integer. By using the notation in Sect. 4.3, let (tγ )c √ denote the Lie algebra of the maximal compact subgroup (Tγ )c of Tγ . Put (tγ )R := −1(tγ )c . Recall that the space Vγ = H 0 (X, L⊗γ ) is written as a direct sum (see Sect. 4.3) Vγ =

nγ 

Vγ ,i ,

i=1

where Vγ ,i = {v ∈ Vγ ; θ σ = χγ ,i (θ )σ for all θ ∈ tγ }. For a Kähler metric ω in the class c1 (L), we choose a Hermitian metric h for L such that Ric(h) = ω. As in (2.2), we have a natural Hermitian structure ργ on Vγ . The concept of balanced metrics introduced in Chap. 5 plays a very important role in the study of CSC Kähler metrics. A similar concept exists in the study of extremal Kähler metrics. Actually, we have a relative version of balanced metrics: Definition 7.1 ω is called a balanced metric for L⊗γ relative to T if there exist a θγ ∈ (tγ )R and a positive constant C such that 1 − χγ ,i (θγ ) > 0 for all i and that nγ qi  

(1 − χγ ,i (θγ ))|vi,α |h2 = C,

i=1 α=1

where {vi,α ; α = 1, 2, · · · , qi , i = 1, 2, · · · nγ } is an orthonormal basis for (Vγ , ργ ) such that each {vi,α ; α = 1, · · · , qi }, i = 1, · · · , nγ , is a basis for Vγ ,i . As in Sect. 5.1, let X ⊂ P(Vγ∗ ) be an abstractly defined Kodaira embedding associated to the complete linear system |L⊗γ | on X. As a relative version of Theorem 5.1, we have the following facts (see for instance [48, 50, 54] for more details): Fact 1 X ⊂ P(Vγ∗ ) is Chow polystable relative to T if and only if there exists a balanced metric ω for L⊗γ relative to T . √ # |2 is written Fact 2 χγ ,i (θγ ) = γ −2 χγ ,i ( −1yγ ) + O(γ −2 ), and in Sect. 4.4, |vi,α −2 2 −2 as {1 − χγ ,i (θγ ) + O(γ )}|vi,α | , where O(γ ) is a quantity whose absolute value is bounded from above by Cγ −2 for some positive constant C independent of i, γ , α. √ √ Put (yγ )R := −1yγ and bi,α := γ −2 χγ ,i ( −1yγ ) = O(γ −1 ). Then by the basis {vi,α ; α = 1, 2, · · · , qi , i = 1, 2, · · · , nγ } for Vγ , we can write γ −2 (yγ )R ∈ tγ as a diagonal matrix with the (i, α)-th component equal to bi,α . Note also that √ # 2 |vi,α | = {1 − γ −2 χγ ,i ( −1yγ )}|vi,α |2 = (1 − bi,α )|vi,α |2 .

(7.2)

62

7 Stability Theorem

7.3 Strong Relative K-Semistability of Extremal Kähler Manifolds In this section, for a polarized algebraic manifold (X, L), we consider the situation that X admits an extremal Kähler metric in the class c1 (L). Then by a result of Calabi [9], the centralizer G0 in Aut(X) of an extremal vector field (cf. Sect. 9.2) is a reductive algebraic group. For the center Z(G0 ) of G0 , we choose an arbitrary algebraic torus T satisfying Z(G0 ) ⊂ T ⊂ G0 , where S 1 generated by the extremal vector field, being a subgroup of the maximal compact subgroup of the center Z(G0 ), sits also in the maximal compact subgroup of T . The following generalization of a result of Donaldson [19] is known: Fact (cf. [54, 72]; see also [71]) If X admits an extremal Kähler metric in the class c1 (L), then (X, L) is asymptotically Chow stable relative to T . Let ω be an extremal Kähler metric in the class c1 (L). Choose a Hermitian metric h for L such that Ric(h) = ω. Then we also have the following: Theorem 7.1 If X admits an extremal Kähler metric ω in the Kähler class c1 (L), then (X, L) is strongly K-semistable relative to T above. Proof For an extremal Kähler metric ω in the class c1 (L), let y := gradC ω Sω be the associated extremal vector field. By using the notation in Sects. 4.4 and 4.5, we consider the twisted Bergman kernel Bγ# (ω) := (n!/γ n )

Nγ 

|vk# |2h .

k=1

Put r0 := {2c1 (L)n [X]}−1 {nc1 (L)n−1 c1 (X)[X] + [48] (see also [50, 51, 54]), we obtain

√  −1 −1 X h (yh)ωn }. Then by

Bγ# (ω) = 1 + r0 γ −1 + O(γ −2 ).

(7.3)

Let {μj } ∈ MT , i.e., μj = (Xσj , Lσj ) for some σj ∈ 1PS(Tγ⊥j ) such that γj → +∞ as j → ∞. Let 0 = uj ∈ t⊥ γj be the fundamental generator for σj . By setting t := exp(s/|uj |∞ ), s ∈ R, we consider a real-valued function fj (s) on R defined # } for V , by (4.4). Then for a suitable choice of an admissible basis {v1# , · · · , vN γj j we can write σj (t) · vk# = t −βk vk# ,

k = 1, 2, · · · , Nj ,

7.3 Strong Relative K-Semistability of Extremal Kähler Manifolds

63

where −β1 , · · · , −βNj are the weights of the C∗ -action on Vγj via σj satisfying the equality β1 + · · · + βNj = 0. By taking dd c log of both sides of (7.3), we obtain ωFS − γj ω = O(γ −2 ), where ωFS := dd c log f˙j (0) =

Nγ

# 2 k=1 |vk | .

1 γ −n (n + 1) |uj |1 j

In view of (7.3) and (7.4),

β |v # |2 + · · · + β |v # |2 1 1 h Nj Nj h # |2 |v1# |2h + · · · + |vN j h

X

1 = γ −n (n + 1)! |uj |1 j =

(7.4)

β |v # |2 + · · · + β |v # |2 1 1 h Nj Nj h Bγ#j (ω)

X

# |2 γj−n (n + 1)! β1 |v1# |2h + · · · + βNj |vN j h

|uj |1

1 + r0 γj−1 + O(γj−2 )

X

n ωFS

{ω + O(γj−3 )}n

{ω + O(γj−3 )}n .

# } as in Sect. 4.4. Similarly, By renumbering {vk# ; k = 1, 2, . . . , Nj }, we obtain {vi,α the weights βk , k = 1, 2, . . . , Nj , are renumbered as βi,α , α = 1, 2, . . . , qi , i = 1, 2, . . . , nγ . Then by (7.2),

# 2 | = β1 |v1# |2h + · · · + βNj |vN j h

nγ qi  

βi,α (1 − bi,α )|vi,α |2h .

i=1 α=1

Hence, in view of the fact that {vi,α ; α = 1, 2, . . . , qi , i = 1, 2, . . . , nγ } is an orthonormal basis for (Vγj , ργj ) and that bi,α = O(γj−1 ), we obtain f˙j (0) =

=

nγ qi 2 γj−n (n + 1)! i=1 α=1 βi,α (1 − bi,α )|vi,α |h |uj |1 γj−n (n + 1)!

1 + r0 γj−1 + O(γj−2 )

X

 nγ q i i=1

|uj |1

α=1 βi,α (1 − bi,α )

1 + r0 γj−1

+

{ω + O(γj−3 )}n

nγ qi  

 |βi,α | O(γj−2 ) .

i=1 α=1

Since uj ∈ t⊥ γj is a diagonal matrix with (i, α)-th diagonal element −βi,α , and since

γj−2 (yγj )R ∈ (tγj )R is a diagonal matrix with (i, α)-th diagonal element bi,α , we nγ qi see from the perpendicularity t⊥ γj ⊥ tγj that i=1 α=1 βi,α bi,α = 0. We also have nγ qi i=1 α=1 βi,α = β1 + · · · + βNj = 0. Hence f˙j (0) = O(γj−1 ) ·

nγ qi i=1

α=1 |βi,α | n+1 γj

·

1 = O(γj−1 ). |uj |1

64

7 Stability Theorem

In view of the condition γj → +∞ as j → ∞, we see that limj →∞ f˙j (0) = 0. Since limj →∞ f˙j (s) is a non-decreasing function of s, it follows that F1 ({μj }) =

lim

lim f˙j (s) ≤ lim f˙j (0) = 0.

s→−∞ j →∞

j →∞

 

Hence (X, L) is strongly K-semistable, as required.

7.4 K-Polystability of Extremal Kähler Manifolds For a polarized algebraic manifold (X, L), let Tmax be a maximal algebraic torus in Aut0 (X). Then the following theorem of Stoppa and Székelyhidi [75] holds: Theorem 7.2 If X admits an extremal Kähler metric in the class c1 (L), then (X, L) is K-polystable relative to T = Tmax . As the algebraic torus T gets smaller, the condition of relative K-polystability gets stronger. Hence, as far as the stability of extremal Kähler manifolds is concerned, we want to choose T as small as possible.

7.5 A Reformulation of the Definition of the Invariant F ({μj }) Let μj := (Xσj , Lσj ), j = 1, 2, · · · , be a sequence of test configurations for (X, L) associated to nontrivial special one-parameter groups σj : C∗ → SL(Vγj ), where γj → +∞ as j → ∞. From now on, (Xσj , Lσj ) will be written simply as (Xj , Lj ). By setting t = exp(s/|uj |∞ ), we consider the function fj (s) :=

|uj |∞ −n γ log σj (t) · Xˆ j CH(ργ ) , j |uj |1 j

s ∈ C,

where uj is the fundamental generator of σjSL , and 0 = Xˆ j ∈ Wj := Symdj (Vγ∗j )⊗n+1 is the Chow form as in (4.4) for the image Xj of the twisted Kodaira embedding. Let N := Lj , Lj , · · · , Lj Xj /A1 be the Deligne pairing of (n + 1)-pieces of Lj . Fix a Hermitian metric h for L such that ω := Ric(h) is Kähler. For a suitable orthonormal basis {v1 , · · · , vNj } for Vγj , we view P(Vγ∗j ) as the projective space PNj −1 (C) := {(z1 : z2 : · · · : zNj )}. Since

7.5 A Reformulation of the Definition of the Invariant F ({μj })

65

Xj ⊂ P(Vγ∗j ) × A1 , by viewing (Xj )1 as X, we have the identification (Xj )t = σj (t) · X ⊂ PNj −1 (C),

t ∈ C∗ .

(7.5)

Let pr1 be the restriction to Xj of the projection P(Vγ∗j ) × A1 → P(Vγ∗j ) to the first factor. Then zj , j = 1, 2, · · · , Nj , are viewed as sections for Lj = pr∗1 OPNj −1 (C) (1), and (|z1 |2 +· · ·+|zNj |2 )−1 defines a Hermitian metric hFS := (|v1 |2 +· · ·+|vNj |2 )−1 for Lj such that the associated Ricci form Ric(hFS ) := −dd c log hFS is the Fubini– Study form ωFS on Xj over A1 . We then consider the Deligne pairing φj := φFS , · · · , φFS Xj /A1 over A1 of (n + 1)-pieces of Kähler potentials φFS := − log hFS . Since the blow-up of Wj at the origin is viewed as the tautological line bundle OP(Wj ) (−1), the Chow norm for Wj induces a Hermitian metric hCH for N¯ := OP(Wj ) (1). In view of (7.5), we put Z(t) := σj (t) · Xˆ j for t ∈ C∗ . The map sending t ∈ C∗ to the Chow point [Z(t)] ∈ P(Wj ) extends to a C∗ -equivariant algebraic map Z¯ : A1 → P(Wj ). Then by a theorem of Zhang [93], the pullback by Z¯ defines an isomorphism Z¯ ∗ : N¯ ∼ =N inducing an isometry between (N¯ , hCH ) and (N , hj ), where hj := e−φj . Let 0 = w ∈ N1 , where N1 is the fiber of N over 1 ∈ A1 . We consider the C∗ -action on N induced by the C∗ -action on Lj . Let λj be the weight of the C∗ -action on the fiber N0 over the origin. For each t ∈ C∗ , we put νj (t) := t −λj {σj (t) · w}.

(7.6)

The mapping A1 \ {0}  t → νˆ j (t) := σj (t) · w defines a nowhere vanishing holomorphic section of N over A1 \ {0}, which extends to a rational section for N over A1 . Let μ be the order of νˆ at the origin, i.e., μ = ordz=0 νˆj (z). Then by the same argument as in the proof of Theorem 2.1, we obtain μ = λj .

66

7 Stability Theorem

Hence the mapping A1 \ {0}  z → νj (z) ∈ Nz extends to a nowhere vanishing section of N over A1 (which trivializes N ). Since the Chow form 0 = Xˆ j ∈ Wj is viewed as a point = 0 in the fiber of OP(Wj ) (−1) over [Xˆ j ] ∈ P(Wj ), we have 0 = Xˆ j−1 ∈ N¯ sitting over [Xˆ j ] ∈ P(Wj ). Hence by setting w := Z¯ ∗ Xˆ j−1 = 0, we obtain σj (t) · Xˆ j −1 CH(ργ

j)

= σj (t) · Xˆ j−1 hCH = σj (t) · whj = t λj νj (t)hj , t ∈ R+ ,

in terms of the notation (7.6). Put t = exp s˜ for s˜ ∈ R. Then from the equality just above, it follows that log σj (t) · Xˆ j CH(ργj ) = − λj s˜ − log νj (t)hj .

(7.7)

By νj (0) = 0 together with a result of Moriwaki [60], we see that νj (z)hj is a positive continuous function in a neighborhood of the origin. Then by the same argument as in (2.11) and (2.12), lim

s˜ →−∞

d log σj (t) · Xˆ j CH(ργj ) = − λj . d s˜

(7.8)

We now put s˜ := s/|uj |∞ . By convexity of the left-hand side of (7.7) as a function of s˜ , we see that κj := − log νj (t)hj is also a convex function κj (s) of s, so that its derivative κ˙ j (s) with respect to s is a nonnegative function satisfying lim κ˙ j (s) = 0.

s→−∞

Since t = exp(s/|uj |∞ ), in terms of the function fj (s) at the beginning of this section, we can rewrite (7.7) as follows: fj (s) = −

|uj |∞ −n 1 γj−n λj s + γ · κj . |uj |1 |uj |1 j

By differentiating this equality with respect to s, we obtain the following: 1 1 1 γ −n λj + γ −n κ˙ j (s) ≥ − γ −n λj . f˙j (s) = − |uj |1 j |uj |1 j |uj |1 j Let j → ∞ and then let s → −∞. Since limj →∞ f˙j (s) is non-decreasing in s, we have F1 ({μj }) =

lim

lim f˙j (s) ≥ φ({μj }),

s→−∞ j →∞

(7.9)

7.5 A Reformulation of the Definition of the Invariant F ({μj })

67

where φ({μj }) ∈ R ∪ {−∞} is defined by φ({μj }) := lim

j →∞

  1 γj−n λj . − |uj |1

−n Since lims→−∞ f˙j (s) = −|uj |−1 1 γj λj by (7.8), we can write φ({μj }) as

φ({μj }) = lim

lim f˙j (s).

j →∞ s→−∞

Hence F1 ({μj }) = φ({μj }) in (7.9) if the double limit in the definition of F ({μj }) commutes. By comparing (7.8) with (5.14) (or by the definition of λj ), we see that − λj = λσj ,

(7.10)

which is the Chow weight for X ⊂ P(Vγ∗j ) for the C∗ -action via σj , i.e., the weight of the C∗ -action on lim|t |→0 σ (t)Xˆ j via σj . The Kodaira embedding and the twisted Kodaira embedding define the same abstract image (see the explanation at the beginning of Sect. 5.1). Hence the weight λσj doesn’t change even if we replace the twisted Kodaira embedding by the ordinary Kodaira embedding in the definition of Xj in Sect. 4.5, where σj (t), t ∈ C∗ , commutes multiplicatively with the diagonal matrix of order Nγj with the (i, α)-th diagonal element √ {1 − γj−2 χγj ,i ( −1yγj )}1/2 by the notation in Sect. 4.4. Moreover, the weight λσj has nothing to do with the choice of the reference Hermitian metric h and ω := Ric(h). Hence: Theorem 7.3 The right-hand side φ({μj }) in (7.9) is independent of the choice of the reference Hermitian metric h for L. Moreover, as far as the value of φ({μj }) is concerned, we may set Xj = Φγj (X) in Sect. 4.5 by replacing the twisted Kodaira embedding by the ordinary Kodaira embedding. It is very plausible that the equality F1 ({μj }) = φ({μj }) holds. Here by using the inequality (7.9), we give a proof for (5.29) in Problem 5.1. Proof of (5.29) In Problem 5.1, replacing L by L⊗γ , we may assume γ = 1 without loss of generality. Then by Theorem 2.2, we obtain λσj =

lim f˙j (s) = (n + 1)!a0

s→−∞

∞  α=1

DFα (X , L )j n+1−α ,

(7.11)

68

7 Stability Theorem

where j  1. Note that γj = j by γ = 1. Since a0 = c1 (L)n [X]/n!, it follows from (7.9), (7.10) and (7.11) that F1 ({μj }) ≥ φ({μj }) = lim ( j →∞

1 −n j λσj ) |uj |1

 1−α (n + 1)c1 (L)n [X] ∞ (n + 1)c1 (L)n [X] α=1 DFα (μ) j = lim = DF1 (μ), |uj |1 μasymp j →∞ where the last equality follows from limj →∞ |uj |1 = μasymp (cf. Sect. 4.2).

Problems 7.1 For a polarized algebraic manifold (X, L), assume that X admits a CSC Kähler metric in the class c1 (L). Show that (X, L) is K-semistable. 7.2 (cf. [66]) Let (X, L) be a polarized algebraic manifold such that X admits a CSC Kähler metric in the class c1 (L). Is (X, L) always asymptotically Chow stable?

Chapter 8

Existence Problem

Abstract In this chapter, we give some remarks on the existence of extremal Kähler metrics on polarized algebraic manifolds: • For the existence of extremal Kähler metrics, as mentioned in Sect. 8.1, a result of He states the following: a polarized algebraic manifold (X, L) admits an extremal Kähler metric in the class c1 (L) if the modified K-energy is proper modulo the action of the centralizer G0 of the extremal vector field in Aut0 (X). Hence the existence of an extremal Kähler metric is reduced to showing the properness of the modified K-energy modulo the action of G0 . • In Sect. 8.2, we give some observations on the existence of extremal metrics, which gives an idea of how strong K-stability is useful for the existence. Keywords The theorem of Chen, Donaldson and Sun and of Tian · The results of Chen, Cheng and He

8.1 A Result of He on the Existence of Extremal Kähler Metrics A Kähler–Einstein metric always exists on a K-polystable Fano manifold by the affirmative solution of the Yau–Tian–Donaldson conjecture (see Conjecture 6.1 in Sect. 6.2) by Chen et al. [13–15] and Tian [80], Recently, for a general polarization, a breakthrough by Chen and Cheng [10– 12] on the existence of CSC Kähler metrics has been developed by He [29] to the case of the existence problem of extremal Kähler metrics. Namely, given a polarized algebraic manifold (X, L), Theorem 9.7 in Sect. 9.6 below is valid even if we replace c1 (X) by an arbitrary polarization class c1 (L), i.e., X admits an extremal Kähler metric in the class c1 (L) if the modified K-energy κˆ is proper modulo the action of the centralizer G0 of the extremal vector field in Aut0 (X).

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0_8

69

70

8 Existence Problem

8.2 Some Observations on the Existence Problem We consider here a polarized algebraic manifold (X, L) which is strongly K-stable relative to a maximal algebraic torus Tmax in Aut0 (X). For simplicity, assume that Tmax = {1}. Fix a Hermitian metric h for L such that ω := Ric(h) is Kähler. Then by Theorem 5.5, (X, L) is asymptotically Chow stable, i.e., there exists an increasing sequence 1 < γ1 < γ2 < · · · < γj < · · · with a balanced metric ωj = Ric(hj ) for L⊗γj , j = 1, 2, · · · , where hj is a Hermitian metric for L. Hence we have an orthonormal basis {v1 , v2 , · · · , vNj } for (Vγj , ργj ) such that (|v1 |2 + |v2 |2 + · · · + |vNj |2 )hj is a constant, where ρj = ργj is such that ργj (v, v ) = (v, v )hj ωjn , v, v ∈ Vγj . X

On the other hand, another Hermitian inner product ρ0 for Vγj is defined by ρ0 (v, v ) =

X

(v, v )h ωn ,

v, v ∈ Vγj .

By choosing a suitable orthonormal basis {w1 , w2 , · · · , wNj } for (Vγj , ρ0 ), and by replacing hj by constant times hj if necessary, we can view ρ0 as an Nj ×Nj identity matrix, and ρj is also written as an Nj ×Nj diagonal matrix in SL(Nj ; C) with each α-th diagonal element λα > 0. Put bα := (1/2) log λα . Then b1 + b2 + · · · + bNj = 0. Approximating each bα by a sequence of rational numbers, we may assume from the beginning that each bα is a rational number (see [53]). By choosing a positive integer mj , we have that bα := mj bα ,

α = 1, 2, · · · , Nj ,

are all integers. Let uj (reps. uj ) in sl(Vγj ) be the diagonal matrices with each α-th diagonal element bα (resp. bα ). Let σj : C∗ → SL(Vγj ) be the special oneparameter group with the fundamental generator uj , so that σj (t) = t uj . In the definition fj (s) :=

|uj |∞ −n γ log  exp(suj /|uj |∞ ) · Xˆ j CH(ργj ) |uj |1 j

8.2 Some Observations on the Existence Problem

71

of fj (s), the function fj (s) doesn’t change even if we replace uj by uj , and hence the function fj (s) is rewritten as fj (s) :=

|uj |∞ |uj |1

γj−n log exp(suj /|uj |∞ ) · Xˆ j CH(ργj ) .

On the other hand, |uj |∞ (see (4.1)) can be viewed as the distance between ργj and ρ0 . Let μj := (Xσj , Lσj ) be the test configuration associated to σj . Put d∞ := sup |uj |∞ . j

Then the following cases are possible: (Case 1) (Case 2)

d∞ = +∞; d∞ < +∞.

We first consider Case 1. Note that σj (t) = exp(suj /|uj |∞ ), t = exp(s/|uj |∞ ). Here the α-th diagonal element bα of the diagonal matrix uj is (1/2) log λα . Hence σj (t) · ρ0 |s=0 = ρ0 , σj (t) · ρ0 |s=−|u

j |∞

= exp(−uj ) · ρ0 ,

where exp(−uj ) is a diagonal matrix with α-th diagonal element e−bα , and it acts on Vγ∗j by the contragradient representation. Since ρ0 sitting in Vγ∗j ⊗ V¯γ∗j is written as an Nj × Nj identity matrix I in terms of the basis {w1 , w2 , · · · , wNj }, we can view σj (t) · ρ0 |s=−|u |∞ as a diagonal matrix j

t

exp(uj ) · I · exp(uj )

whose α-th diagonal element is e2bα = λα . Thus we obtain σj (t) · ρ0 |s=−|u

j |∞

= ρj .

Since the Chow norm takes the critical value at the balanced metric ωj (which corresponds to ρj ), the derivative of the function fj (s) at s = −|uj |∞ vanishes: f˙j (−|uj |∞ ) = 0

for all j.

By the assumption of Case 1, d∞ = +∞. Hence replacing {uj } by its subsequence if necessary, we may assume that |uj |∞ ,

j = 1, 2, · · · ,

72

8 Existence Problem

is a monotone increasing sequence diverging to +∞. For the time being, fix an arbitrary j . Then by the monotonicity of the sequence, |uj |∞ ≥ |uj |∞

for all j ≥ j.

Hence −|uj |∞ ≤ −|uj |∞ . Since f˙j (s) is a non-decreasing function of s, we obtain 0 = f˙j (−|uj |∞ ) ≤ f˙j (−|uj |∞ )

for all j ≥ j.

y

y = f˙j (s) y = f˙j (s)

y-coordinate ≥ 0 ⇒

−|uj |∞

−|uj |∞

s O

By letting j → ∞, we obtain limj →∞ f˙j (s) ≥ 0 for all s such that −|uj |∞ ≤ s < +∞. Now let j → ∞. Then by |uj |∞ → +∞, it follows that lim f˙j (s) ≥ 0

j →∞

for all s with − ∞ < s < +∞.

Here we let s → −∞. Then F1 ({μj }) =

lim

lim f˙j (s) ≥ 0.

s→−∞ j →∞

However, by |uj |∞ → +∞, we may assume that all μj are nontrivial, and therefore it follows from the strong K-stability of (X, L) that the inequality F1 ({μj }) < 0 holds in contradiction. Hence Case 1 cannot occur. We now consider Case 2. In this case by d∞ < +∞, we intuitively see that ρj is not so far away from ρ0 (and hence ωj is not so far away from ω). For instance we have an a priori C 0 bound for the sequence of balanced metrics (cf. [53]). It is then expected that ωj converges to a CSC Kähler metric.

Problems

73

The above argument goes through also for relative versions including the case of the existence problem of extremal Kähler metrics.

Problems 8.1 Recall the following theorem of Matsushima [59]: Theorem Let G be a connected linear algebraic group and let H be its closed algebraic subgroup. If G is reductive, then G/H is affine if and only if H is reductive. In view of this theorem, what can you say about the relationship between the asymptotic Chow polystability for (X, L) and the Matsushima–Lichnerowicz obstruction (cf. [38, 58]) for the existence of CSC Kähler metrics in the class c1 (L)? 8.2 Since P2 (C) and P1 (C) × P1 (C) are homogeneous spaces, they admit Kähler– Einstein metrics (the Fubini–Study metric or a product of the Fubini–Study metrics). Enumerate all other compact complex connected surfaces with Kähler–Einstein metrics of positive Ricci curvature.

Chapter 9

Canonical Kähler Metrics on Fano Manifolds

Abstract In this chapter, we study canonical Kähler metrics in the class c1 (X) for Fano manifolds with no Kähler–Einstein metrics. Typical examples are: • Kähler–Ricci solitons, • Extremal Kähler metrics, • Generalized Kähler–Einstein metrics. Since Kähler-Ricci solitons and extremal Kähler metrics are well-known, we focus on the recent developments of the studies of generalized Kähler–Einstein metrics. • In Sects. 9.1 and 9.2, we discuss basic facts concerning holomorphic vector fields (including the extremal vector field) on Fano manifolds. • In Sects. 9.3 and 9.4, we discuss obstructions to the existence of generalized Kähler–Einstein metrics. • In Sects. 9.5, 9.7, 9.8, and 9.9, we discuss the existence problem of generalized Kähler–Einstein metrics including both Yao’s result and Hisamoto’s result. • In Sect. 9.6, we shall show that a generalized Kähler–Einstein manifold always admits an extremal Kähler metric in the class c1 (X). Keywords Generalized Kähler–Einstein metrics · Extremal vector fields · Yao’s result · Hisamoto’s result

9.1 Kähler Metrics in Anticanonical Class A compact complex connected manifold X is called a Fano manifold if c1 (X) > 0. For an n-dimensional Fano manifold X, let ω be a Kähler form in the class c1 (X). Then the corresponding Ricci potential is the real-valued smooth function fω ∈ C ∞ (X)R on X defined uniquely by C

dd fω = Ric(ω) − ω



fω

and

e ω X

n

=

ωn . X

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0_9

75

76

9 Canonical Kähler Metrics on Fano Manifolds

Since c1 (X) > 0, by the Kodaira vanishing theorem, we obtain h1,0 (X) = h0,1 (X) = hn−1 (X, O(KX )) = 0. Hence all holomorphic vector fields on the Kähler manifold (X, ω) are Hamiltonian (cf. Sect. 3.2) in the sense that the Lie algebra   n ψ is holomorphic and ψω = 0 gω := ψ ∈ C ∞ (X)C ; gradC ω X

endowed with the Poisson bracket is identified with the space g := H 0 (X, O(T X)) of holomorphic vector fields on X by the isomorphism gω ∼ = g,

ψ ↔ gradC ω ψ.

Definition 9.1 (cf. [45]) ω above is called generalized Kähler–Einsten if the fω fω ∈ g . complex vector field gradC ω ω (1 − e ) is holomorphic, i.e., 1 − e Recall that ω is Kähler–Einstein if fω is constant, whereas ω is called a Kähler– Ricci soliton if gradC ω fω is holomorphic. Obviously, a Kähler–Einstein form ω is always generalized Kähler–Einstein. Recall also that ω is called extremal Kähler C if the complex vector field gradC ω Sω (= gradω (Sω − n)) associated to the scalar curvature function Sω of ω is holomorphic.

9.2 Extremal Vector Fields In this section, we start from a general situation that (X, L) is a polarized algebraic manifold such that L is an ample line bundle which is not necessarily very ample. Here X is not necessarily a Fano manifold, nor is L necessarily the anticanonical bundle KX−1 . For a Kähler metric ω in the class c1 (L), we put kω := {ψ ∈ gω ; ψ is real-valued}, where gω is as defined in Sect. 9.1. Then for the space k := {gradC ω ψ ; ψ ∈ kω } of the Killing vector fields on (X, ω), we have the isomorphism kω ∼ = k. Let k⊥ ω be the orthogonal complement   2 n ϕ ∈ L (X, ω)R ; ϕψ ω = 0 for all ψ ∈ kω X

of kω in the space L2 (X, ω)R of all real-valued L2 functions on the Kähler manifold (X, ω). For the projection pr : L2 (X, ω)R (= kω ⊕ k⊥ ω ) → kω to the first factor, by viewing pr(Sω − n) as an element yω of k by the isomorphism k ∼ = kω , the

9.2 Extremal Vector Fields

77

holomorphic vector field yω ∈ k on X is called the extremal vector field on (X, L) for ω. Then by [26], the associated real vector field yωR := yω + y¯ω on X satisfies exp(2πmyωR ) = idX

(9.1)

for some positive integer m, where yω = 0 if and only if the Futaki character F (cf. Sect. 3.1) vanishes. If yω = 0, then yωR generates an isometric S 1 -action on (X, ω). From now on, we assume that L = KX−1 , i.e., the Kähler class c1 (L) is c1 (X). Theorem 9.1 pr(Sω − n) = pr(1 − efω ).  ¯ Proof For the Laplacian Δω := α,β g βα ∂ 2 /∂zα ∂zβ¯ , we put Dω := Δω +



¯

g βα

α,β

∂fω ∂ . ∂zα ∂zβ¯

Then a formula in [24, p. 41], shows that





(Dω ϕ1 )ϕ¯2 efω ωn =

ϕ1 (Dω ϕ2 ) efω ωn = −

X

X

X

¯ 1 , ∂ϕ ¯ 2 )ω e f ω ω n (∂ϕ

for all ϕ1 , ϕ2 ∈ C ∞ (X)C , where ( , )ω denotes the pointwise Hermitian inner product by ω for 1-forms on X. Moreover, KerR (Dω + 1) ∼ = kω ,

ϕ ↔ ϕ − Cϕ ,

(9.2)

∞ where Ker ω + 1) := {ϕ ∈ C (X)R ; (Dω + 1)ϕ = 0 }, and Cϕ :=   R (D n/ n . Hence for every ψ ∈ k , there exists ϕ ∈ Ker (D + 1) such ϕω ω ω ω R X X that ψ = ϕ − Cϕ . Then



ψ pr(1 − e )ω fω

X



ψ(1 − e )ω fω



=



=

n



X

(Dω ϕ)ωn + X



⎝Δω ϕ +

= − X





X

 α,β

(Dω ϕ)ωn +



(Dω ϕ)efω ωn X

(Dω ϕ)ωn X

⎞ ⎛ ⎞  ∂f ∂ϕ ∂f ∂ϕ ¯ ¯ ω ω ⎠ ωn = − ⎝ ⎠ ωn g βα g βα ∂zα ∂zβ¯ ∂zα ∂zβ¯ X

X



¯ ∂f ¯ ω )ω ωn = (∂ψ,

X



ψ(Sω − n) ωn = X

X

ϕ (Dω 1) efω ωn = −

¯ ∂f ¯ ω )ω ωn = − (∂ϕ,

=− =

⎛

(ϕ − Cϕ )(1 − efω )ωn X

(−Dω ϕ)(1 − efω )ωn = −



= −

=

X

ϕ(1 − efω )ωn = X

n

ψ pr(Sω − n) ωn . X



α,β

X

ψ(Δω fω ) ωn

78

9 Canonical Kähler Metrics on Fano Manifolds

Since both pr(1 − efω ) and pr(Sω − n) are in kω , it therefore follows that pr(1 − efω ) coincides with pr(Sω − n), as required.  

9.3 An Obstruction of Matsushima’s Type If X admits a generalized Kähler–Einstein metric, we have a decomposition theorem of the Lie algebra g as shown for extremal Kähler metrics by Calabi [9]. For each nonnegative rational number λ, we consider the linear subspace gλ of g defined by √ gλ := { u ∈ g ; [ −1 yω , u] = λu }. Then for λ = 0, the associated Lie subalgebra g0 of g is nothing but the centralizer Zg (yω ) of the extremal vector field yω in g. Let kC be the complexification of k in g. Theorem 9.2 If X admits a generalized Kähler–Einstein metric, then g0 coincides with kC , and there exists a sequence of rational numbers 0 = λ0 < λ1 < · · · < λr such that g viewed as a vector space is written as a direct sum g =

r 

gλ i .

(9.3)

i=0

Proof By h1,0 (X) = 0, the identity component G := Aut0 (X) of the group of all holomorphic automorphisms of X is linear algebraic. Then the Chevalley decomposition allows us to write the associated Lie algebra g as a semi-direct sum g = kC  u, for the unipotent radical u of g. Let T be the algebraic torus generated by yω . Then by setting t = es for s ∈ C, we have an isomorphism i : C∗ ∼ = T such that √ i(t) := exp(s −1 myω ), for m as in (9.1). Since the T -action on u by the adjoint representation is completely reducible, an increasing sequence of integers k1 < k2 < · · · < kr exists such that u viewed as a vector space is a direct sum u1 ⊕ u2 ⊕ · · · ⊕ ur , where √ uα := { u ∈ u ; Ad(i(t))(u) = t kα u } = { u ∈ u ; [ −1 myω , u] = kα u }, for α = 1, 2, . . . , r. Then by setting λα := kα /m, we can write √ √ uα = { u ∈ u ; [ −1 yω , u] = λα u } = { u ∈ g ; [ −1 yω , u] = λα u }. (9.4)

9.3 An Obstruction of Matsushima’s Type

79

Recall that yω sits in the center of kC (cf. [26]). Hence kC ⊂ Zg (yω ). Then the proof is reduced to showing that α = 1, 2, . . . , r.

λα > 0,

(9.5)

Because if (9.5) is true, then by g = kC u, the equality (9.3) is straightforward from the decomposition u = u1 ⊕ u2 ⊕ · · · ⊕ ur . Furthermore, by writing each element w in Zg√ (yω ) as a sum κ +u for some κ ∈ kC and u ∈ u, we immediately see from (9.5) and [ −1yω , w] = 0 that the equality u = 0 holds, i.e., Zg (yω ) ⊂ kC . Thus we obtain the equality Zg(yω ) = kC , as required. We shall now show (9.5). Let 0 = u ∈ uα . For Cϕ as in (9.2), we have the following isomorphisms (cf. [24]): KerC (Dω + 1) ∼ = g, = gω ∼

ϕ ↔ ϕ − Cϕ ↔ gradC ω ϕ,

where KerC (Dω + 1) is the space of all ϕ ∈ C ∞ (X)C such that (Dω + 1)ϕ = 0. Hence there exist ϕ1 , ϕ2 ∈ KerC (Dω + 1) such that u = gradC ω ϕ1 and yω = gradC ϕ . Since ω is generalized Kähler–Einstein, we obtain ω 2 1 − efω = pr(1 − efω ) = ϕ2 − Cϕ2 .

(9.6)

/ KerC (Dω + 1), because if ϕ¯ 1 ∈ KerC (Dω + 1), then First by u = 0, we obtain ϕ¯ 1 ∈ by ϕ1 ∈ KerC (Dω + 1), both the real part Re ϕ1 and the imaginary part Im ϕ1 of ϕ1 would belong to KerR (Dω + 1), implying that u ∈ kC , in contradiction. Secondly,





ϕ¯1 efω ωn = X

ϕ 1 e fω ω n = − X

(Dω ϕ1 ) efω ωn = 0

(9.7)

X

  because X (Dω ϕ1 ) efω ωn = X ϕ1 Dω (1) efω ωn = 0. Recall that, by Futaki [24], all eigenvalues of the operator −Dω are nonnegative real numbers, where its first positive eigenvalue is 1. Moreover, all eigenfunctions of −Dω with eigenvalue 0 are constant. Hence by ϕ¯ 1 ∈ / KerC (Dω + 1) and (9.7),

(−Dω ϕ¯ 1 )ϕ1 e ω fω

n

>

X

|ϕ1 |2 efω ωn .

(9.8)

|ϕ1 |2 efω ωn .

(9.9)

X

On the other hand, by ϕ1 ∈ KerC (Dω + 1),

(−Dω ϕ1 )ϕ¯1 efω ωn = X

X

80

9 Canonical Kähler Metrics on Fano Manifolds

  Since the left-hand side X (−Dω ϕ¯1 )ϕ1 efω ωn of (9.8) is X ϕ¯ 1 (−D¯ ω ϕ1 ) efω ωn , by subtracting (9.9) from (9.8), we see that

{(Dω − D¯ ω )ϕ1 }ϕ¯1 efω ωn > 0.

(9.10)

X

Let [ , ]ω be the Poisson bracket with respect to the Kähler form ω. In view of the definition of the operator Dω , we obtain {(Dω − D¯ ω )ϕ1 }efω

⎛ ⎞  ¯ ∂fω ∂ϕ1  ¯ ∂fω ∂ϕ1 ⎠ = e fω ⎝ g βα − g αβ ∂zα ∂zβ¯ ∂zα¯ ∂zβ α,β

=



g

α,β



 ¯ ∂ϕ1  αβ ∂efω ∂ϕ1 − g¯ = g βα ∂zα ∂zβ¯ ∂zα¯ ∂zβ

¯ ∂e βα

α,β

fω

α,β

α,β

∂efω ∂ϕ1 ∂ϕ1 ∂efω − ∂zα ∂zβ¯ ∂zα ∂zβ¯



√ √ = − −1 [ efω , ϕ1 ]ω = −1 [ϕ2 − Cϕ2 , ϕ1 − Cϕ1 ]ω , ∼ g, it follows where in the last√equality, we use (9.6). By the isomorphism gω = from (9.4) that [ −1(ϕ2 − Cϕ2 ), ϕ1 − Cϕ1 ]ω = λα (ϕ1 − Cϕ1 ). Hence {(Dω − D¯ ω )ϕ1 }efω = λα (ϕ1 − Cϕ1 ). From this together with (9.10), it follows that λα X

Now by Cϕ1 =

 X

ϕ1 ω n /

 X

(ϕ1 − Cϕ1 )ϕ¯1 ωn > 0.

ωn and the Schwarz inequality, we obtain

(ϕ1 − Cϕ1 )ϕ¯1 ω =

2 n

n

X

(9.11)

X

 =

X

|ϕ1 | ω − Cϕ1 |ϕ1 |2 ωn



ϕ¯1 ωn X

% %2 ω n − % X ϕ1 ω n % ≥ 0. n Xω

X 

Here by u = 0, ϕ1 is non-constant, so that in the last inequality, the equality cannot occur. Hence by (9.11), we obtain λα > 0, as required.  

9.4 An Invariant as an Obstruction to the Existence For a Kähler metric ω in the class c1 (M), let μω : X → R be the function on X defined by μω (x) := pr(Sω − n)(x) = pr(1 − efω )(x), x ∈ X. Then by yω = gradC ω μω , ¯ ω, 2π iyω ω = ∂μ

9.5 Examples of Generalized Kähler–Einstein Metrics

81

so that μω defines a moment map for the C∗ -action on X generated by yω , where the vertices of the image of the moment map are Q-rational (cf. [26]). Hence γX := max μω (x) ∈ Q X

is a number independent of the choice of ω in the class c1 (X), where by μω ∈ kω , the equality X μω ωn = 0 always holds, and hence γX ≥ 0. This γX then gives an obstruction [45] to the existence of generalized Kähler–Einstein metrics: Theorem 9.3 If X admits a generalized Kähler–Einstein metric, then γX < 1. Proof Assume that X admits a generalized Kähler–Einstein metric ω. Then μω = pr(1 − efω ) = 1 − efω . Let x0 ∈ X be the point at which the function μω attains its maximum. Then γX = μω (x0 ) = 1 − efω (x0 ) < 1, as required.  

9.5 Examples of Generalized Kähler–Einstein Metrics Let N be a k-dimensional Fano manifold with a Kähler–Einstein form φ in the class c1 (N) such that Ric(φ) = φ. Let L be a holomorphic line bundle over N with a Hermitian metric h for L such that all eigenvalues β1 , β2 , . . . , βk of Ric(h) with respect to φ are constant satisfying −1 < βi < 1 for all i = 1, 2, · · · , k. For the vector bundle ON ⊕ L of rank 2 over N, we consider the associated P1 -bundle X := P(ON ⊕ L)

(9.12)

over N consisting of all lines through the origin in the fibers of the vector bundle. Hence n := dim X is nothing but k + 1. For the line subbundles ON ⊕ {0}, {0} ⊕ L of E, we put D0 := P. (ON ⊕ {0}), D∞ := P({0} ⊕ L), respectively. Then D0 ∼ =N and D∞ ∼ = N. Moreover, we have the identification L = X \ D∞ .

(9.13)

Let ρ : L   → ρ() := h ∈ R≥0 be the Hermitian norm for L induced by h. Then by (9.13), ρ is viewed as a function on X\D∞ . Define x ∈ C ∞ (X\D0 ∪D∞ )R by x = log(ρ 2 ). Let p : X → N be the natural projection. For some real-valued smooth function λ(x) of x, we consider the volume form on X \ D0 ∪ D∞ , Ω :=

√ −1 −λ(x) ∗ k ¯ ne (p φ) ∧ ∂x ∧ ∂x, 2π

(9.14)

82

9 Canonical Kähler Metrics on Fano Manifolds

 which is supposed to extend to a volume form on X such that X Ω = c1 (X)n [X]. Put ω := Ric Ω. Then by (9.14) and Ric(φ) = φ,√the same computation as in ¯ In [43] allows us to write ω as p∗ φ − λ (x) Ric(h) + { −1/(2π)} λ (x)∂x ∧ ∂x. particular,   k

−1 ¯ n λ (x) (1 − βi λ (x)) (p∗ φ)k ∧ ∂x ∧ ∂x. ωn = 2π √

(9.15)

i=1

  Since dd c fω = Ric(ω) − ω = dd c log(Ω/ωn ), and since X efω ωn = X ωn =   c1 (X)n [X] = X Ω = X (Ω/ωn )ωn , we have efω = Ω/ωn . By (9.14) and (9.15),

e

fω

 −1 k

Ω −λ(x) λ (x) (1 − βi λ (x)) = n = e . ω

(9.16)

i=1

Note that the C∗ -action on the line bundle L extends naturally to a holomorphic C∗ -action on X = P(ON ⊕ L) induced by C∗ × (ON ⊕ L) → ON ⊕ L,

(t, x ⊕ y) → x ⊕ ty.

Then λ (x), mapping X onto the interval [−1, 1], defines a moment map for the C∗ action on X. Actually, we have λ (x) ∈ KerR (Dω +1) and u := gradC ω λ (x) ∈ k (see for instance [43]). Hence, in order to show that ω is a generalized Kähler–Einstein metric on X, it suffices to solve the following differential equation in λ = λ(x): R.H.S. of (9.16) = C1 + C2 λ (x),

(9.17)

for some real constants C1 and C2 such that Ω extends to a volume form on X and that ω extends to a Kähler form on X. Put bα :=

1

−1

qα

k

(1 − βi q) dq,

α = 0, 1, 2.

i=1

Since 1 − βi q is positive for all i with −1 < q < 1, both b0 and b2 are positive. Then by the Schwarz inequality, b12 < b0 b2 . Let z be a system of holomorphic local coordinates on N centered at a point p in N. For a local base σ = σ (z) for L on a neighborhood of p, by writing  = tσ (z) for fiber coordinate t for L, we have ex = ρ 2 = 2h = a(z)|t|2 ,

9.5 Examples of Generalized Kähler–Einstein Metrics

83

√

where a(z) := σ (z)2h . Then we set t = re −1 θ for polar coordinates (r, θ ). For a suitable choice of σ , we may assume that (da)(0) = 0 and a(0) = 1. Hence ¯ = ∂r ¯ at the point p. Then ∂ρ = ∂r and ∂ρ √ √ ¯ ¯ = −1 dt ∧ d t = 2 dr ∧ dθ = dx ∧ dθ −1 ∂x ∧ ∂x 2 |t| r

(9.18)

at p. For simplicity, put q := λ . Then the Jacobian for the mapping q : R → [−1, 1], x → q(x), is λ (x). In view of (9.15) and (9.18), we obtain λ (x)α ωn = c0 bα , α = 0, 1, 2, (9.19) X



   where c0 := n N φ k . Now by (9.17), X {C1 + C2 λ (x)}ωn = X efω ωn = X ωn . This together with (9.19) allows us to obtain C1 b0 + C2 b1 = b0 .

(9.20)

 On the other hand, by (9.17) and λ (x) ∈ KerR (Dω + 1), we have X λ (x){C1 +   C2 λ (x)}ωn = X λ (x)efω ωn = − X {Dω λ (x)}efω ωn = 0. Hence by (9.19), C1 b1 + C2 b2 = 0.

(9.21)

If b1 = 0, then by (9.20) and (9.21), (C1 , C2 ) = (1, 0). Hence by the vanishing b1 = 0 of the Futaki character, we see from Koiso and Sakane’s result [36] that ω is a Kähler–Einstein metric on X, since in this case (9.17) above reduces to the equation for ω to be a Kähler–Einstein metric (see for instance [43]). Hence we may assume b1 = 0. Now by (9.20) and (9.21), we obtain (C1 , C2 ) = (b0 b2 /(b0 b2 − b12 ), −b0 b1 /(b0 b2 − b12 )). For the time being, assuming that |b1 | < b2 ,

(9.22)

we shall show that ω is generalized Kähler–Einstein. By 1 ≤ λ (x) ≤ 1, the righthand side of (9.17) is bounded from below by a positive real constant as follows: C1 + C2 λ (x) ≥ C1 − |C2 | =

b0 (b2 − |b1|) > 0. b0 b2 − b12

s * Put A(s) := − −1 q(C1 + C2 q) ki=1 (1 − βi q) dq, s ∈ R. By (9.21), A(1) = A(−1) = 0, where the order of zeroes of A(s) at s = ±1 is 1 (cf. [43]). Then 0 < A(s) ≤ A(0) and A (s)/s < 0 for 0 = |s| < 1. Hence −A (s)/(s A(s)) is a positive rational function free from poles and zeroes on the open interval (−1, 1),

84

9 Canonical Kähler Metrics on Fano Manifolds

and by (9.22), has poles of order 1 at s = ±1. It then follows that the function x : (−1, 1) → R defined by

s

x(s) := −

A (q)/(q A(q)) dq

0

is monotone increasing, and maps (−1, 1) diffeomorphically onto R, since in a neighborhood of s = 1 (resp. s = −1), x(s) is expressible as − log(1 − s) + real analytic function (resp. log(1 + s) + real analytic function). Let s = s(x) : R → (−1, 1) be the inverse function of x = x(s) : (−1, 1) → R above. Let us now consider the function λ = λ(x) of x defined by λ(x) := − log(A(s)). Then λ (x) = −

A (s) sA(s) A (s) ds · = · = s. A(s) dx A(s) A (s)

In particular, we obtain the equality λ (x) = s (x) = 1/x (s) = −sA(s)/A (s), where on the right-hand side, * the numerator −s A(s) is −s e−λ(x), and the denomi nator A (s) is −s (C1 + C2 s) ki=1 (1 − βi s). Hence by s = λ (x),  e

−λ(x)

k

λ (x) (1 − βi λ (x))

−1



= C1 + C2 λ (x),

i=1

i.e., for the function s = s(x) above, λ(x) = − log(A(s)) satisfies (9.17). Since e−λ(x) = A(s) has zeroes of order 1 at s = ±1, and since the real analytic function λ (x) = −sA(s)/A (s) of s = s(x) is non-vanishing on the interval (−1, 1) with zeroes of order 1 at s = ±1, we see from (9.18) that Ω extends to a volume form on X, and ω extends to a Kähler form on X (cf. [43]). Thus we obtain: Theorem 9.4 If |b1 | < b2 , then X in (9.12) admits a generalized Kähler–Einstein metric, and in particular γX < 1. Remark 9.1 Actually, by using the generalized Kähler–Einstein metric ω, we see from (9.16) and (9.17) the following: ) ( γX = max μω = max(1 − efω ) = max (1 − C1 ) − C2 λ (x) X

X

= 1 − C1 + |C2 | =

X

b0 |b1 | − b12 b0 b2 − b12

< 1.

Finally, we consider the case |b1 | ≥ b2 . Let Ω in (9.14) be chosen in such a way that it extends to a volume form on X and that the associated ω extends to a Kähler form on X. Here λ(x) is not necessarily assumed to satisfy (9.17). Since every holomorphic automorphism of X induces a holomorphic automorphism of N, and since every infinitesimal holomorphic action on N lifts to an infinitesimal

9.5 Examples of Generalized Kähler–Einstein Metrics

85

holomorphic action on L, we can write g = H 0 (X, O(T X)) as a semidirect sum g = H 0 (N, O(T N))  h, where h is the Lie algebra of all holomorphic automorphisms of X preserving the fibers of X over N. Since the C∗ -action on L commutes with any infinitesimal holomorphic action on N, the holomorphic vector field u := gradC ω λ (x) ∈ k on X ∗ 0 associated to the C -action on L commutes with the Lie algebra H (N, O(T N)). Let K0 be the identity component of the group of isometries of (N, φ). Since (N, φ) is Kähler–Einstein, H 0 (N, O(T N)) is a complexification kC 0 of the Lie algebra k0 := Lie(K0 ) of K0 such that k = k0 ⊕ Ru, where K0 is assumed to fix the Hermitian metric h for L. Since the extremal vector field yω on X is in the center of kC , by (9.1), we can write yω as a sum w + α1 u for some w ∈ k0 and a real constant α1 . Since w = gradC ω ϕ for some ϕ ∈ kω , we have pr(1 − efω ) = ϕ + α1 λ (x) + α2  for some real constant α2 . For α0 := { X λ (x)ϕ ωn }(b0/c0 )/(b0 b2 − b12 ), we put ϕ0 := ϕ − α0 {λ (x) − (b1 /b0 )}, C3 := α2 − α0 (b1 /b0 ) and C4 := α0 + α1 . Then pr(1 − efω ) = ϕ0 + C4 λ (x) + C3 .

(9.23)

Since λ (x) − (b1 /b0 ) ∈ kω , we have ϕ0 ∈ kω . Then by (9.19),





λ (x)ϕ0 ω = X





λ (x)ϕ ω − α0

n

X





=

λ (x)ϕ ω X

 n





λ (x) ω − (b1 /b0 )

n

2 n

X

b2 − (b1 /b0 )b1 1 − b0 · b0 b2 − b12





λ (x)ω

 n

X

= 0.

(  ) ⊥ n For k˜ω := kω ⊕ R, let k˜⊥ ω := ϕ ∈ kω ; X ϕω = 0 be its orthogonal complement ˜ in L2 (X, ω)R. Then for the natural projection p+r : L2 (X, ω)R (= k˜ω ⊕ k˜⊥ ω ) → kω , we see from X (1 − efω )ωn = 0 that pr(1 − efω ) = p+r(1 − efω ) = 1 − p+r(efω ).

(9.24)

86

9 Canonical Kähler Metrics on Fano Manifolds

Now by (9.23) and (9.24), p+r(efω ) = −ϕ0 − C4 λ (x) + (1 − C3 ), and we write efω ˜ as a sum p+r(efω ) + ζ for some ζ ∈ k˜⊥ ω . By λ (x) ∈ KerR (Dω + 1) ⊂ kω and (9.19),

λ (x){−ϕ0 − C4 λ (x) + (1 − C3 )}ωn

{−C4 b2 + (1 − C3 )b1 } c0 = X



λ (x) p+r(efω ) ωn =

= X



λ (x) efω ωn =

=

λ (x) (efω − ζ )ωn X



X



−{Dω λ (x)} efω ωn = 0. X

Hence C4 b2 = (1 − C3 )b1 . On the other hand, by (9.19) and ϕ0 ∈ kω ,

{−ϕ0 − C4 λ (x) + (1 − C3 )}ωn

{−C4 b1 + (1 − C3 )b0 } c0 =

X



=

p+r(efω )ωn = X





(efω − ζ ) · 1 · ωn = X

e fω ω n = X

ω n = b 0 c0 , X

and hence −C4 b1 + (1 − C3 )b0 = b0 . From these, we obtain (C3 , C4 ) = (−b12 /(b0 b2 − b12 ), b0 b1 /(b0 b2 − b12 )). This together with (9.24) and p+r(efω ) = −ϕ0 − C4 λ (x) + (1 − C3 ) implies that pr(1 − efω ) = ϕ0 + C3 + C4 λ (x) = ϕ0 +

b0 b1 λ (x) − b12 b0 b2 − b12

(9.25)

.

Let us now assume that Aut0 (N) is semisimple. For instance, such a condition is satisfied if N is a rational homogeneous space or if Aut(N) is discrete. For λ = λ (x) ∈ KerR (Dω + 1) above, by setting gN := H 0 (N, O(T N)), we consider  gˆ ω :=



n

φ ∈ gω ;

φλ ω = 0 and X

gradC ωφ

 ∈ gN ⊕ Cu ,

which is a Lie subalgebra of gω by the Poisson bracket [ , ]ω in terms of ω as follows: Let of gN ⊕ Cu, we obtain [φ2 , λ ]ω =  φ1 , φ2 ∈ gˆ ω . nSinceu is in the center n 0. Hence X [φ1 , φ2 ]ω λ ω = X φ1 [φ2 , λ ]ω ω = 0. This means that [φ1 , φ2 ]ω ∈ ˆ ω } of g, we have a Lie algebra gˆ ω . For the Lie subalgebra gˆ := {gradC ωφ; φ ∈ g isomorphism pr1 : gˆ ∼ = gN , where pr1 is the restriction to gˆ of the projection: gN ⊕ Cu → gN to the first factor. By the semisimplicity of gN , we have [gN , gN ] = gN , and hence [ˆg, gˆ ] = gˆ . In

9.5 Examples of Generalized Kähler–Einstein Metrics

87

particular, for the Futaki character F : g → C, we have F (ˆg) = F ([ˆg, gˆ ]) = 0. ˆ ⊂ Ker F . Then by (9.23), have gradC Since ϕ0 in (9.23) belongs ω ϕ0 ∈ g  to gˆ ω , we n since ϕ0 ∈ kω , and since X ϕ0 λ ω = 0, it follows that 0 = F (gradC ω ϕ0 ) =

ϕ0 pr(1 − efω )ωn X





=

ϕ0 (ϕ0 + C4 λ + C3 )ω

n

=

X

X

ϕ02 ωn ,

and hence ϕ0 = 0. Substituting this into (9.25), we obtain pr(1 − efω ) = {−b12 + b0 b1 λ (x)}/(b0b2 − b12 ). Note that λ |D0 = −1 and λ |D∞ = 1. Hence γX = max pr(1 − efω ) =

b0 |b1 | − b12 b0 b2 − b12

X

≥ 1,

(9.26)

where for the last inequality, the condition |b1 | ≥ b2 is used. Then by Theorem 9.3, X admits no generalized Kähler–Einstein metrics. By summing up, we obtain: Theorem 9.5 For X in (9.12), assume that Aut0 (N) is semisimple. Assume further that K0 fixes the Hermitian metric h for L. Then γX =

b0 |b1 | − b12 b0 b2 − b12

.

Hence in this case γX < 1 if and only if |b1 | < b2 , and furthermore X admits a generalized Kähler–Einstein metric if and only if γX < 1. Example 9.1 (cf. [45]) Let N = Pk (C) and L = OPk (1), k ≥ 1. By c1 (N) = 1 L⊗k+1 , we have bα = −1 y α {1 − (k + 1)−1 y}k dy. Note that b1 < 0. It is easy to check that b2 − |b1 | = b2 + b1 ≥

1 2 · > 0. 3 k+1

Hence X := P(OPk ⊕ OPk (1)) admits a generalized Kähler–Einstein metric, where by b1 = 0, X admits no Kähler–Einstein metrics. For instance, if k = 1, then X is the Hirzebruch surface F1 , and by (b0 , b1 , b2 ) = (2, −1/3, 2/3) and Remark 9.1, γX =

b0 |b1 | − b12 b0 b2 − b12

=

5 < 1. 11

88

9 Canonical Kähler Metrics on Fano Manifolds

Example 9.2 (cf. [45]) Let N = P2 (C) and L = OP2 (2). Since bα = (2y/3)}2dy, we have (b0 , b1 , b2 ) = (62/27, −8/9, 38/45). Then |b1 | =

1

−1 y

α {1

−

38 8 > = b2 . 9 45

Hence X := P(OP2 ⊕ OP2 (2)) admits no generalized Kähler–Einstein metrics. Moreover by (9.26), we have the inequality γX ≥

b0 |b1 | − b12 b0 b2 − b12

=

380 > 1. 349

However, it is also known (cf. [27, 33]) that X admits an extremal Kähler metric in the class c1 (M). This last fact was pointed out to the author by S. Nakamura.

9.6 Extremal Metrics on Generalized Kähler–Einstein Manifolds In this section, we briefly discuss a fact concerning the existence of extremal Kähler metrics on generalized Kähler–Einstein manifolds. The proof down below is due to S. Nakamura. For an n-dimensional Fano manifold X, let K be its Kähler class c1 (X). By fixing a reference metric ω0 in K , we can write K = { ωϕ ; ϕ ∈ C ∞ (X)R is such that ωϕ is Kähler }, where ωϕ := ω0 + dd c ϕ. Let μω := pr(1 − efω ) = pr(Sω − n) be as in Sect. 9.4. Put V := X ω0n = c1 (X)n [X]. We now consider Aubin’s functional J : K → R≥0 , Ding’s functional D : K → R, and the K-energy κ : K → R defined by 1 J (ϕ) := V

1 

0

1 D(ϕ) := − V 1 κ(ϕ) := − V

X

1 





 ϕ˙ s (ω0n

0

1 

0

− ωsn ) 

X

X

ϕ˙s ωsn

ϕ˙s (Sωs

ds, 

1 ds − log V  − n)ωsn ds,

e X

f0 −ϕ

 ω0n

.

9.6 Extremal Metrics on Generalized Kähler–Einstein Manifolds

89

where {ϕs }0≤s≤1 is a piecewise smooth path in C ∞ (X)R as in Sect. 6.3 such that ϕ0 = 0 and ϕ1 = ϕ. Here by ϕ˙s , we mean ∂ϕs /∂s, while ωϕs and fωs are written as ωs and fs , respectively. The corresponding modified energies are 1 Jˆ(ϕ) := V



1 

0

1 ˆ D(ϕ) := − V 1 κ(ϕ) ˆ := − V

1 





ϕ˙s ((1 − X

0

1 

0

μω0 )ω0n

− (1 − μωs )ωsn ) 

X

1 ds − log( V  n − n − μωs )ωs ds.

ϕ˙s (Sωs

(9.27)

ef0 −ϕ ω0n ),

(9.28)



ϕ˙s (1 − μωs )ωsn

X

ds,

X

(9.29)

First by (9.29), ω is a critical point of κˆ if and only if Sωs − n = μωs , i.e., ω is an extremal Kähler metric. Next by differentiating (9.27), we obtain d ˆ 1 D(ϕs ) = − ds V



X

ϕ˙s (1 − μωs )ωsn

ef0 −ϕs ϕ˙ s ω0n . f0 −ϕs ωn 0 Xe

 + X

(9.30)

For the time being, we assume the following: Claim

' &  fs = f0 − ϕs + log(ω0n /ωsn ) + log V / X ef0 −ϕs ω0n .

By this claim, we immediately see that ef0 −ϕs = efs (ωsn /ω0n )( Substituting this into (9.30), we obtain d ˆ 1 D(ϕs ) = − ds V

X

ϕ˙s (1−μωs )ωsn +

1 V

X

efs ϕ˙s ωsn = −

1 V

 X

ef0 −ϕs ω0n /V ).

X

ϕ˙s (1−efs −μωs )ωsn .

Hence ω is a critical point of Dˆ if and only if 1 − efω = μω , i.e., ω is a generalized Kähler–Einstein metric. We shall now prove the above claim. Proof Note that dd c fs = Ric(ωs ) − ωs = {Ric(ω0 ) − ω0 } − dd c ϕs + {Ric(ωs ) − Ric(ω0 )} = dd c {f0 − ϕs + log(ω0n /ωsn )}. Hence fs = f0 − ϕs + log(ω0n /ωsn ) + C for some real constant C. Then Claim follows from fs n f0 −ϕs +log(ω0n /ωsn )+C n C V = e ωs = e ωs = e ef0 −ϕs ω0n . X

X

X

  with respect to s is Δωs ϕ˙ s . Then by differentiNote that the derivative of ating the equality in Claim with respect to s, we have log ωsn

 f −ϕ e 0 s ϕ˙ s ω0n ˙  fs = −ϕ˙s − Δωs ϕ˙s + X . f0 −ϕs ωn 0 Xe

90

9 Canonical Kähler Metrics on Fano Manifolds

We integrate this over X by the volume form (1/V )ωsn . It then follows that 1 V



1 f˙s ωsn = − V X



X

ϕ˙s ωsn

ef0 −ϕs ϕ˙s ω0n . f0 −ϕs ωn 0 Xe

 + X

Now by comparing this with (9.30), we obtain d ˆ 1 D(ϕs ) = ds V



1 f˙s ωsn + V X

X

ϕ˙ s μωs ωsn .

(9.31)

    Since (d/ds)( X fs ωsn ) − X f˙s ωsn = X fs (Δωs ϕ˙ s )ωsn = X ϕ˙ s (Δωs fs )ωsn , and since Sωs − n = Δωs fs , we see from (9.29) and (9.31) the following:

d 1 κ(ϕ ˆ s) = − ds V

ϕ˙s (Sωs − X



n − μωs ) ωsn

1 = − V

X

ϕ˙ s (Δωs fs − μωs ) ωsn

 1 1 fs ωsn + ϕ˙ s μωs ωsn f˙s ωsn + V X V X X   d d ˆ 1 n = − fs ωs + D(ϕ s ). ds V X ds = −

d ds

1 V



Integrate this equality over the interval [0, 1]. Then in view of ϕ0 = 0 and ϕ1 = ϕ, ˆ since κ(0) ˆ = 0 and D(0) = 0, we obtain 1 1 ˆ κ(ϕ) ˆ = − fωϕ ωϕn + fω ωn + D(ϕ) (9.32) V X V X 0 0 for all ωϕ in K . Since x + 1 ≤ ex for all real numbers x, we have

X

(fωϕ + 1)ωϕn

≤

e X

fω ϕ

ωϕn

= X

ωϕn ,

  so that X fωϕ ωϕn ≤ 0 for all ωϕ in K . Hence by setting C0 := −(1/V ) X fω0 ω0n , it follows from (9.32) that ˆ κ(ϕ) ˆ ≥ D(ϕ) − C0

(9.33)

for all ωϕ in K , where C0 is a nonnegative real constant independent of ϕ. Let ω be a generalized Kähler–Einstein metric. Then the following cases are possible: Case 1: yω = 0. Then 1 − efω = μω = 0. Hence ω is Kähler–Einstein, and in particular extremal Kähler.

9.6 Extremal Metrics on Generalized Kähler–Einstein Manifolds

91

Case 2: yω = 0. In this case, by taking the circle group Tc := exp(RyωR ), we choose a Tc -invariant Kähler metric in K as the reference metric ω0 above. Let G0 be the centralizer of Tc in G = Aut0 (X). Put H Tc = { ϕ ∈ C ∞ (X)R ; ωϕ ∈ K and ϕ is Tc -invariant }. Since ω is generalized Kähler–Einstein, we have γX < 1. Recall the following theorem of Li and Zhou [41]: Theorem 9.6 (1) If γX < 1, then there exist positive constants C1 and C2 independent of ϕ such that 0 ≤ C1 J (ϕ) ≤ Jˆ(ϕ) ≤ C2 J (ϕ) for all ϕ ∈ H Tc . (2) If X admits a generalized Kähler–Einstein metric, then there exist positive ˆ constants C3 and C4 independent of ϕ such that D(ϕ) ≥ C3 infg∈G0 Jˆ(ϕg )−C4 Tc , where ϕ ∈ H Tc is such that g ∗ (ω ) = ω + dd c ϕ and that for all ϕ ∈ H g ϕ 0 g  n X ϕg ω0 = 0. Since ω is generalized Kähler–Einstein, by (2) of Theorem 9.6, there exist positive constants C3 and C4 as above such that ˆ D(ϕ) ≥ C3 inf Jˆ(ϕg ) − C4 , g∈G0

ϕ ∈ H Tc .

This together with (9.33) allows us to obtain ˆ κ(ϕ) ˆ ≥ D(ϕ) − C0 ≥ C3 inf Jˆ(ϕg ) − (C4 + C0 ). g∈G0

(9.34)

Hence the functional κ(ϕ), ˆ ϕ ∈ H Tc , is bounded from below by a real constant independent of the choice of ϕ, since in (9.34), we see from (1) of Theorem 9.6 that inf Jˆ(ϕg ) ≥ C1 inf J (ϕg ) ≥ 0.

g∈G0

g∈G0

(9.35)

We now apply the following theorem of He [29, Theorem 2]. Here, we adapt his result to the anticanonical class for Fano manifolds. Theorem 9.7 The Kähler class K on X admits an extremal Kähler metric if the following conditions are satisfied: (1) κ(ϕ), ˆ ϕ ∈ H Tc , is bounded from below. (2) For every sequence ϕi , i=1,2,. . . , in H Tc such that infg∈G0 J ((ϕi )g ) → +∞ as i → ∞, we always have κ(ϕ ˆ i ) → +∞ as i → ∞. Remark 9.2 The modified K-energy is called proper modulo the action of G0 if the above conditions (1) and (2) are satisfied.

92

9 Canonical Kähler Metrics on Fano Manifolds

As we saw above, by (9.34) and (9.35), condition (1) of Theorem 9.7 is satisfied. For condition (2), again by (9.34) and (9.35), we see that κ(ϕ ˆ i ) ≥ C3 C1 inf J ((ϕi )g ) − (C4 + C0 ). g∈G0

Since infg∈G0 J ((ϕi )g ) → +∞ as i → ∞, it then follows that κ(ϕ ˆ i ) → +∞ as i → ∞. Hence (2) is also satisfied. Now by applying Theorem 9.7, the Kähler class K on X admits an extremal Kähler metric. By summing up, we obtain: Theorem 9.8 If a Fano manifold X admits a generalized Kähler–Einstein metric, then the Kähler class c1 (X) admits an extremal Kähler metric.

9.7 The Product Formula for the Invariant γX The results in this section are due to Y. Nitta and S. Saito. For the product X = X1 × X2 of Fano manifolds X1 and X2 , we have the following formula for γX : Theorem 9.9 γX = γX1 + γX2 . Proof For i = 1, 2, we choose a Kähler form ωi on Xi in the class c1 (Xi ), and let πi : X (= X1 × X2 ) → Xi , i = 1, 2, be the projection to the i-th factor. Put ω := π1∗ ω1 + π2∗ ω2 ,

(9.36)

where we also put n1 := dim X1 and n2 := dim X2 . Hence for n := n1 +n2 , we have ωn = {n!/(n1 ! n2 !)}(π1∗ ω1n ) (π2∗ ω2n ), so that Ric(ω) = −dd c log ωn is expressible as Ric(ω) = −π1∗ dd c log ω1n − π2∗ dd c log ω2n = π1∗ Ric(ω1 ) + π2∗ Ric(ω2 ). (9.37) Note that Sω = Trω Ric(ω) and Sωi = Trωi Ric(ωi ), i = 1, 2. Hence by comparing the equalities (9.36) and (9.37), we obtain Sω = π1∗ Sω1 + π2∗ Sω2 .

(9.38)

Let pr : L2 (X, ω) → kω be the orthogonal projection as in Sect. 9.2. Similarly, we consider the orthogonal projections pri : L2 (Xi , ωi ) → kωi , i = 1, 2. Then by (9.36), kω = π1∗ kω1 ⊕ π2∗ kω2 . It then follows from (9.38) that pr(Sω ) = pr(π1∗ Sω1 ) + pr(π2∗ Sω2 ) = π1∗ (pr1 Sω1 ) + π2∗ (pr2 Sω2 ).

(9.39)

9.7 The Product Formula for the Invariant γX

93

Let a point x = (x1 , x2 ) run through the set X = X1 × X2 . In view of the equalities pr(Sω − n) = pr(Sω ) and pri (Sωi − ni ) = pri (Sωi ), i = 1, 2, we obtain γX = max pr(Sω )(x) = max{π1∗ (pr1 Sω1 )(x) + π2∗ (pr2 Sω2 )(x)} x∈X

x∈X

= max (pr1 Sω1 )(x1 ) + max (pr2 Sω2 )(x2 ) = γX1 + γX2 . x1 ∈X1

x2 ∈X2

 

From (9.38) and n = n1 + n2 , we obtain the equality Sω − n = π1∗ (Sω1 − n1 ) + ∗ π2 (Sω2 −n2 ), while by (9.39), pr(Sω −n) = π1∗ (pr1 (Sω1 −n1 )) + π2∗ (pr2 (Sω2 −n2 )). Hence if ω1 is extremal Kähler, and in addition if ω2 is extremal Kähler, then we have both Sω1 −n1 = pr1 (Sω1 −n1 ) and Sω2 −n2 = pr2 (Sω2 −n2 ), and consequently Sω − n = pr(Sω − n), i.e., ω is extremal Kähler. Thus we obtain: Theorem 9.10 If ω1 is an extremal Kähler metric in the class c1 (X1 ), and if ω2 is an extremal Kähler metric in the class c1 (X2 ), then ω in (9.36) is an extremal Kähler metric in the class c1 (X) for X = X1 × X2 . However, the same thing is not true for generalized Kähler–Einstein metrics. Suppose that a Fano manifold X0 admits a generalized Kähler–Einstein metric ω0 which is not Kähler–Einstein. Then by Theorem 9.3, 0 < γX0 < 1. Here we have γX0 = 0 as follows. For contradiction, assume that γX0 = 0. Then the Hamiltonian function μω0 = pr(1 − efω0 ) for the extremal vector field on X0 satisfies 0 = γX0 = max μω0 . X

Since ω0 is generalized Kähler–Einstein, we can write μω0 = 1 − efω0 , which has   maximum 0 on X, i.e., fω0 ≥ 0 on X. Then by X efω0 ω0n = ω0n , we obtain fω0 = 0 on X in contradiction to the fact that ω0 is not Kähler–Einstein. We now choose arbitrarily an integer k such that kγX0 ≥ 1. Let us consider the direct product of k copies of X0 , X := X0k = X0 × X0 × · · · × X0 . Then by Theorem 9.9, we have γX = kγX0 ≥ 1. It then follows from Theorem 9.3 that X admits no generalized Kähler–Einstein metrics.

94

9 Canonical Kähler Metrics on Fano Manifolds

Example 9.3 Let X0 be the Hirzebruch surface F1 := P(OP1 ⊕ OP1 (1)). This is just the case k = 1 in Example 9.1. In view of γX0 = 5/11, X0 admits a generalized Kähler–Einstein metric. Moreover by Calabi [8] (see also [27, 33]), X0 admits an extremal Kähler metric in the class c1 (X0 ). Let X be the direct product of k copies of X0 . Then the following cases are possible: Case 1: k = 2. In this case γX = 10/11 < 1. Since X is a toric Fano manifold, the result in the next section shows that X admits a generalized Kähler–Einstein metric. By Theorem 9.10, X admits an extremal Kähler metric in the class c1 (X). Case 2: k ≥ 3. In this case γX = 5k/11 > 1. Then by Theorem 9.3, X admits no generalized Kähler–Einstein metrics, whereas by Theorem 9.10, X admits an extremal Kähler metric in the class c1 (X).

9.8 Yao’s Result for Toric Fano Manifolds We discuss here Yao’s result [90] on the existence of generalized Kähler–Einstein metrics on toric Fano manifolds. In this section, by T , we mean the n-dimensional algebraic torus (C∗ )n endowed with the multiplicative action of T itself: T × T  (t, t ) → t · t ∈ T , where t · t := (t1 t1 , . . . , tn tn ) for t = (t1 , · · · , tn ) ∈ T and t = (t1 , · · · , tn ) ∈ T . Recall that a Fano manifold X is called toric if X is a T -equivariant compactification of T itself such that T is a Zariski open dense subset of X. Theorem 9.11 A toric Fano manifold X admits a generalized Kähler–Einstein metric if and only if γX < 1. Proof In view of Theorem 9.3, the proof is reduced to showing that there exists a generalized Kähler–Einstein metric on X provided that γX < 1. For the T equivariant inclusion: T = {(t1 , . . . , tn )} → X, we define a real-valued functions xα on the open subset T of X by |tα |2 = exα , α = 1, 2, . . . , n. Let Ω := n! e

−λ(x)

n

α=1

√

−1 dtα ∧ dtα¯ · 2π |ta |2

 (9.40)

be a volume form on X, where λ = λ(x) is a function of x = (x1 , . . . , xn ) such that the right-hand side of (9.40) extends to a volume form on X. Put qα := ∂λ/∂xα . Then q(x) := (q1 (x), q2 (x), . . . , qn (x)) extends to a moment map q : X → Rn = {(u1 , . . . , un )}

(9.41)

9.8 Yao’s Result for Toric Fano Manifolds

95

for the T -action on X such that qα = q ∗ uα for all α, where u = (u1 , . . . , un ) is the standard coordinates on Rn . Assuming that ω := Ric Ω is Kähler on X, we have   √ dtα ∧ dtβ¯ ∂ 2λ −1  ω = −dd log Ω = · 2π ∂xα ∂xβ tα tβ¯ c

(9.42)

α,β

and qα ∈ KerR (Dω + 1), α = 1, 2, . . . , n. Then by the same argument as in obtaining the equality (9.16), we see from (9.40) and (9.42) that

e

fω

  −1  Ω ∂ 2λ −λ = n = e . det ω ∂xα ∂xβ 1≤α,β≤n

(9.43)

Hence, in order to show that ω is a generalized Kähler–Einstein metric on X, it suffices to solve the following differential equation in λ: R.H.S. of (9.43) = C0 +

n 

Cα

α=1

∂λ , ∂xα

(9.44)

for suitable real constants Cα , α = 0, 1, . . . , n, such that Ω extends to a volume form on X and that ω extends to a Kähler form on X. Note that T is a maximal algebraic torus in G = Aut0 (X) whose maximal compact subgroup acts isometrically on (X, ω). In particular, the Lie algebra t of T is a Lie subalgebra of kC . Since the extremal vector field yω is in the center of kC (cf. [26]), yω belongs to t. Let p+r : L2 (X, ω)R → k˜ω be as in (9.24). Again by (9.24), p+r(efω ) = 1 − pr(1 − efω ) = 1 + pr(efω ),

(9.45)

fω where as remarked above, the holomorphic vector field yω := − gradC ω pr(e ) is in t. Hence we have real constants Cα , α = 0, 1, . . . , n, such that (9.45) is written as

p+r(efω ) = C0 +

n  α=1

Cα

∂λ . ∂xα

(9.46)

If ω is generalized Kähler–Einstein, then pr(1 − efω ) = 1 − efω , and from this equality together with (9.45), we obtain p+r(efω ) = efω , and in this special case the real constants Cα , α = 0, 1, . . . , n, in (9.44) above reduce to Cα , α = 0, 1, . . . , n, respectively, Hence for a general ω, where Ω is as in (9.40), Cα are chosen such that Cα := Cα ,

α = 0, 1, . . . , n.

(9.47)

96

9 Canonical Kähler Metrics on Fano Manifolds

As observed in Remark 9.3 below, these Cα are constants depending only on the image P := Im(q) of the moment map q : X → Rn in (9.41), where P is a convex body in Rn independent of the choice of ω. In (9.41), we have ∂λ/∂xα = qα = q ∗ uα . Hence by (9.45), (9.46) and (9.47), we obtain pr(1 − efω ) = 1 − p+r(efω ) = 1 − q ∗ σ,  where we define σ := nα=0 Cα uα by setting u0 := 1. Then σ¯ := maxP σ and σ := minP σ are constants independent of the choice of ω, and we obtain γX = max pr(1 − efω ) = 1 − σ

min pr(1 − efω ) = 1 − σ¯ .

and

X

X

From the assumption γX < 1, it follows that σ > 0. Since the right-hand side of (9.44) is written as q ∗ σ , we have the following a priori bounds: 0 < σ ≤ C0 +

n 

Cα

α=1

∂λ ∂xα

≤ σ¯ .

(9.48)

Hence (9.44) is solved by the continuity method as in Wang and Zhu [83], where their method is applied to our situation by replacing the term of the form exp(C0 +  n n C ∂λ/∂x ) in [83] by the term C + C   α α 0 α=1 α=1 α ∂λ/∂xα above. Remark 9.3 The constants Cα , α = 0, 1, . . . , n, in (9.44), (9.46), and (9.47) above are computed as follows: For simplicity, put q0 := 1. Let H = (hαβ )1≤α,β≤n be the Hessian matrix for the function λ = λ(x) defined by hαβ := ∂ 2 λ/∂xα ∂xβ = ∂qα /∂xβ . n If we view q as a function on √R = {(x1 , . . . , xn2 )}, then det H is the Jacobian of q. For θα := arg(tα ), we have −1dtα ∧ dtα¯ /|tα | = dxα ∧ dθα as in (9.18). In view of (9.42) together with (9.46) and (9.47), we obtain

b00 =

du1 ∧ · · · ∧ dun = (2π) P

= (2π)−n

−n

(dq1 ∧ dθ1 ) ∧ · · · ∧ (dqn ∧ dθn ) X



(det H ) (dx1 ∧ dθ1 ) ∧ (dx2 ∧ dθ2 ) · · · ∧ (dxn ∧ dθn ) X





=



ωn = X

⎛



⎝

= P

e fω ω n = X

n 

β=0

⎛



⎝

p+r(efω ) ωn = X

⎞ Cβ uβ ⎠ du1 ∧ · · · ∧ dun =

X n  β=0

n 

β=0

b0β Cβ ,

⎞ Cβ qβ ⎠ ωn

9.9 Hisamoto’s Result on the Existence Problem

97

 where we set bαβ := P uα uβ du1 ∧· · ·∧dun for all α, β ∈ {0, 1, . . . , n}. Similarly, by qα = ∂λ/∂xα ∈ KerR (Dω + 1), α = 1, . . . , n, we obtain



0 = − X

= X



(Dω qα ) efω ω n = ⎛

qα ⎝

n 

β=0

⎞

X

Cβ qβ ⎠ ω n =

qα efω ω n = P

⎛ uα ⎝

n 

X

qα p+r(efω )ω n ⎞

Cβ uβ ⎠ du1 ∧ · · · ∧ dun =

β=0

n 

bαβ Cβ

β=0

for α = 1, . . . , n. Since B = (bαβ )0≤α,β≤n is a positive-definite square matrix of order (n + 1), we consider its inverse matrix B −1 = (bαβ ). Then Cα = bα0 b00,

α = 0, 1, . . . , n.

9.9 Hisamoto’s Result on the Existence Problem For a Fano manifold X, let G0 be as in Sect. 9.6. We consider the center Z of G0 . For Aubin’s functional J (ϕ) in Sect. 9.6, we put JZ (ϕ) := inf J (ϕg ), g∈Z

ϕ ∈ H Tc .

(9.49)

Let (X , L ) be a test configuration for (X, KX−1 ), where in this section, the ample anticanonical line bundle L = KX−1 is not necessarily assumed to be very ample. ˆ For the modified Ding functional D(ϕ) in (9.28) and the functional JZ (ϕ) in (9.49), we have the non-Archimedean version [3, 4] of the functionals Dˆ and JZ : Dˆ NA (X , L ) ∈ R

and

JZNA (X , L ) ∈ R≥0 .

In the Yau–Tian–Donaldson conjecture, Dˆ NA (X , L ) corresponds to the Donaldson-Futaki invariant of (X , L ), while JZNA (X , L ) corresponds to the asymptotic L1 - norm of the test configuration (X , L ). A recent result of Hisamoto [32] together with Yao’s work [91] shows that: Theorem 9.12 A Fano manifold X admits a generalized Kähler–Einstein metric if and only if the following conditions are satisfied: • The obstruction of Matsushima’s type vanishes, i.e., G0 is reductive; • (X, L) is uniformly Ding stable relative to Z. In the above theorem, (X, L) is called uniformly Ding stable relative to Z if there exists a positive real constant δ > 0 such that Dˆ NA (X , L ) ≥ δ JZNA (X , L )

98

9 Canonical Kähler Metrics on Fano Manifolds

for all G0 -equivariant test configurations (X , L ) for (X, L). Here a test configuration (X , L ) is called G0 -equivariant if the natural G0 -action on X1 = X extends to a G0 -action on (X , L ) which covers the trivial action on A1 and commutes with the C∗ -action of the test configuration (X , L ).

Problems 9.1 Put N := P1 (C) × P2 (C) and L := ON (1, −1), where for integers p and q, ON (p, q) denotes the line bundle pr∗1 OP1 (p) ⊗ pr∗2 OP1 (q) over N with natural projections pri : P1 (C) × P2 (C) → Pi (C), i = 1, 2. Show that X := P. (ON ⊕ L) admits no Kähler–Einstein metrics, and that X admits a generalized Kähler–Einstein metric. 9.2 For X in Problem 9.1, let Xk := X × · · · × X be the direct product of k copies of X. Find the smallest positive integer k such that k · γX ≥ 1. Note that, for such an integer k, the direct product Xk admits no generalized Kähler–Einstein metrics.

Appendix A

Geometry of Pseudo-Normed Graded Algebras

A.1 Differential Geometric Viewpoints Graded algebras (such as canonical rings) coming from the space of sections of polarized algebraic varieties are studied by many mathematicians. Differential geometrically, such a study begins with a work of Royden [69], followed by the pseudo-norm project proposed by S.-T. Yau and C.-Y. Chi [16, 17]. These allow us to obtain a new aspect of the Torelli-type theorem. In this appendix, we discuss how the geometry of Lp -spaces allows us to obtain a natural compactification of the moduli space of pseudo-normed graded algebras of a certain type. In contrast to the GIT-limits in algebraic geometry (or to the Gromov–Hausdorff limit in Riemannian geometry), we have some straightforward compactification (cf. [44]) of the moduli space of pseudo-normed graded algebras. For stable curves in the Deligne–Mumford compactification, the notion of an orthogonal direct sum of pseudo-normed spaces comes up naturally.

A.2 Lp -Spaces Let p be a real constant such that p = 2 or 0 < p ≤ 1. We consider a complex vector space V of complex dimension N, where we assume N < +∞ throughout subsequent sections, though only in this section, N can possibly be infinite. Definition A.1 (V ,  ) is called an Lp -space, if V  v → v ∈ R≥0 is a continuous function, called a pseudo-norm of order p, satisfying the following conditions: • For v ∈ V , we have v = 0 if and only if v = 0; • homogeneity: c · v = |c| · v for all c ∈ C and v ∈ V ;

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2021 T. Mabuchi, Test Configurations, Stabilities and Canonical Kähler Metrics, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-16-0500-0

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100

A

Geometry of Pseudo-Normed Graded Algebras

• subadditivity: If 0 < p ≤ 1, then u + vp ≤ up + vp for all u, v ∈ V . If p = 2, then   is a Hermitian norm for V . Hence if p = 2, there exists a positive definite Hermitian inner product ( , ) such that v2 = (v, v) for all v ∈ V , and in particular u + v ≤ u + v for all u, v ∈ V. Example A.1 Let (X, dμ) be a measure space. Then V = Lp (X, dμ) with the Lp -norm   is an Lp -space, where Lp (X, dμ)  denotes the space of all complex measurable functions f on (X, dμ) such that X |f |p dμ < +∞. To check the subadditivity for 0 < p ≤ 1, let f , g ∈ Lp (X, dμ). Recall that the inequality (a + b)p ≤ a p + b p holds for all nonnegative real numbers a and b. Then by applying this inequality to a = |f (x)| and b = |g(x)|, x ∈ X, we obtain the inequality |f (x) + g(x)|p ≤ (|f (x)| + |g(x)|)p ≤ |f (x)|p + |g(x)|p , and hence



f + gp =

|f + g|p dμ ≤ X

(|f |p + |g|p )dμ = f p + gp . X

Example A.2 Let X be an n-dimensional compact complex connected manifold. For the canonical bundle KX of X, we consider the dualizing sheaf ωX := O(KX ) of X. For every positive integer m, we put p = 2/m. Put |σ |p := (σ ∧ σ¯ )1/m . Then ⊗m V := H 0 (X, ωX ) has the structure of an Lp -space by the pseudo-norm 1/p

 σ m :=

|σ |

p

σ ∈ V.

,

X

A.3 An Orthogonal Direct Sum of Lp -Spaces Let (Vλ ,  λ ), λ ∈ Λ, be Lp -spaces for a fixed p, where Λ = {1, 2, · · · , k}, and p is a real number such that 0 < p ≤ 1 or p = 2. Definition A.2 (V ,  ) is called an orthogonal direct sum of Lp -spaces (Vλ ,  λ ), λ = 1, 2, · · · , k, if the following conditions are satisfied: , • V = λ∈Λ Vλ as a vector , space. • If vλ ∈ Vλ , then for v = λ∈Λ vλ , we have vp =



p

vλ λ .

λ∈Λ

If (V ,  ) is an orthogonal direct sum as above, then we write (V ,  ) =

 (Vλ ,  λ ). λ∈Λ

A Geometry of Pseudo-Normed Graded Algebras

101

Note that an orthogonal direct sum of Lp -spaces is again an Lp -space. An Lp space (V ,  ) is called irreducible, if it is not written as an orthogonal direct sum (V ,   )⊕(V ,   ) of nontrivial Lp -subspaces (V ,   ), (V ,   ). Every Lp space is an orthogonal direct sum of irreducible Lp -subspaces (cf. [44]). If p = 2, every irreducible Lp -space is 1-dimensional, and the decomposition into irreducible Lp -spaces is given by a choice of an orthonormal basis. Hence for p = 2, the decomposition is not unique. However, for p = 2, the decomposition is unique. Example A.3 (C,  ) is an Lp -space by setting z := |z| for z ∈ C. We then consider its orthogonal direct sum (CN ,  N ) :=

N 

(C,  ).

Hence for all z = (z1 , · · · , zN ) ∈ CN , we have zN = (|z1 |p + · · · + |zN |p )1/p . For an Lp -space (V ,  ), the set Σ := {v ∈ V ; v ≤ 1 } is called the indicatrix for (V ,  ). In the above example, let N = 2. Let 0 < p ≤ 1. Then Σ = {(z1 , z2 ) ∈ C2 ; |z1 |p + |z2 |p ≤ 1}. Then Σ is, when restricted to the real points (x1 , x2 ) in R2 , described as in the following figures: p=1

0