Lectures on Riemann Surfaces: Jacobi Varieties 9781400872695

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LECTURES ON RIEMANN SURFACES, JACOBI VARIETIES

BY R. C. GUNNING

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1972

Copyright

©

1972 by P r i n c e t o n U n i v e r s i t y A l l Rights Reserved L.C. Card: I.S.B.N.

72-9950

0-691-08127-1

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

Press

Preface. These are notes based on a course of lectures given at Princeton University during the Spring Term of 1972. They are intended as a sequel to the notes Lectures on Riemann Surfaces (Mathematical Notes, Princeton University Press, 1966), and are principally devoted to a continuation of the discussion of the dimensions of spaces of holomorphic cross-sections of complex line bundles over compact Riemann surfaces contained in §7 of the earlier notes; but whereas the earlier treatment was limited to results obtainable chiefly by one dimensional methods, a more detailed analysis requires the use of various properties of Jacobi varieties and of symmetric products of Riemann surfaces, and consequently serves as a further introduction to these topics as well. The reader is assumed to be familiar with the material covered in the first 8 chapters of the earlier notes, and the terminology and notation introduced there are used with no further explanation.

The first chapter of these notes consists of the

rather explicit description of a canonical basis for the Abelian differentials on a marked Riemann surface, and of the description of the canonical meromorphic differentials and the prime function (in more or less the sense used in Hensel and Landsberg "Theorie der algebraische Funktionen einer Variabeln") of a marked Riemann surface. Although most of this material is used only to a minor extent in the present volume of notes, it is important for later

-i-

developments and this is a convenient point at which it can be treated, particularly in connection with the introduction of the very useful concept of a marked Riemann surface, which concept is employed systematically throughout the remainder of these notes. The second chapter contains a discussion of Jacobi varieties of compact Riemann surfaces and of various subvarieties which arise in determining the dimensions of spaces of holomorphic cross-sections of complex line bundles. The Jacobi variety of a marked Riemann surface is introduced in an explicit but convenient manner, and the usual invariance properties follow easily from an identification of the Picard and Jacobi varieties. The third chapter contains a discussion of the relations between Jacobi varieties and symmetric products of Riemann surfaces which are relevant to the determination of dimensions of spaces of holomorphic cross-sections of complex line bundles. The analytical and topological properties of symmetric products are not really discussed in general, although an indication of the usefulness of some of these properties in studying the problem at hand is indicated in the appendix. The final chapter consists of the derivation of Torelli's theorem following A. Weil but in an analytic context. The results which may be new are somewhat scattered and follow fairly naturally from the lines of development of the subject, so there is no purpose served in attempting to point them out explicitly. However I have tried in the notes and references

to indicate rather explicitly the sources used for the material covered here, but without attempting a complete bibliography or detailed and accurate history of the subject. As is the case with so many other publications on this subject the debts owed to the work of A. Weil are immense and obvious.

I have also been heavily

influenced by the work of Η. H. Martens and have followed his treatment at several

places indicated in the notes. Finally I

should like to express my thanks to the students

and colleagues

who attended these lectures, particularly to Robert H. Risch, for many helpful suggestions and references; and to Elizabeth Epstein for her customary beautiful job of typing these notes.

R. C. Gunning Princeton, New Jersey September, 1972

-111-

§1.

Marked Riemann surfaces and t h e i r canonical differentials

Pag

a. Markings of a Riemann surface (l) b. Canonical holomorphic Abelian differentials (7) c. Canonical meromorphic Abelian differentials (15) d. Prime function of a marked Riemann surface (23) Notes for §1 (29) §2.

Jacobi varieties and their distinguished subvarieties

3^-

a. Jacobi mapping and Jacobi homomorphism (3^) b. Subvarieties of positive divisors (39) c. Subvarieties of special positive divisors (U6) d. Hyperelliptic surfaces and Clifford's theorem (55) Notes for §2 (62) References (70) §3. Jacobi varieties and symmetric products of Riemann surfaces a. Symmetric products of Riemann surfaces (72) b. Symmetric products and Jacobi mappings (75) c. Singularities of subvarieties of positive divisors (87) d. Hyperelliptic surfaces and the extension of Clifford's theorem (98) e. Dimensions of subvarieties of special positive divisors (99) f. Jacobi mappings and analytic fibrations (105) g. Jacobi fibrations and associated vector bundles (113) Notes for §3 (127) References (139)

72

Page Intersections in Jacobi varieties and Torelli's theorem a. b. c. d.

ll+l

Intersections of special translates of hypersurfaces (lUl) Intersections of general translates of hypersurfaces (1U3) Surfaces of low genus (159) Torelli*s theorem (169)

Notes for & (173) References (176)

Appendix.

On conditions ensuring that

W^ ^ 0

177

Index of symbols

188

Index

I89

§1. Marked Riemann surfaces and their canonical differentials.

(a)

At several points in the more detailed study of Riemann sur­

faces, the explicit topological properties of surfaces play an impor­ tant role; and it is convenient to have these properties established from the beginning of the discussion, to avoid the necessity of inserting topological digressions later. Since the universal cover­ ing space of a connected orientable surface of genus

g > 0

is a

cell, the fundamental group carries essentially all the topological properties of the surface; so it is also convenient to introduce from the beginning and to use systematically henceforth the repre­ sentation of a Riemann surface in terms of its universal covering space. Let Μ let Μ

be a compact Riemann surface of genus

be its universal covering space.

inherits from Μ hence Μ

g > 0 , and

The topological space Μ

in an obvious manner a complex analytic structure,

is itself a Riemann surface; and Μ

is topologically

quite trivial, being homeomorphic to an open disc. transformations form a group Γ

of complex analytic homeomorphisms

Τ: Μ — > Μ j and the Riemann surface Μ quotient space M/r .

The covering

can be identified with the

It will be assumed that the reader is familiar

with the topological properties of covering spaces, so that no fur­ ther details need be given here.

-1-

Select a base point lying over ρ

ρ

eΜ

and also a base point

ζ

eΜ

. Having made these selections, there is a canonical

isomorphism between the covering transformation group Γ fundamental group 7Γ. (M,p ) of the surface Μ

and the

based at ρ

is the isomorphism which associates to any transformation

j this

Τ eΓ

the class of loops in ττ- (Μ,ρ ) represented by the image in Μ any path from

ζ

to

Tz

of

in Μ . Again the details will be omit­

ted, since they can be supplied in a quite straightforward manner by anyone familiar with the topological properties of covering spaces; but it should be noted that this isomorphism does depend on the choice of the base point point in Μ

lying over ρ

ζ

, since the selection of another base alters the isomorphism by an inner

automorphism of the group τ. (M,p ) . Now for the more detailed properties, recall that topologically Μ

is just a sphere with

g handlesj hence

Μ

can be dis­

sected into a connected contractible set by cutting along 2g paths, as indicated in the accompanying diagram. handle 2 / / / ^ ~ N V , \

Each loop a.

lifts to a unique path a.

or β.

beginning at the base point Α.ζ , where A. e Γ l ο ' ι topy class of a.

ζ

j the path a.

or β.

runs from

in Μ ζ

to

is the transformation associated to the homo-

in τν. (Μ,ρ ) under the isomorphism introduced

above, and the path ' ^

β. ι

κ

runs from

ζ ο

to Β.ζ , where ι ο '

Β. e Γ ι

is

associated to β.

in the corresponding manner.

Μ - ^ (α. U β.) X i=l 1

is simply connected, hence lifts homeomorphically

to a number of disjoint open subsets of Μ

The complement

which are permuted by

the action of any element of the covering transformation group.

It

follows readily, upon tracing out the boundary of this complement in the preceding diagram, that one of these liftings has the form indicated in the following diagram.

c

C

iA2zo

cα

C

lzo

lA2B2Zo

Α Β ζ

1 1 Ο

-3-

In this diagram the elements

axe the commutators

(1) and. the domain

as indicated is hameamorphic to the complement The point set closure

polygonally shaped subset of

of this domain is a

with boundary consisting of the

closed curve

As usual in the discussion of the groupoid of paths, a product of paths is the path obtained by traversing first

and then

this multiplication is noncommutative, but is defined only when the end point of

coincides with the initial point of

notation used in (2) should lead to no confusion. r

the group

, hence the

As is well known,

is generated by the 2g elements

and these generators are subject to the single relation (3) where again

denotes the commutator (l).

The fundamental group

is of course correspondingly generated by the loops subject to the corresponding relation; that this relation does hold is obvious from (2) , although it does require some more work to show that all other relations axe consequences of (3).

- k -

A selection of base points cuts α ,...,α ,β ,...,β ι g -^g

ρ e Μ, ζ ε Μ and a set of ο ' ο of the above canonical form will be called

a marking of the Riemann surface

Μ . A marked Riemann surface thus

has a specified base point, a fixed isomorphism between the covering transformation group of its universal covering space and the funda­ mental group at its chosen base point, a canonical set of generators for that covering transformation group and hence for the fundamental group, and a canonical dissection of the universal covering space into polygonally shaped subsets ΓΔ = (ΤΔ|τ e Γ) . The set Δ

will

be called the standard fundamental domain for the action of the covering transformation group Γ

on the universal covering space Μ j

and the notation introduced above for the canonical generators of Γ will be used consistently in the sequel. There are of course a vast number of possible markings for any given Riemann surface. As noted above, the choice of another base point in Μ between Γ

has merely the effect of altering the isomorphism

and ττ. (Μ,ρ ) by an inner automorphism, hence of alter­

ing the canonical generators of Γ

by an inner automorphism of Γ

while of course leaving the canonical generators of 7Γη (Μ,ρ ) unchanged, and of replacing the standard fundamental domain Δ a suitable translate of Δ

by

under the action of Γ . An arbitrary

orientation preserving homeomorphism of the topological space Μ clearly transforms any marking into another marking, modulo choices of base points in Μ ; and it is easy to see that conversely any two markings of Μ

can be transformed into one another by some

orientation preserving homeomorphism of the underlying topological

-5-

space

Μ , again modulo choices of base points in Μ , since the

corresponding standard fundamental domains in Μ

are evidently

homeomorphic under an orientation preserving homeomorphism commuting with Γ . Thus, holding the base points

ρ

ε Μ, ζ

eΜ

fixed for

simplicity, all possible markings arise from a given marking by apply­ ing suitable orientation preserving homeomorphisms of Μ leaving

ρ

to itself

fixed. Note that any such homeomorphism determines in

turn an automorphism of the fundamental group ir, (Μ,ρ ) , taking the canonical set of generators corresponding to one marking into the canonical set of generators corresponding to another marking.

A

homeomorphism which is homotopic to the identity (through homeomor­ phisms leaving the base point fixed) clearly yields the identity automorphism of the fundamental group, so that the canonical sets of generators of the fundamental group π. (Μ,ρ ) associated to two markings so related actually coincide; two markings so related will be called equivalent, and the reader should be warned that in much of the literature only these equivalence classes of markings are really considered.

The more detailed investigation of these ques­

tions is an interesting subject in its own right, but must be left aside at present. The paths α ,...,α ,β ,...,β -L

g

-i-

can be viewed as singular g

cycles on Μ , and as such represent a basis for the singular homology group

Ηη(Μ,Ζ) ; thus a marked Riemann surface also has a

canonical set of generators for its first homology group.

The inter­

section properties of these one-cycles can be read off immediately

-6-

from the diagram on page 2. Note that the choice of another marking on the surface, by the application of a homeamorphism of Μ

to it­

self, determines an automorphism of the homology group H. (M,Z) J this automorphism can also be determined directly from the corre­ sponding automorphism of 7Γ. (M,Z) , recalling that just the abelianization of ττ. (M,Z) .

tL (M,Z) is

It should be mentioned that

there are nontrivial automorphisms of 7Γη (Μ,Ζ) which induce trivial automorphisms of IL (M,Z) ; there is a real and important distinc­ tion between properties of the surface which depend on homological and those which depend on homotopical properties, as will become evident later.

(b)

The Abelian differentials on Μ

ential forms of type

are the holomorphic differ­

(l,0) j they form a g-dimensional complex

vector space r(M, Θ- ' ) . Note that any Abelian differential ω e r(M, &• ' ) can be viewed as a Γ-invariant holomorphic differ­ ential form of type

(l,0) on the universal covering surface Μ ;

this form will also be denoted by

ω , a notational convention that

really leads to no confusion since Μ

can be identified with the

quotient space M/r . Since

Μ

such Abelian differential

is closed, there must exist a holo­

morphic function w

on Μ

ω

is simply connected and since any

such that

ω = dw j such a function is

called an Abelian integral for the Riemann surface. Note that the function

w

is determined uniquely up to an additive constant.

-7-

For a Riemann surface with a specified base point

ζ

e Μ , the

associated Abelian integral can be normalized so that w(z ) = 0 , and can thus be viewed as determined uniquely by the Abelian differ­ ential

ω ; indeed the Abelian integral is then given explicitly by

the integral ζ

w(z) = / z

ο

ω .

Note further that since the Abelian differential

ω

the Abelian integral w(z) has the property that ω(Τζ) - ω(ζ) = 0 for some constant readily that

for any

Τ e Γ , hence that

is r-invariant,

d[w(Tz) - w(z)] =

w(Tz) = w(z) - ω(τ)

ω(τ) e C depending on Τ e Γ .

It then follows

co(ST) = u)(S) + ω(τ) for any two elements

S, Τ e Γ ;

hence the set of these constants can be viewed as an element ω e Hom(r,C) , which will be called the period class of the Abelian differential

ω e r(M, $· ' ) . This terminology is suggested by Tz the observation that ω(Τ) = - / ° ω for any Τ e Γ . Note ο finally that an Abelian differential is determined uniquely by its period class; for if

ω, ,u„ e r(M, ©

class, their difference

' ) have the same period

ω, - ω ? = d[w - w 2 ]

is the derivative of

a holomorphic Γ-invariant function on Μ , hence that difference must vanish since the only holomorphic functions on the compact Riemann surface M/r

are constants.

Now select a marking for the Riemann surface Μ ω.,...,ω

and a basis

for the Abelian differentials on Μ . The period classes

-8-

of these Abelian differentials are determined by their values ω. (Α.), ω. (Β.) formation group

on the canonical generators for the covering transΓ ; these values can be grouped together to form

the associated

g χ 2g

fi" = (ω. (Β.)}

are

period matrix

gxg

(Ω' 3 Ω")

}

where

Ω'={ω.(Α.)} ,

matrices.

Theorem 1. The period matrix

(Ω',Ω")

of a basis for the

Abelian differentials on a marked Riemann surface

Μ

satisfies the

conditions (i) (ii)

Ω'· Ω" - Ω"· Ω' = 0 , (Riemann's equality), and ΪΩ'· Ω" - ΪΩ"· Ω'

is positive definite Hermitian,

(Riemann's inequality). Proof. Although this was proved in the earlier lecture notes (Theorem 17) > it is perhaps worthwhile repeating that proof to show how the intersection matrix of the canonical basis for the one-cycles on a marked Riemann surface can be calculated and to serve as a model for several quite similar later calculations. The essential point in deriving Riemann's equality is that viewing these as forms on Μ

ω.~ ω. = 0 ; J

and integrating over the standard

fundamental domain Δ , and recalling that the boundary of Δ the form given in (2), it follows that:

-9-

has

which establishes (l).

The essential point in deriving Riemann's

inequality is that for any Abelian differential a local coordinate system

hence that

with equality holding only when

In particular, putting

for arbitrary complex constants

t^ , it follows

that

with equality holding only when

thus the matrix

, where

is positive definite Hermitian.

To

determine this matrix explicitly, it follows as above that

which establishes (ii) and concludes the proof. Consider then another basis differentials on

M , where

complex matrix

for the Abelian for some nonsingular

The period matrix associated to this

basis evidently has the form

Note that an

immediate consequence of Riemaim's inequality is that the matrices and

are nonsingularj for if

possible to choose a nonzero row vector hence such that

were singular it would be such that contradicting Riemann's

inequality, and similarly of course for

-11-

Thus there is a unique

basis

ou,...,(j

for the Abelian differentials on Μ

associated period matrix has the form

(l,fl) , where

such that the I

is the iden­

tity matrix; this will be called the canonical basis for the Abelian differentials on Μ

associated to the given marking.

For such a

basis the period matrix is of course determined by the second com­ ponent

Ω = {ω..} j where lj

ω.. = ω.(Β.) , while -*-0

terms of the Kronecker symbol.

1

ω.(A.) = δ. in

υ

^-

u

J

In this case, Riemann's equality

reduces to the assertion that the matrix

Ω

is symmetric, and

Riemann's inequality reduces to the assertion that the imaginary part of the matrix

Ω

is positive definite.

The matrix

Ω

itself

will be called the canonical period matrix for the marked Riemann surface. The canonical period matrix for a marked Riemann surface does depend on the choice of the marking; a more detailed discussion of this will be left aside at present, but it is at least worth pointing out here that the canonical period matrix is only a homological invariant.

If two markings of the surface determine the

same canonical generators for Ηη(Μ,Ζ) , then the canonical period matrices associated to these two markings obviously coincide; thus the only analytic invariants of the surface

Μ

that can possibly

be expressed as functions of the canonical period matrix

Ω are

those which are also homological invariants, in the same sense. In the proof of Theorem 1, Riemann's equality was demon­ strated as a direct consequence of the equalities

-12-

ω.Λω. = 0

for

any pair of Abelian different!alsj but a more careful examination of the proof shows that Riemann's equality is really rather weaker than the equalities

ω.Λω. = 0 , indeed is equivalent to the -^- J equalities /. ω.Λω. = 0 . That is to say, Riemann's equality ι J merely reflects the fact that the differential forms ω.Λω. are -^- J homologous to zero, and not the stronger assertion that these forms actually vanish.

This suggests looking more closely into this situ­

ation, to see what further properties can be deduced from the equalities

ω.Λω. = 0 .

Choosing then a canonical basis

ω, ,...,ω

differentials on the marked Riemann surface morphic differential forms

for the Abelian

M, introduce the holo-

σ. . = w.co. e r(M, 0- ' )

are the normalized Abelian integrals. The equalities

, where w. ω.Λω. = 0 1

υ

are of course precisely equivalent to the conditions that these forms σ.. be closed, since do. . = ω.Λω. : and since the universal 10 10 ι 0 ' covering space Μ is simply connected, these conditions are in turn equivalent to the existence of holomorphic functions s. . e r(M, Θ- ) such that ij

σ. • = ds. . . The functions

10

10

s. . are

ij

of course only determined up to an additive constant, so can be fixed uniquely by the normalization

s..(z ) = 0 ; and this normal­

ization will be adopted systematically henceforth. any element

Τ eΓ

the differential forms

σ.. ij

hence the functions

s.. satisfy 10

-13-

Note that for

satisfy

(5)

for some complex constants

depending on

; the set

of these constants can be viewed as a mapping

and

these mappings will be called the quadratic period classes of the canonical basis for the Abelian differentials on the marked Riemann surface. It follows immediately from (5) that these quadratic period classes satisfy the formal algebraic condition (6)

for any elements

S,T £ r j this is the analogue for the quadratic

period classes of the condition that the ordinary period classes of Abelian differentials are homomorphisms from

r

into

C . Note

that as a consequence of (6),

(7) for any commutator for

r

and since the canonical generators

satisfy

where

it follows further that

Thus Riemann's equality is a formal consequence of the existence of quadratic period classes satisfying (6), reflecting the observation made above that Riemann's equality is weaker than the equations

-ih-

involved in the construction of the quadratic period classes. A more detailed discussion of these quadratic period classes must be left until the later discussion of general multiplicative properties.

(c)

The meromorphic Abelian differentials on Μ

morphic differential forms of type

are the mero­

(ΐ,θ) , which compose a vector

space ΓίΜ,Ί^ ' ) . These forms are of course determined by their singularities, up to additive holomorphic Abelian differentials; and there arises almost immediately the problem of deciding what these singularities can be.

holomorphic differential forms of type sheaf 7)( '

(S- '

Viewing the sheaf

of germs of

(1,0) as a subsheaf of the

of germs of meromorphic differential forms of type

(1,0) , the natural quotient sheaf

f

λ

'°

= IT)1'0/

d? 1 ' 0 describes

these singularities locally; this will be called the sheaf of germs of principal parts of meromorphic differential forms of type In terms of a local coordinate

ζ

in a coordinate neighborhood of

Μ , any meromorphic differential form of type in the form

f(z)dz

(l,0).

(l,0) can be written

for some meromorphic function

f(z) ; and

evidently the principal parts of the Laurent expansion of that mero­ morphic function at each of its poles describes the residue class of that differential form in the quotient sheaf

/Μ

ί

1 ο ' . These local

expansions can thus be used to describe elements of the sheaf

f

Recall that the residue of a meromorphic differential form

at

one of its poles

ρ

is defined to be the integral of

φ

φ

' .

around a

closed path encircling that pole once in a counterclockwise direction

-15-

and containing no other singularities of will be denoted tion to

this residue, which

clearly remains unchanged by the addi-

of any holomorphic differential form of type

(1,0) ,

hence can be extended to be defined on the elements of the sheaf

From the exact sequence of sheaves

there results an exact cohomology sequence including the segment

hence given any collection of principal parts the surface M

on

M , there exists a meromorphic Abelian differential on

having those principal parts as its singularities precisely when From-16the Serre duality theorem it follows

that

is canonically dual to

. and

for a compact Riemann surface; thus meromorphic to fication equated canthe be following identified Theorem to can the Abelian betotal 2. so if result. with chosen On differentials and residue a the only compact that complex if ofthe the Riemann with numbers element total a prescribed surface residue C . Indeed M of leading principal there this is exist thereby zero identipart is.

Proof.

As noted, it is only necessary to show that can be identified with the total residue of

under an appropriate isomorphism covering

of

M

.

by contractible open coordinate neigh-

borhoods, such that each singularity of only one of the sets any singularity of

Choose a

is contained within

indeed such that an open neighborhood of contained within

other sets of the covering.

is disjoint from the

In each of these sets

the section

can be represented by a meromorphic function ; and in each intersection

the differences

are holomorphic differential forms, and the cocycle represents the cohomology class Then to apply Serre duality it is enough to choose forms

_

differential

such that in

; for

, so these differential forms represent a

global form

, and the isomorphism

can be taken as that associating to the cocycle constant

. Now for each set

the complex

choose a

function

which is identically zero in an open neighborhood of any singularity of

in

and which is identically one in

It is then possible to take

so that

whenever j and

is identified with the total residue of

and the proof is thereby concluded.

-17-

over

M

For any principal part

φ e r(M, Τ

' ) having total residue

zero there exist meromorphic Abelian differentials having the singu­ larities specified by

φ .

It is customary to call these forms

Abelian differentials of the second kind if their residue at each point of Μ

is zero, and Abelian differentials of the third kind if

their total residue is zero but they have nonzero residues at some points.

The holomorphic Abelian differentials are called the Abelian

differentials of the first kind. It is in some ways convenient to begin the more detailed discussion of meromorphic Abelian differentials by considering Abelian differentials of the third kind; and the simplest such differentials are those having two simple poles with opposite residues. any two distinct points

on Μ , there exists a meromorphic

φ e r(M, 7Ϊ\ ' ) having as sole singularities

Abelian differential simple poles at ρ

ρ ,ρ

Selecting

with residue

-1 and at ρ

with residue

+1.

The most general such meromorphic Abelian differential is of course φ+ω

for an arbitrary holomorphic Abelian differential ω£Γ(Μ, ff ' )

and a unique canonical such form can be specified by imposing some suitable conditions on the periods. Some care must be taken in defin­ ing the periods of such a differential form, though, since it has non­ zero residues; but the difficulties can easily be avoided as follows. Select a simple arc

6

from

ρ

to

ρ

on Μ , and let

δ be any

lifting of this arc to the universal covering space Μ . The dif­ ferential form

φ

can be viewed as a Γ-invariant meromorphic

differential form on H ; and as such it is a Γ-invariant holomorphic

-15-

differential form on Μ - Γδ , and has zero integral around any closed path in Μ - Γδ . This form can therefore be written as the exterior derivative of a holomorphic function f

function

f on Μ - Γδ , the

being unique up to an additive constant. Wow as in

the case of the holomorphic Abelian differentials it follows that f(Tz) = f(z) - φ&(τ)

for some complex constants

φδ(Τ)

depending

Τ e Γ ; and the set of these constants can be viewed as an

on

element

cp_ e Hom(r3C) , which will be called the period class of

the meromorphic Abelian differential

φ with respect to the arc δ .

Having chosen a marking of the Riemann surface

Μ , this period

class is of course determined uniquely by its values

φ(Α.), φ(Β.)

on the canonical generators of Γ . There is a unique holomorphic Abelian differential also having the periods cal generators form

Α-^,.,.,Α

; hence there is a unique differential

φ + ω having zero periods on the canonical generators

A^...jA

. This will be called the canonical Abelian differential

of the third kind associated to the arc uu . Thus

the points

ρ

having as sole singularities simple poles at

with residue

is an arc from

such that

δ , and will be denoted

ut e r(M, Ύΐ[ ' ) is the unique meromorphic Abelian

differential on Μ

δ

1 , and the covering

has a correspondingly explicit form.

How-

ever this identification really plays no role whatsoever in most of the present discussion, and will generally be ignored here.

(b)

The period classes of Abelian differentials can also be

introduced as follows.

From the exact sequence of sheaves

there follows an exact cohomology sequence, which for a compact Riemann surface reduces to the sequence

as noted in §8(a) of the earlier lecture notes.

and the connecting homomorphism

Now

can thereby be identified with

the mapping which assigns to any Abelian differential its period class.

The remainder of this portion of the exact cohomology

sequence appears as a condition determining which elements of

-29-

Η (M,C) = Hom(r,C) are the period classes of the Abelian differ­ entials on Μ ; the condition is just that an element of tt (M,C) > Η (Μ, Φ . The resulting condition is a

trivialityj but it is a good exercise to verify that that is the case by tracing this condition through in detail. The algebraic condition (6) satisfied by the quadratic period classes should be familiar to anyone acquainted with the cohomology theory of abstract groups. That condition is just that the two-cocycle

c(S,T) = (O.(S)IO.(T) e Ζ (r,C) is the coboundary

of the one-cochain of the group Γ

σ..(τ) e C (r3C) , in terms of the cohomology ιJ with coefficients in the trivial Γ-module C .

Actually the two-cocycle

c(S,T) = ω. (S)oo.(T) is Just the cup pro-

duct of the one-cocycles

ω.(Τ) and ι

i

J

ω.(Τ) , and the condition J

that this two-cocycle be cohomologous to zero is the algebraic reflection of the fact that the differential form

ω.(ζ)Λω.(ζ) is 1

t)

cohomologous to zero on Μ ; so the algebraic condition that there exist some mappings

σ..(Τ) satisfying (6) is really equivalent

to Riemann's equality. (c)

That a function of several complex variables is meromorphic

if it is meromorphic in each variable separately is a result of W. Rothstein, extending the well known theorem of Osgood and Hartogs with the corresponding assertion for holomorphic functions. For a

-30-

survey of these results, see H. Behnke & P. Thullen, Theorie der Furiktionen mehrerer komplexer Ver&nderlichen (second edition, Springer, 1970). If then points of

is identified with a subset of the complex plane, M

can be viewed as complex numbers and the canon-

ical Abelian integral of the second kind can be defined simply as the derivative

(13)

and the corresponding canonical Abelian differential as

(1*0 This is in many ways a great convenience, especially when considering Abelian differentials of the second kind with higher order poles; and the principal parts of meromorphic Abelian differentials can then be described without the need for specifying a choice of local coordinate system.

The canonical Abelian differentials of the

second kind can also be viewed as a symmetric differential form

on the product manifold ingly as the homomorphism

M x M , and the period class correspondHorn

described by

For such a simple situation as this, though, it seems unnecessary

-31-

to worry unduly with such additional machinery. When considering meromorphic functions rather than meromorphic differential forms, the appropriate sheaf of germs of principal parts is the quotient sheaf

; and from the

exact sequence of sheaves

there results an exact cohomology sequence including the segment

In this case the Serre duality theorem asserts that canonically dual to

.

is

; and it is easy to verify, paral-

leling the proof of Theorem 2, that this pairing is that which associates to the section

and to any hclomorphic

Abelian differential

the complex number

Consequently there exists a global meromorphic function on principal part

f

if and only if

M

with for

j = l,...,g , in terms of any basis for the space of Abelian differentials

. Actually the more detailed analysis of

this result can be reduced directly to the study of the meromorphic differentials, though. note that

df

For given any principal part necessarily has zero residue, hence

there exists a global iheromorphic differential form

-32-

having the principal part

df ; and this form is unique up to an

additive Abelian differential form

ω e r(M, Θ

' ) . Then there

exists a global meromorphic function with the principal part and only if there exists an Abelian differential such that the form

φ+ω

f if

ω e r(M, CP . Note that for any covering transformation φ(Τζ) - φ(ζ) = Cw^Tz) - v±(z)} consequently

φ

Τ eΓ ,

= {OJ^T"1)} e %

induces a complex analytic mapping

;

φ: Μ

> J(M) ,

which will he called the Jacobi mapping of the marked Riemann sur­ face

Μ

into its Jacobi variety. Theorem 5.

Jacobi mapping between

Μ

For a Riemann surface Μ > J(M)

φ: Μ

of positive genus the

is a complex analytic homeomorphism

and a complex analytic submanifold

Proof.

The Jacobian of the mapping

W-, C J(M) .

φ

is the vector con­

sisting of the canonical holomorphic Abelian differentials on Μ , and is nonsingular since these differentials have no common zeros at any point of Μ notes).

Since

(as observed on page 119 of the earlier lecture

Μ

is compact, it follows immediately that the image

Wj = φ(Μ) C J(M)

is a complex analytic submanifold of

that the mapping

φ: Μ — > W-,

morphism. mapping p., p_

φ

is locally a complex analytic homeo­

It is a simple consequence of Abel's theorem that the is one-to-one. For if

on Μ

the vector

J ( M ) , and

and

δ

φ(ρ,) = φ(ρ?)

is any simple arc from

u = {u.} with components

lattice 7 , and consequently

ζ

ρ

for two points to

p ? , then

u. = /„ ω. belongs to the

= ζ_

(upon recalling Corollary

2 to Theorem 18 of the earlier lecture notes); but

ζ p

a surface of positive genus implies that

ρ

= p.

l

= ( * ^2

for

(upon recalling

Lemma 16 of the earlier lecture notes and the subsequent discussion).

-35-

This then serves to conclude the proof. Since the Jacobi variety

j(M)

has a natural group struc-

ture, it is evident that the Jacobi mapping

extends

uniquely to a homomorphism from the free Abelian group generated by the points of

M

to

J(M) .

the points of

M

can be identified with the group of divisors on

M , the group

Of course this free Abelian group on

where

germs of divisors on

is the sheaf of

M j and the group homomorphism is clearly given by

where

Recall that two divisors

in

are called linearly equivalent, written

if

their difference is the divisor of a meromorphic function on

M ;

that this is the equivalence relation naturally corresponding to the homomorphism

which associates to a

divisor

the complex line bundle , where

ciated to

p^ e M ; and that the group of linear equivalence

classes of divisors on

M , the divisor class group of

isomorphic to the group M

is the point bundle asso-

M , is

of complex line bundles over

under the induced homomorphism. of the line bundle

Note that the Chern class associated to the divisor

is just equal to the degree divisor.

-36-

of that

Theorem 6.

For a Riemann surface

M

of positive genus the

Jacobi homomorphism

is surjective, and its kernel consists of those divisors such that point of

where

is the base

M .

Proof.

Note that for the base point

it follows

from the definition of the Jacobi mapping that and since

is a group homomorphism, for any divisor

.

Now the divisor

has degree zero, so can be written in the form for some points selecting a r c s f r o m

p. J

t

o

i

n

i

of s

M j and

evident that

is represented by the vector having components

for

precisely when

Then

, and by Abel's theorem that is

in turn equivalent to the condition that the kernel of

has the form asserted.

hence For a divisor

of

degree

it follows in particular that

cisely when

, or equivalently precisely when the associated

line bundle restriction of

i pre-

is analytically trivial; consequently the to the subgroup of divisors of degree zero

induces an infective homomorphism from the Picard variety to the Jacobi variety

-37-

J(M) .

Since both

P(M)

and J(M)

are complex tori of dimension

homomorphism must evidently be an isomorphism; so of course

g

this

φ

is

surjective, and the proof is thereby completed. As an immediate corollary of this result, note that the map­ ping which associates to any divisor cp( $ ) e J(M),

/$• e r(M, «0 ) the pair

\ I& I e Ζ , is a surjective homomorphism from

r(M, JS ) to the direct sum group

J(M) θ Ζ , and its kernel con­

sists precisely of those divisors linearly equivalent to zero; this homomorphism thus determines an isomorphism between the divisor class group of Μ

and the group

homomorphism r(M, iS ) Λ?

= Σ.ν.ρ. 0 JO

J(M) Φ Ζ . Recalling that the

> π (Μ, #

the line bundle

f V

) associating to a divisor

= Π. t J Pi

induces a canonical

isomorphism between the divisor class group and the group of line bundles over Μ , the composition of these two isomorphisms leads to a canonical isomorphism ^(M, Θ *) = J(M) θ Ζ . In particular, there results a canonical isomorphism Picard variety variety

P(M>· = (ξ e Η (Μ, Θ )|c(|) = 0}

between the

and the Jacobi

J(M) ; and this canonical isomorphism will be used quite

freely in the sequel to identify the Picard variety Jacobi variety

P(M) with the

J(M) . This is of course really just an explicit

form for the isomorphism discussed in §8 of the earlier notes; it can be viewed as providing an explicit coordinate representation of the Picard variety as a complex torus. At the same time it

-38-

demonstrates that the Jacobi variety as an analytic group is really independent of the marking on the surface, except for the choice of "base point on the surface and corresponding identity element of the group. Note that the image under the Jacobi mapping of a point

p eM

is the same as the image under the Jacobi homo-

morphism

of the divisor

divisor corresponds to the line bundle as a subset of the Picard variety,

j and this , Thus viewed

consists of those line

bundles

which can be written in the form

for

some point

, This is equivalent to the condition that the

line bundle

have a nontrivial holomorphic section; for does have such a section, and conversely if

a nontrivial section with divisor

p

then

has Writing

for any complex line bundle

it follows

that

(b)

The image under the Jacobi homomorphism

of the set of positive divisors of degree a well defined subset

r

on the surface

This subset

M

is

can be described

equivalently as the image of the complex analytic mapping defined by , where copies of the Riemann surface

is the Cartesian product of M

-39-

r

and has the obvious structure of

a compact complex analytic manifold of dimension

r . It is a well

known result in the theory of functions of several complex variables (the proper mapping theorem) that the image of any such mapping is a complex analytic subvariety of the complex manifold the subsets

W C j(M)

J(M) ; hence

axe complex analytic subvaxieties. Actually

these axe also irreducible subvaxieties, in the sense that any meromorphic function on set of W

J(M) which vanishes on a relatively open sub­

necessarily vanishes identically on W

j that is an

immediate consequence of the identity theorem for functions of several complex variables, since the restriction to W morphic function on

J(M)

function on the manifold

of a mero-

can be identified with a meromorphic W

terms of local coordinates

by means of the mapping ζ . near the points J > J(M)

Jacobian of the analytic mapping

φ: Μ

(z..,...,ζ ) e W

r χ g matrix

is clearly the

φ . In

ρ. e Μ , the J at the point (w?(ζ.)) , where

ω.(ζ.) = w.'(z.)dz. are the canonical holomorphic Abelian differentials on Μ

expressed in these local coordinates; and since the

Abelian differentials are linearly independent, it is evident that the Jacobian matrix will have maximal rank on a dense open subset of Μ

, indeed on the complement of a proper complex analytic sub-

variety of Μ

.

lecture notes.) φ: Μ

> J(M)

(See the discussion on page 119 of the preceding In particular, for r = l,...,g , the mapping will be a nonsingular local homeomorphism on a

dense open subset of the manifold

-U0-

Μ

j and calling once again on

some results from the theory of functions of several complex variables , it follows that the irreducible subvarieties of dimension

r , for

oxe

Of course for

it has

already been demonstrated that the Jacobi mapping is everywhere nonsingular, is indeed a complex analytic homeomorphism between of

M

For

and r = g

, so that

is an analytic submanifold

it follows that

, since

a manifold (hence irreducible) and dimension

= dimension

. . is .

;

this assertion is traditionally known as the Jacobi inversion theorem. For

the mapping

ties, and the image

does have singulari-

may also have singularitiesj a more

detailed analysis will be postponed to a later section of these notes. where

Note finally that is the base point of the marked surface for any

r .

M , hence that

There is consequently a chain of irreduc-

ible analytic subvarieties

where dimension Viewed as a subset of the Picard variety note that the subvariety consists precisely of those complex line bundles can be written in the form on the surface

W^ which

for some points M .

This is of course in turn equiva-

lent to the condition that the line bundle

- i a -

have a nontrivial

holomorphic section; for conversely if

does have such a section, and

has such a section with divisor

then

Writing

complex line bundle

for any

it thus follows that

(1) Incidentally, since

it is a familiar consequence of

the Riemann-Roch theorem that hence that

whenever

whenever

, and

for any

this then

provides an alternative proof of the Jacobi inversion theorem, that whenever An obvious and useful question to ask is what effect the natural algebraic operations of the group of irreducible analytic subvarieties.

J(M)

have on this chain

For any subset

it

is possible to introduce the inverse set the translate

, and

; and for any pair of subsets it is possible to introduce the sum and the quotient This last set must of course b e

distinguished from the difference

; the construc-

tion is a familiar o n e , in the context of ideals. analytic subvariety of they arise from

S

J(M)

then so are

and

If

S

is an since

b y the application of an analytic homeomorphism

-1+2-

of the manifold of

J(M) .

If

S and Τ are analytic subvarieties

J(M) then as a consequence of the proper mapping theorem so i s

S + Τ , since i t can be viewed as the image of the analytic mapping from the compact analytic variety ( s , t ) —> s + t j and i f S © Τ for any subset (2)

S

S χ Τ into

J(M) defined by

is an analytic subvariety so also i s

Τ , since

S©T = Π

(S-t)

t e Τ

is an intersection of analytic subvarieties of J ( M ) . These oper­ ations can be applied to the analytic subvarieties W following resultsj for simplicity define W

, with the

= 0 , the identity

element of the group J ( M ) , noting that as a subset of the Picard variety

P ( M ) this set can be described as

W = {ξ e P(M)|/(|) > 1} , paralleling the alternative description given for the other subvarieties Wr . Β Lemma 1. For any integers (3) \-J/

r,s > 0 ,

Wr + Ws = Wι r+s

and for any integers (k) \ '

r,s such that

0 < r < s < g-1 ,

W ©W =W w s r s-r Proof.

The first assertion is a trivial consequence of the

definition. As for the second assertion, viewing these sets as subvarieties of the Picard variety

M-

P ( M ) , note that

Now for any line bundle

such that

it follows easily though that precisely when

for all points

p e M

and b y iterating this observation it

then follows that

as desired.

To complete the proof b y demonstrating this auxiliary

statement, note firstly that for all points

obviously implies that p e M .

On the other hand if

it follows from the Riemann-Roch theorem that and that the canonical bundle.

, where Now selecting a point

is

p e M

least one of the holomorphic sections of the bundle z e r o , there axe evidently at most holomorphic sections of

p

which vanish at

is

p ; but the dimen-

, hence

that p o i n t , and the proof is concluded.

-kk-

is non-

linearly independent

sions of the space of holomorphic sections of at

at which at

which vanish at

The condition that

in the second assertion of the

lemma is of course quite necessary, since for and consequently

necessarily as well.

Note that

having proved Lemma 1 , the corresponding statements for translates of these subvarieties follow quite trivially; hence

(5) for any integers

and any points

, and

(6) for any integers

and any points

A special case of the lemma which merits particular mention is the assertion that , since thus the subsets being preserved b y translations.

are quite far from

Another special case which also

merits note is the assertion that

(7)

this equation shows that the submanifold merely from the terminal portion

can be recovered of this

sequence of subvarieties, hence that the original Riemann surface is determined b y that portion of the sequence.

-45-

(c)

When viewed as a subset of the Picard v a r i e t y , the sub-

variety of positive divisors was characterized as e Ρ(Μ)|> 1} ; and from t h i s point of view i t P o only n a t u r a l to introduce the further subsets

W = {|

(8)

is

W^ = £ fi e Ρ ( Μ ) | 7 ( ξ ζ ' ) > V )

for arbitrary integers

ν > 1 . These subsets will be called the

subsets of special positive divisors for

ν > 1 , and form a de­

scending filtration W

r

= W 1 D W 2 D W 3 D ... r — r — r —

of the subvariety W

of positive divisors of degree r.

They

can also be characterized quite conveniently as'follows. Lemma 2. For any integers (9)

W^ =vr_v+1Q(.-vv-1)

(10)

W Proof.

= 0

r > 0

and

whenever

whenever

ν > 1 , ν < r+1 , and

ν > r+1 .

Letting χ e j(M) correspond to a line bundle

ξ £ P(M) , it follows from the definition that χ e W when

γ(ξζ

) > ν . Now the existence of at least o independent holomorphic sections of the bundle ξζ

ν

precisely linearly

P

is clearly o equivalent to the condition that there exists a nontrivial holoP

morphic section of the bundle

ξζ

with zeros at an arbitrarily o V-1 points of the Riemann surface, hence to the p

specified set of

-k6-

condition that

for arbitrary points

on the Riemann surface. Chern class ever

Since this line bundle has

the last condition is evidently impossible whenp so that

in that case.

Otherwise the last

condition can be rewritten in the form , and is thus in turn equivalent to the condition that

or in

terms of the Jacobi variety, that for any points

on the Riemann surface; but this is

precisely the definition of the subsel

, and the

proof is thereby concluded. It is an immediate consequence of this lemma that the subv sets

W^

are complex analytic subvarieties of the Jacobi variety

j(M) , indeed that they can be written in the form

(11) whenever extended b y putting

and are otherwise empty. whenever

If the notation is

r < 0 , then equation (10)

can b e subsumed as a special case of equation (9).

The assertions

of the lemma can also b e inverted, yielding the equation

(12) for any integers

-h7-

If

it follows immediately from the Riemann-Roch

theorem that

, so that

and

when w

h

e

n

a

when

and from Lemma 2 it follows that n

when

d 2 .

that

when

and

These cases being quite trivial, it

is evidently only reasonable to restrict attention to those subvarieties

with indices in the range

As a useful bit of further notation, note that the divisors of any two holomorphic or meromorphic Abelian differentials are linearly equivalent, hence that the image any such divisor

of

is a well defined point of the Jacobi variety;

this point will b e called the canonical point of of the Picard variety, the canonical point the line bundle

where

of the Riemann surface.

k

K

J(M) .

In terms

is represented b y

is the canonical, bundle

It follows from the Riemann-Roch theorem

that

for any line bundle

; note that if

a point k- x e J(M) .

represents the point Now

precisely when

cisely when and

represents

hence pre. ; and if

this last inequality is just the condition that Thus the Riemann-Roch theorem leads immediately

-1+8-

to the identity V

k - W = i/^

(13) K

'

r

s

where s = 2g-2-r and μ = g-1 - (r-v) , provided of course that r, s > 0 and ν, μ > 1 so that both sides are well defined. Note first of all that when g-1 < r < 2g-2 then necessarily 0 < s < g-1 j thus when considering the analytic properties of the separate subvarieties W

, rather than their locations and inter­

relations in J(M) , it is really sufficient to restrict attention to those subvarieties with indices in the range 1 < ν < r < g-1 . The most symmetric case of formula (13) is that in which r = s = g-1 , and in this case k - WV . = WV . g-1 g-1

(lU)

for 1 < V < g-1. In particular, when ν = 1 , it follows that k - W g-1, = Wg-1 ., ' . hence that k = Wg-1 _ θ (-Wg-1' . ) ': and this condition of course determines the canonical point uniquely. Now Lemma 2 asserts that W = W 1 < V < r , and i f i n a d d i t i o n W

.

and

-W

,n θ (-W ,) whenever

r r-v+1 r-v+1 < g and

a r e a n a l y t i c s u b v a r i e t i e s of

v

v-1' v-1 < g

then

J ( M ) of diaienV

sions

r-v+1

whenever

and

v-1

respectively;

W = jfl

v-1 > r - v + 1 , s i n c e i n t h a t case no t r a n s l a t e of

can p o s s i b l y be c o n t a i n e d i n (15)

and c l e a r l y t h e n

V

W =φ

W

. .

whenever

-k9-

Thus 2v > r+2

Λ

-W

where

and

line ease that

Furthermore, in the border-

2v = r+2 , the subvariety

consists of those points

such that

equivalently such that

since

or and its

translates are irreducible subvarieties; and there is obviously at most one such point (16)

Thus

either point

consists of the unique u e J(M)

such that

, when-

ever Of course in the special case

v = g , the set

consists precisely of the canonical p o i n t , as noted earlier. Note that (15) can b e rephrased as the inequality for all points

(

whenever

; the bundles the line bundles of Chern class

r .

2v > r+2

and

are of course all

This is easily seen to be

equivalent to the inequality

(17)

for all line bundles with Chern classes in the range

where as usual

denotes the greatest integer function.

This

represents a significant improvement in the estimate of the maximal values for

derived in the earlier lecture notes; the table

on page 113 of those notes can thus b e replaced the the following

-50-

table of inequalities relating the Chern class sion

and the dimen-

for any line bundle

(18) 0 m min

a

x

1

1

2

3

2

2

2

^

5

3

6

3

^

7

.

2g-b

2g-3

m a x g - 2

g-1

min

g-3

g-U

Note b y the way that if

.

g

+i ....

0

1

2g-2

2g-l

2g

g-1

g

g

g+1

g-2

g-1

g

g+1

£

g

H ...

0 0 0 0 0 0 0 0 . . .

2g-5

.

2

is a line bundle with even Chern class

and with the maximal number of holomorphic sections as listed in this table

, then

a point of the subvariety

represents

but as noted in (16) this sub-

variety is either empty or consists of but a single point.

Thus

on any Riemann surface there can exist at most one line bundle with Chern class is actually attained.

|

such that this maximum It will appear later that there are

surfaces (indeed the hyperelliptic

Riemann

Riemann surfaces) for which

this maximum value is attained; and of course tensoring such a bundle with an arbitrary point bundle w i l l yield a line bundle of odd Chern class for which the maximum is also attained.

Thus the estimates

provided b y this table (18) or b y the inequality (17) are the best

-51-

possible, in general. Consider next a fixed point ρ e Μ

with image

φ(ρ) = χ e W. C J(M) , represented by the line bundle ξ = ζ ζ" e P(M) ; and note that for any integer Ρ P0 r-p has image

the divisor

cp(r-p) = r-x e r-W C W C j(M) , represented by the

ξ Γ e P(M) . The subvariety

line bundle

r > 1

r-W C »

thus corre­

sponds to the subset of positive divisors that can be represented as based at a single point on the Riemann surface Μ ; and these divisors were considered in some detail in the earlier lecture In particular, in Theorem Ik of those lecture notes it was

notes.

demonstrated that

r+1 - γ(ζ ) is equal to the number of Weier­

strass gap values at the point

ρ

in the sequence

1,2,...,r j

that is of course equivalent to the assertion that equal to the number of nongaps at the point sequence γ(ζ

P

v

1,2,...,r . Now r-x e W

) = γ(,ί ζ

PQ

(19)

precisely when

V-1 nongaps at the point

= (r-cp(p)| there are at least at ρ

ρ

among

in

v-1 nongaps

(l,2,...,r)} .

ρ e Μ which is not a Weierstrass point the gap sequence

is precisely r-W.. Π W

occurring in the

1,2,...,r ; thus

r-W Π W

At a point

is

) > ν , hence using the preceding observation, pre-

cisely when there are at least the sequence

ρ

/(ζ )-1

1,2,...,g j hence for 1 < V < r < g

the intersection

can only consist of multiples of Weierstrass points on

W.. , so is necessarily a finite set of points. Some at least of

-52-

these intersections are necessarily nonempty, depending on the gap structure at the Weierstrass points. Turning now to more general properties of these subvarieties of special positive divisors, Lemma 2 would seem to indicate that they should b e particularly well behaved under the operation and that is indeed the case. Lemma 3»

For any integers

(20) (21)

(22) Proof.

These assertions are immediate consequences of

Lemmas 1 and 2 and of the elementary observation that for any subsets

A, B, C

verify this observation, note that for all for all

b e B

and

, hence precisely when

of

J(M) .

To

precisely when x + c + b

e A

and the last condition is just that

Wow to prove (20) note that

to prove (21) note that

and to prove (22) note that as a consequence of (20) and (21) it follows that thus completing the proof. -53-

Although the subsets

are analytic subvarieties of the

Jacobi variety, they may not b e irreducible subvarieties if

v > 1 .

It is still possible to speak of the dimensions of these subvarieties though, where as usual the dimension of an analytic subvariety is defined to b e the maximum of the dimensions of its irreducible components; this dimension will b e denoted d

i

m

w

h

i

l

dim

e

Recall that

dim

for Lemma (23)

d

(2l+)

dim

If the subvariety

i

m

w

h

e

n

is nonempty, then e

v

e

r

,

whenever

Proof.

Note that

whenever

, so that

is then necessarily a proper analytic subvariety of If

J(M) .

the first assertion of the lemma is trivial.

wise select an irreducible component w

and

i

t

h

a

n

d

since

ducible variety

of the subvariety

note that b y Lemma 3-

analytic subvariety of

V

Other-

Now

V-W^

is an irreducible

J(M) , since it is the image of the irre-

V x W^

under the obvious analytic mapping

and since

either

V = V - W^

or

which is impossible since

V

variety of

is contained in J(m) ; therefore

and dim

is a proper analytic sub-

which is the first assertion of the lemma.

The proof of the second

assertion is quite similar.

select an irreducible

component

V

such that

If

dim V

= dim

and note that

since again V

= V

V" + W ^

b y Lemma 3;

is irreducible, and it is impossible that

+ W 1 , so necessarily

dim

That suffices to con-

clude the proof. Theorem 7.

The subvarieties

of special positive

divisors on a Riemann surface are analytic subvarieties of the Jacobi variety such that

Proof.

By successive application of the inequality (23)

cf Lemma U it follows that

Each of these

v-1

least 1 , so that

inequalities must cut the dimension b y at dim

and since it follows immediately that

(d)

This maximum is actually attained for hyperelliptic Riemann

surfaces, which are particularly rich in functions and are exceptional in several ways.

Recall that a hyperelliptic Riemann sur-

face is one that can be represented as a two

-55-

sheeted branched

analytic covering of the projective line; and that each such surface

M

has a unique analytic automorphism

of period

2 corresponding to the interchange of sheets in any representation of

M

as a two sheeted branched analytic covering of the projec-

tive line.

The meromorphic functions on

M

invariant under this

automorphism can b e identified with the meromorphic functions on the projective line; hence for any two points

p, q

of

M

there

w i l l exist a meromorphic function having divisor Thus any two divisors on

M

of the form

p + 0p

equivalent; the common image

are linearly of all such

divisors is a w e l l defined point of the Jacobi variety, which w i l l be called the hyperelliptic point of

J(M) .

In terms of the

Picard variety, the hyperelliptic point is represented b y any line bundle of the form

Such a line bundle

has the property that versly, whenever

, so that

is nonempty its unique point is represented

b y a line bundle

such that

hence the surface is hyperelliptic and the line bundle sents the hyperelliptic point.

unique point in

t| repre-

Thus hyperelliptic Riemann sur-

faces can be characterized as those for which

Now if

and con-

, and the

is then the hyperelliptic point. e

is the hyperelliptic point on the Jacobi

variety of a hyperelliptic Riemann surface it follows from (l6) that

-W^ = W ^ - e ; and iterating this relation,

-56-

Thus

is nonempty, indeed consists of the point

whenever

(v-l)-e ,

In particular, the canonical point is given

in terms of the hyperelliptic point b y the formula

k = (g-l)•e .

It farther follows, as already noted as a consequence of the condition

that the maximum values for

given b y

formula (17) on the table (18) are actually attained on hyperelliptic Riemann surfaces.

Finally note also that

on a hyperelliptic Riemann surface

(25) whenever

so the subvarieties of special position

divisors are irreducible analytic subvarieties of the maximal possible dimension in this case. The hyperelliptic Riemann surfaces not only provide examples of surfaces for which the maximal values described in the preceding discussion are actually attained, but are even characterized by the attainment of these maximal values.

A partial result in this direc-

tion is the following. Theorem 8 (Clifford's Theorem). in the range

If

index

v

genus

g , then that surface is hyperelliptic.

for some

for a Riemann surface of

-57-

Proof.

A s noted earlier, hyperelliptic Riemann surfaces

are characterized b y the condition that

so in order to

prove the theorem it is sufficient to show that if some index

v

in the range

some index

\

in the range

for

then

for

Note first of all that

from the Riemann-Roch theorem in the form (13) it follows that where assumed that value

and it can of course be (Note that this is the point at which the

v = g

must b e excluded, since then

p. = 1 .)

choose a point bundle

j and l

represented b y a line e

t

b

e

b y a line bundle bundle.

The point

x

can b e described as the image under the

M , and correspondingly

a divisor

of degree

y

2v-2

on the sur-

can b e described as the image of

of degree

Since

two points of the divisor

with

can b e specified quite arbitrarily,

so it can be assumed that at least one point of in

represented is the canonical

Jacobi homomorphism of a divisor face

Assuming that

and that at least one point of

also appears

does not appear in

the set o f common points (counting multiplicities) of the two divisors of degree

and

can b e viewed as forming a divisor

r , where

as a consequence of these

choices, and that divisor has as image under the Jacobi homeomorphism a point

.

Now recall that

can

also b e described as the dimension of the complex vector space

-58-

consisting of those meromorphic functions mann surface

M

such that

f

on the Rie-

, and correspondingly

is the dimension of the complex vector space associated to the divisor

as described on page 57 and the

following pages in the earlier lecture notes.

Clearly

so that

and since it is also evident that

it follows that

Rewriting these observations in terms of the dimensions of the spaces of holomorphic sections of the appropriate line bundles, it then follows that

where

(; e P(M)

is the line bundle corresponding to the point

z e J(M) j and since

-59-

the Riemann-Roch theorem can be applied once again to rewrite this inequality in the form Since the left hand side in this last inequality is

and from the upper bound provided b y (17), the right hand side in this inequality is

It thus

follows that

this can only happen when and when

r

is an even integer, say

This in turn means that

that

for some index

hence

in the range

and

as already noted, that suffices to conclude the proof of the theorem. An immediate corollary of this theorem is the observation that the inequality (17) can b e improved on a Riemann surface that is not hyperelliptic; for any line bundle Chern class

it follows that

with even whenever

since a line bundle with represents a point in

Thus

(26)

for all line bundles with Chern classes in the range

, when

the Riemann surface is not hyperelliptic. Correspondingly, when

M

is not hyperelliptic the table of inequal-

ities (18) can b e improved a l s o , as follows.

(27) 0 m

a

mill

x

1

1 2

0

2 2

0

3 2

0

4 2

0

5 3

0

0

6 3 0

7

.

.

.

^

.

.

.

0

.

.

.

2g-l

g+1

1

2

2g-5

Zg-k

2g-3

m a x . . .

g-2

g-2

g-1

g

g

g+1

min

g-4

g-3

g-2

g-1

g

g+1

...

2g-2

g

2g

Notes for §2. (a)

This section is really just a review, although from a

slightly different point of view, of material that can be found in §8 of the earlier lecture notes on Riemann surfaces.

For most if

not all of the results in §2, it is not really necessary to select a particular coordinate representation of the Picard variety

P(M) ,

such as the representation given here in terms of the Jacohi variety; indeed, the discussion in the remainder of §2 can be rephrased entirely in terms of the Picard variety, continuing more directly the treatment begun in §8 of the earlier lecture notes. However there are some later points at which the explicit representation of the Picard variety is useful, if not essential, particularly when discussing homotopy rather than homology invariants; and the change in point of view may serve as a useful element in reviewing the relevant portions of the earlier lecture notes. The Jacobi mapping

φ: Μ

> J(M) has a useful functorial

property, which can conveniently be described as follows. Consider an arbitrary compact complex analytic torus by a lattice subgroup ψ: Μ —5» J mapping

the Riemann surface Μ

, defined

C (Γ , and a complex analytic mapping

J

from the universal covering surface into the universal covering space

Μ

of

Cr

of

; and the component functions of the lifted mapping

are complex analytic functions Τ eΓ

= (Γ / X

. This mapping obviously lifts to a complex analytic

ψ: Μ — > C

the torus

^f

J

the values

ψ.(ζ)

ψ.(Tz) - ψ.(ζ)

-62-

on Μ

such that for any

are constants, indeed are the

components of a vector belonging to the lattice tions

ψ.(ζ)

5*

. The func­

are consequently Abelian integrals on Μ , so can be

expressed in terms of the canonical Abelian integrals in the form S ΦΊ·(ζ) = Σ for some constants stants

c..

c w (ζ) + c

and

,

1 < i < η ,

c. : and it is clear that the con-

c.. must have the property that

Σ

c..ω.(Τ; , 1 < i < η ,

are the components of a vector in the lattice formation

Τ e Γ . The matrix

5f

homomorphism

C: J(MJ

such that

> J . Conversely it is clear that any

a linear transformation

ψ = Ccp + c , where

> J(Μ)

analytic mapping

ψ: Μ

C: J(M)

lytic group homomorphism,

> J

> J(M)

> J(M)

arises from

c( X ) C

^

.

must be of the form

is an arbitrary complex ana-

is the Jaeobi mapping. ψ: Μ

> J

c is any point in the group

J

, and

In particular, any complex

> j(M) of a Riemann surface Μ

Jaeobi variety must be of the form C: J(M)

C: J(M)

C: C^ — > «ί1 such that

Thus altogether the mapping

phism

C: Φ — > Cr

; and this in turn induces a complex analytic group

complex analytic group homomorphism

φ: Μ

for any trans­

C = (c..) can now be viewed as

determining a linear transformation C( X ) C

^

into its

ψ = Ccp + c for some endomor-

and some point

c e J(M) .

It is fairly

evident how this observation can be used to give a functorial characterization of the Jaeobi variety, but details will be left to the reader; the compact complex torus defined in this functorial manner is often called the Albanese variety of the Riemann surface M.

-63-

The Jacobi homomorphism

can of

course b e described correspondingly. analytic mapping

Consider an arbitrary complex

, for some index

that for any fixed points

Note

the restriction

can be viewed as a complex analytic mapping from M into

J(m) , hence can b e written in the form

for some endomorphism and the matrix

C^

and some point

and the point

on the points

c^

evidently depend analytically

Now it follows immediately that must actually b e a constant function.

(This function

is a complex analytic mapping from the compact complex analytic manifold

into the complex vector space

of

g X g

com-

plex matrices, so that b y the proper mapping theorem its image is a compact complex analytic subvariety of

; but it follows

directly from the maximum modulus theorem, and is proved in most of the standard texts on functions of several complex variables, that the only such subvarieties must consist of isolated points, from which the desired result follows immediately. endomorphism

C: J(m)

> J(M)

g X g

must be determined b y a

complex matrix

C

integer matrix

D , as an immediate consequence of the condition

that

such that

Alternatively, any

for some

2g x 2g

; thus the endomorphisms are necessarily a dis-

crete subset of the vector space

-6b-

matrices,

so that even any continuous mapping from Μ

into the space of

all such endomorphisms must be a constant mapping.) The function c.(pp,...jP ) is a complex analytic mapping

1ΥΓ

> J(M) , so

that by induction r ψ(ρ ,.-.,ρ ) = Σ C φ (ρ ) +c ι J J ί = 1 for some endomorphisms If the mapping

C : J(M) J

> j(M) and some point

e e J(H) .

f merely depends on the divisor p. + ... + ρ

,

hence is invariant under permutations of the variables, it follows further that the endomorphisms

C. must all be equal, so that

t(p 2 + . ·. + p r ) = Ccp^ + ... + p r ) + c for some endomorphism where (b)

φ

C: J(M)

> J(M)

and some point

c e J(M) ,

is the Jacobi homomorphism.

That the image of a complex analytic mapping from a compact

complex manifold into any analytic manifold is an analytic subvariety is a consequence of Remmert's proper mapping theorem, the proof of which can be found in most texts on functions of several complex variables.

It should be emphasized that the image is an

analytic subvariety, but not necessarily an analytic submanifold} an analytic subvariety is a closed subset which can be described locally as the set of common zeros of finitely many analytic functions.

-65-

(c)

The subset

is an analytic subvariety of

v W^

-which will b e called the subvariety of gap points of

the gap subvariety of set

; its complement in

or

is an open sub-

which will be called the subset of nongap points of Of course all points of

are gap points when

, since

so this concept is really only

interesting for indices points then

v = 1 ,

If all the points of

dim

are gap

provided that

a result that nicely complements Lemma U and indicates some of the usefulness of these concepts; but this is unfortunately not generally the case, so the situation is rather more complicated and not yet completely straightened out. If a point

x e J(M)

then

is represented b y a line bundle

precisely when

a gap point of

and

precisely when

Consequently

representing

x

has the properties that

is represented as the image r

on

M

for some

if and only if the line bundle

for all points

degree

is

for sane point

or equivalently, precisely when point

x

p e M .

and If a nongap point of a divisor of

under the Jacobi homomorphism, then

and hence for any such index

for any index i

there must exist a meromorphic function

-66-

such that

but It is then obvious

that a general linear combination of these functions a meromorphic function on

M

w i l l be

having as polar divisor precisely

the divisor

On the other hand if

gap point and

then either

b y a line bundle

or

x

such that for some point

p^ e M

is a is represented and

; and in the latter case

can b e represented as the image of degree

r

on

M

x

of a divisor

containing the point

p ^ , since

and

This last condition means that for any meromorphic function

such that

necessarily

as

w e l l , and consequently there cannot exist any meromorphic function having as polar divisor precisely the divisor That explains the terminology,

- (p^ +... + p ) .

and indicates the extent

to which

this notion is a natural generalization of the classical notion of the Weierstrass gap sequence. As an immediate consequence of the Riemann-Roch theorem, if a nongap point

is represented as the image of a divisor of degree

r

on

M , then for any

index

i ; hence for any Abelian differential

that

such it necessarily follows

-67-

that

Λ3 (ω) > ρ η +...+ ρ

as well.

ι v+1 χ f. W

a gap point such that

represented as the image gree

r

on Μ

and if

χ = φ(ρ

such that

γ(κζ~

Correspondingly if χ e W ν > 2 , then

χ

is

can be

+...+ ρ ) of a divisor of de­ ... ζ" ) = g-r+v-1

and

P l ? r -1 γ(κζ ... ζ ) = g-r+v j consequently there must exist an Abelian Pp P-v. differential u ε r(M, (5 ' ) such that J (ω) > p 2 +... + ρ but

-1

Λ9 (ω) £ ρ, +p„ +...+ ρ

. Now the natural next step is to express

these conditions in terms of the ranks of the matrices formed from the values of the Abelian differentials of a basis for r(M, Θ at the various points

')

p. , paralleling the classical form of the

discussion of the Weierstrass gap sequence; but there are some dif­ ficulties in doing so, since the case in which all the points p. are distinct and the cases in which there are some coincidences require separate treatment, so the continuation of this discussion will be postponed until these difficulties are resolved later in the course of these lectures.

W

v+1

It is also quite natural to consider the subvariety ν - W C W , and to seek to develop a corresponding notion of

gap points in this context as well.

However it is an immediate

consequence of the Riemann-Roch theorem as expressed in formula (13) that

χ €W

v+1 ν . -W C W

precisely when

k-x

the earlier sense of the subvariety W^ , where and

s = 2g-2-r

and the indices are such that

is a gap point in μ = g-1- (r-v) r, s > 0

and

ν, μ > 1 j thus these two notions of gap points are really dual to

-68-

one another, by appropriate use of the Riemann-Roch theorem, and it suffices to consider merely the one notion.

(d)

On a hyperelliptic Riemann surface it follows from (25)

that W

= W . ,_ - (v-l)*e whenever 1 < ν < r < g-1 and r r-2v+2 ^ ' = =B r-2v+2 > 0 ; hence if in addition r-2v+2 > 1 it follows that

W rV = W 1 + W r-2v+2 „ _,_„ - (v-l) · e = W W V Ί ', so that all the "points 1 n + r-1 of W v

are gap points whenever 1 < ν < r < g-1 and r-2v+2 > 1 .

In the special case that r-2v+2 = 0 , the subvariety W merely of the point

-(v-l)-e

and W

- (v-l)*e is not a gap point of W of W

= φ , so that the point

. Conversely if all the points

are gap points, so long as W

that the surface Μ

consists

/ 0 , it is easy to see

is hyperelliptic; the details will be left to

the reader. This indicates at any rate that there really are montrivial nongap points on surfaces other than hyperelliptic surfaces.

-69-

References.

The material discussed in this chapter has

been taken primarily from the following sources. [1]

Hamilton, Richard S . , Non-hyperelliptic Riemann surfaces, Jour. Differential Geom. 3(1969), 95-101.

[2]

M a r t e n s , Henrik H . , Torelli's theorem and a generalization for hyperelliptic surfaces, Comm. Pure Applied Math. 16 (1963), 97-110.

[3]

5 A new proof of Torelli's theorem, Annals of Math. 78(1963), 107-111.

[4]

,

On the varieties of special divisors on a C u r v e , 1 ,

Jour, reine Angew. Math. 222(1967), 111-120;

I I , Jour,

reine Angew. Math. 233 (1968), 89-IOO. [5]

)

Prom the classical theory of Jacobian varieties,

Proc. X V Scandinavian Congress of Mathematicians, O s l o ,

1968. [6]

,

(Springer lecture notes 1 1 8 ,

7U-98.

Three lectures in the classical theory o f Jacobian

varieties, Mimeographed notes, n . d . [7]

Matsusaka, I . , On a theorem of Torelli, Amer. Jour. M a t h . 80(1958), 784-800.

[8]

M a y e r , A . L . , Special divisors and the Jacobian variety, Math. Annalen

[9]

,

153(1964), 163-167.

On the Jacobi inversion theorem, (Ph.D. thesis,

Princeton University, [10]

1961).

W e i l , A n d r e , Zum Beweis des Torellischen Satzes, Nachrichten Akad. Wissenschaften Gtittingen

-70-

(1957), 33-53.

Other references, particularly to older literature, can be found in the bibliographies of these papers. The notation and properties of the subvarieties of positive divisors and of special positive divisors are given in several of these papers, in one form or another.

The proof of Cliffords theorem used in these

notes is that given in [U]; the extended form of this theorem, the assertion that if the subvariety W

attains the maximal dimension

given in Theorem 7 for any indices in the range

2 < ν < r < g-2

then the Riemann surface is hyperelliptic, can be found in [U] and in [1], This will also be treated at a later point in these notes.

-71-

§3· Jacobi varieties and symmetric products of Riemann surfaces (a)

The restriction of the Jacobi homomorphism to the set of

positive divisors of degree r can be viewed as a complex analytic mapping

φ: Μ

> J(M) J and it is evident that this mapping is

really independent of the order of the factors in the Cartesian χ product Μ

(r) . This suggests introducing the symmetric product Μ = Μ / (s

which is defined to be the quotient space Μ compact complex analytic manifold ΓΓ the symmetric group

(5

of the

under the natural action of

on r letters as the group of permuta­

tions of the factors in the Cartesian product Ά each permutation 7Γ e (5

,

of the integers

sidered as defining a mapping ττ: Μ

. That is to say,

1,2,...,Γ

can be con­

> W , by setting

ir(p15... ,pr) = (ρ^,.-.,ρ^) ; this exhibits

(s r as a group of

analytic homeomorphisms of the manifold Ά , and the quotient space (r) (r) is by definition the symmetric product Μ

' . The points of Μ

can be considered as r-tuples of points of Μ , without regard to order, hence can be identified with positive divisors Λ / = ρ, + ... + ρ (r) of degree r on M'j consequently the symmetric product Μ will also be called the manifold of positive divisors of degree r on the Riemann surface Μ . Theorem 9· The r-fold symmetric product Μ

(r)

of a compact

Riemann surface Μ has the structure of a compact complex analytic manifold of dimension r , such that the natural quotient mapping J τ: Μ fold Μ

(r) >Μ is a complex analytic mapping exhibiting the mani­ as an (r.1)-sheeted branched analytic covering of the

-72-

manifold Proof.

It is evident that if none of the permutations in

leave a point

fixed, then these transfor-

mations will take a sufficiently small open neighborhood of that point into pairvn.se disjoint open subsets of

each of these

sets can be viewed as a local coordinate neighborhood in the quotient space

hence in that neighborhood

will

have the structure of an r-dimensional complex analytic manifold such that the natural quotient mapping -sheeted analytic covering mapping.

is an Thus the only difficulty

lies in the presence of fixed points of some of the transformations in

these fixed points are of course the r-tuples such that not all of the points

If there are

s

p^

are distinct.

distinct points in this r-tuple the points can be

renumbered so that coincidences occur in the form

where

.

about the points

Select open coordinate neighborhoods

such that

whenever

whenever

but

The product

is then an open coordinate neighborhood of the point in

such that any permutation in

-73-

either maps

onto

itself or transforms

U

into an open subset of

indeed the permutations in

mapping

the subgroup

U

disjoint from onto itself form consisting of those

permutations interchanging the first the second

indices among themselves,

indices among themselves, and so on.

In order to

conclude the proof it suffices to show that the quotient space can be given the structure of a complex manifold of dimension

r

such that the naturai mapping is analytic.

local coordinates in the neighborhoods analytic mapping

Letting

be

introduce the complex

defined by coordinate functions

of the form

(2)

The functions

are the elementary symmetric

-7I+-

; and l as is well known, they determine an open complex analytic mapping

functions of the second kind in the variables

v

U

> C

χ U_ χ... χ U

l

1 U

χ U

I

x. ..x U V

(5-

l

ζ ,z?,...,z

V

identifying the quotient space V. with an open subset of C . The same

1

observation can be made for each of the

s blocks of coincident

neighborhoods, and this evidently serves to conclude the proof of the theorem. The particular local coordinates in the neighborhood of the (r) point

ρ Ί + p_ +...+ ρ

W '

in the manifold

described by equa­

tion (2) will be used freely in the ensuing discussion.

It should

be noted that they depend on the choice of local coordinates in an open neighborhood of each of the distinct points

p.

surface Μ . (b) As already noted, the Jacobi homomorphism

ψ

ways the mapping

τ: Μ

r

> Μ

(r) for some complex analytic mapping ψ: Μ ;

φ = ψ° τ

this mapping

> J (Μ)

φ: Μ

can be factored through the natural quotient mapping so that

on the Riemann

will also be called the Jacobi mapping. ψ

is more natural than the mapping

> j(M) ;

In many

φ , even

(r)

though the manifold

fr) ,

j

is more complicated than the manifold W ;

Μ

that will become more apparent as further properties of this mapping ψ

are derived.

For any index

image of the mapping variety W

C J(M)

ψ: Μ

r

in the range

> J(M)

1 < r < g-1

is the proper analytic sub-

of positive divisors of degree

can be viewed as an analytic mapping

-75-

the

ψ: Μ

'

> W

r , so that C J(M) .

ψ

For any index

r

with

the image of the mapping

is the entire Jacobi variety; in particular the mapping

)

sional manifold

is an analytic mapping of the g-dimen-

onto the g-dimensional manifold

, as a

consequence of the Jacobi inversion theorem, and the problem of describing this analytic mapping in more detail can be viewed as an extended form of the Jacobi inversion problem. varieties

The analytic sub-

of special positive divisors in

by the mapping

_ axe transformed

to analytic subvarieties

which will also be called subvarieties of special positive divisors. Note that for a point

viewed as a divisor

, the image

is also represented

by the complex line bundle and this point

in the Picard variety;

lies in

precisely when

Consequently the subvarieties can be defined by (3) These subvarieties furnish a descending filtration

of the complex manifold

by analytic subvarieties, which

eventually terminate in the empty set.

-76-

For any point that

such

the fibre

of the mapping

is an analytic subvariety of

of dimension

•v-1 which can be represented as the image of a one-one analytic mapping of Proof. of

That the fibre

is an analytic subvariety

is an immediate consequence of the analyticity of the map-

ping

.

It then suffices just to show that there

exists a one-one analytic mapping

p:

having as

image precisely this analytic subvariety

the image

must then be an analytic subvariety of dimension

v-1 . Now it follows directly from Theorem 6 that the fibre consists of the positive divisors of degree

linearly

equivalent to the given divisor

and viewing these points as

divisors on the Riemann surface

they axe just the divisors of

arbitrary nontrivial holomorphic sections of the line bundle Letting

be a basis for the space of

holomorphic sections of this line bundle, associate to any point other than the origin the divisor

there results a well defined mapping

from the complement of the

origin in

having as image precisely

the fibre

into the manifold

Note that two nontrivial sections

-77-

have the same divisor if and only if they are constant multiples of one another, since the quotient then a holomorphic function on all of

is

M ; consequently

if and only if for some nonzero complex constant induces a one-one mapping

c , so that

p

having as image pre-

cisely the fibre It remains merely to show that this is a complex analytic mapping.

Fixing a point in

]P

v-1

represented by a vector

be a divisor in which coincidences among the points occur as in (l), so that this divisor can also be written as a divisor of distinct points in the form and choose disjoint coordinate neighborhoods centered at these distinct points in the Rieman surface

M .

In the coordinate neighborhood

the section

can be viewed as an ordinary complex analytic function, and its divisor in stants

is just

and if the con-

are sufficiently close to

, it

follows in a very familiar fashion that the section will be an analytic function in degree

there.

section, where

having a divisor of total

Letting

be the divisor of this

denote the coordinates of these various points

in terms of the chosen coordinate system in

-78-

, it follows as

(

usual from the Cauchy integral formula that

here

z

is the local coordinate in

the sections

viewed as complex analytic functions

in

denotes the derivative of the function coordinate

Now for

are

, and

with respect to the the expression

is precisely one of the standard coordinates of the point in

as introduced in

obvious that the coordinates

and it is then

of the points are complex analytic functions of the

points

in an open neighborhood of

Considering the other coordinate neighborhoods similarly, it finally follows that the mapping

is an analytic mapping in terms of the complex structure

introduced on

, and that suffices to conclude the proof.

As an immediate consequence of this theorem, note that for the analytic mapping ping from the complement of in

is a one-one mapin

and further note that for

analytic mapping dimension

onto the complement of the fibres of the

are analytic subvarieties of which can be represented as images of one-one

analytic mappings of

into the manifold

-79-

Even more

can easily be deduced as follows. Theorem

At any point

that

such

the differential of the analytic mapping has rank given by

Proof. occur as in

If coincidences in the divisor that divisor can also be written as a divisor of

distinct points in the form and choosing disjoint coordinate neighborhoods centered at these distinct points on the Riemann surface metric product

M , introduce the standard coordinates on the symas in

The theorem will be proved simply

by calculating the Jacobian matrix of the analytic mapping at the point on

in terms of these coordinates

and the obvious coordinates on the Jacobi variety.

Note

that an Abelian integral of the first kind can be represented ; uniquely up to an additive constant, by an analytic function of the local coordinate in any of the coordinate neighborhoods chosen on the Riemann surface dinate

z

and in terms of the local coor-

centered at the point

function has a Taylor expansion

-80-

for example, this analytic

Now if

are

points in this coordinate neighborhood,

expressed in terms of the given local coordinate, then

where

are the standard local coordinates in

as

in (2) and the remaining terms in the last series expansion involve higher powers of these local coordinates.

The same construction

can be carried out in the remaining coordinate neighborhoods as well.

Thus for any divisor

sufficiently near

the given divisor dinates

and represented by coorin terms of the standard coordinate system

chosen ina neighborhood the coordinate analytic mapping can beofrepresented by ,the functions

+ higher order terms

for some constants

where

Hie Jacofoism matrix of

-81-

the mapping

, as expressed in terms of these coordinates, is just

the matrix of linear terms in this Taylor expansion; that matrix is just the

g x r

matrix having the following typical rows:

(b)

for Now for this matrix (U) observe that

g - rank

is the

dimension of the vector space consisting of those row vectors such that

For any such

row vector, though, it is apparent from

that

will be a holomorphic Abelian differential such that where

are the basic Abelian differentials of the

first kind.

Therefore

g - rank

and since

as a consequence of the Riemann-Roch theorem, it follows that

rank

which was to

be demonstrated. Several almost immediate consequences of the combination of the two parts of this theorem deserve more detailed discussion. Corollary 1 to Theorem 10. such that of the mapping

For any point the fibre is a complex analytic

/

submanifold of

which is analytically homeomorphic to

-82-

Proof. As a consequence of Theorem 10(b) the Jacobian of the analytic mapping

> J(M) has rank

ψ: NT

r - (v-l) at

each point of the fibre ψ~ ψ($• ) , hence the fibre must locally be contained in an analytic submanifold of Μ

(r)

v-l ;

of dimension

but since as a consequence of Theorem 10(a) the fibre is an analytic subvariety of Μ

(r)

of dimension

v-l , it must locally coincide

with that submanifold, hence is itself an analytic submanifold of Μ

' . The one-one analytic mapping from

manifold ψ

F~

onto this sub­

Ψ(Λ9- ) , as in Theorem 10(a), is necessarily an ana­

lytic homeomorphism, and the desired result is thereby demonstrated. Corollary 2 to Theorem 10. ping

If r > 2g-l the analytic map­

> j(M) has the property that ψ

ψ: Μ

(χ) is an

(r) analytic submanifold of Μ for each point Proof. theorem that

r-g analytically homeomorphic to

Ρ

χ e J(M) . If r > 2g-l then it follows from the Riemann-Roch

7(ζ_

—

ζ„ ) = r+l-g

for any point

(r) /i7 = p. +... + ρ E H j hence the desired result follows immedi­ ately from Corollary 1 to Theorem 10. Actually somewhat more can be said in this case, as will be demonstrated anon later; the mapping

ψ: Μ

(r) ;

> j(M) exhibits

(r) as a complex analytic fibre bundle over J(M) with fibre i?r-e

-83-

Corollary 3 to Theorem 10. mapping

ψ: Μ

> J(M)

If

1 < r < g

the analytic

induces a complex analytic homeomorphism

ψ: M ^ X G 2 — = - > W \W2 . r r r (Here

AVB

denotes t h e s e t - t h e o r e t i c d i f f e r e n c e between s e t s A

and B, t h e complement of t h e s u b s e t Β i n A . ) Proof. necessarily

y(ζ

For any d i v i s o r

/S e Μ '

such t h a t

,3- £ G

) = 1 ; hence as a consequence of Theorem 10(a)

the r e s t r i c t i o n ψ: M ^ A G 2 r

> W \W2 r r

is a one-one analytic mapping between these two sets, and as a con­ sequence of Theorem 10(b) the differential of this restriction has rank

r

phism.

at each point so is locally a complex analytic homeomorThat suffices to prove the desired result. In the course of the proof of Theorem 10(b) the Jacobian

matrix of the analytic mapping

ψ: Μ

(r)

> J(M) was calculated

quite explicitly, in terms of the standard local coordinates intro(r) duced on the manifolds

Μ

' and

j(M) ; the result of that calcu­

lation is useful by itself, and merits explicit mention.

It is only

natural to seek to express that result rather more intrinsically, though, at least avoiding the necessity of making particular choices of local coordinates on the manifolds

Μ

(r)

and

J(M) ; and the

interpretation of that result used in the proof of Theorem 10(b) suggests a convenient approach to such a reformulation.

-8U-

In the representation of the Jacobi variety as a quotient group

, the coordinates

I

in

convenient local coordinates at any point

provide

and in terms

of these coordinates, a natural basis for the complex tangent space of the manifold

at the point

the tangent vectors

and dually a natural basis for

the complex cotangent space point

x

is provided by

of the manifold

is provided by the covectors

ing

at the The dual pair-

is then given by

for arbitrary complex constants

Now any covector

extends to a unique group invariant covector field on the complex Lie group

namely to the holo-

morphic differential form all of

on

J(M) ; and the restriction of this differential form to the

analytic submanifold on

is a holomorphic differential form

which under the Jacobi homeomorphism

induces

the holomorphic differential form There results a linear mapping

which

is readily seen to be an isomorphism; hence the complex cotangent space

to the complex manifold

can naturally be identified with the space morphic Abelian differentials on

M .

-85-

at any point

x

of holo-

At any point

' the analytic mapping

has a well defined differential

which

is just the linear mapping

between the tangent spaces of these two manifolds induced by the mapping

In terms of the natural bases provided in these tan-

gent spaces by the standard local coordinates introduced on the manifolds

and

the linear mapping

described by the Jacobian matrix

thus if

where on the Riemann surface

is that

axe distinct points

M , then the image of the mapping

the linear subspace of

spanned by the vectors

for

and

-where

tives of order

m

first kind on

is

are the deriva-

of the canonical Abelian differentials of the

M , in terms of any local coordinates at the points

and

The linear subspace of

to the image of the mapping covectors for all indices identification of

dual

is thus that consisting of those such that and with

-86-

and under the natural this corresponds to

the lineax subspace of differentials

consisting of those Abelian

such that

, Thus (U) can be inter-

preted in the following form. Corollary U to Theorem 10.

For any point

the

image of the differential of the mapping

) is the

linear subspace space

dual to the subdefined by

(5) with the natural identification

intro-

duced above.

(c)

It is an immediate consequence of Corollary 3 to Theorem 10

that the subvariety of

j(M)

is a regular analytic submanifold

at any point not contained in

whenever

and that leads to the problem of describing the singularities of the various subvarieties

As a matter of notation, the points

of an analytic subvariety lytic submanifold of

at which

is a regular ana-

are called the regular points of that

subvariety, and the set of regular points of by

V

the remaining points of

V

V

will be denoted

are called the singular

points of that subvariety, and the set of singular points of will be denoted by

The singular locus

a proper analytic subvariety of

V .

can be associated the linear subspace

-87-

To any point

is always there spanned by

all covectors of the form

df

, where

in an open neighborhood of the point χ identically on the subvariety

f

is any analytic function

in J ( M ) which vanishes

V j alternatively

Τ (V) can be

described as the linear subspace of the cotangent space formed by the differentials at χ

of all germs of analytic func­

tions in the ideal of the analytic subvariety

V

at the point

"£

χ e j(M) . The natural dual to the subspace a linear subspace

Τ (J(M))

-it-

I (V) C I (J(M)) is

Τ (V) C Τ (J(M)) which will be called the tan­

gent space to the subvariety

V C j(M) at the point

χ ; and the

dimension of the linear subspace

Τ (V) will be called the imbed­

ding dimension of the variety

V

at the point

dimension of V

χ

can also be characterized as the

at the point

smallest dimension of a local submanifold of the subvariety

V

regular points of

χ . The imbedding

J(M) which contains

in some open neighborhood of χ V

Theorem 11.

j(M) ; the

are thus precisely those points at which the

imbedding dimension of V the dimension of V

in

is equal to the local dimension of V ,

in a small open neighborhood of (a) For any index

the singular locus of the subvariety

r

χ .

such that

W C j(M)

1 < r < g-1

is precisely the

2

subvariety W C W (b) For any indices

r > 1, ν > 1

such that W

proper analytic subvariety of J(M) , the subvariety

W

is a is con-

•y

tained in the singular locus of W W

C J(M)

j indeed the analytic subvariety

has imbedding dimension equal to

χ e W V + 1 C WV . r r

g

at each point

Proof. As already noted it follows from Corollary 3 to 2 Theorem 10 that all points of W not contained in W are regular points of W

, hence that J (W ) C W

ο

; it is therefore sufficient

merely to prove assertion (b), indeed merely to prove the last part /~~\ r w of that assertion. Now recall from Lemma 3 that W Vr+ l = W Vr (-) (Wn1-W,1 )\' :

V

V+l

thus selecting a point χ e W

, it follows that

x+l - W

CW

hence that x+cp(p) - cp(q) e W

for any points p, q e Μ . If f

,

is any analytic function in an open neighborhood of χ in j(M) vanishing identically on W v in that neighborhood, and ρ is any point on the Eiemann surface Μ , then

f(x + cp(p) - cp(q)) = 0

identically as a function of q whenever q

is sufficiently near

ρ ; and upon differentiating this identity with respect to q at the point q = ρ , in terms of a local coordinate system near ρ on the Riemann surface Μ , it follows that cp'(p) is the vector with components

d f-cp'(p) = 0 where

{wi(p)) and the product is

the ordinary vector inner product. Since the vectors the full vector space Cr

cp'(p) span

as ρ varies over Μ , as a consequence

of the familiar fact that the Abelian differentials w.' (z)dz = ω. (ζ) are linearly independent, it follows that hold for all analytic functions vanishing on W subvariety W

f

d f = 0 ; but that must

in an open neighborhood of χ

, hence the imbedding dimension of the analytic at the point χ

is equal to g , and the proof is

thereby concluded.

-89-

There remains the question whether the singular locus of is necessarily contained in analytic subvariety of

when

and

is a proper

at least some information can be

obtained quite directly in the special case

As a

useful notational convention, the germ of an analytic subvariety at a point

be denoted by

indicates that the analytic subvariety

V

thus

is only to be con-

sidered in arbitrarily small open neighborhoods of the point

x

in

Recall from Lemma 2 that consequently

. thus providing a

representation of the germ of the analytic subvariety of its points

x

as an intersection of appropriate translations of

the germs of the analytic subvariety Now note that

at the various points for all points

and only if

thus whenever

will exist points that

hence such

is a regular point of the subvariety

p , the germ

if there

such that

as a result of Theorem 11 (a). point

at any one

since For any such

is contained in the germ which is the germ of a complex analytic sub-

manifold of dimension

The tangent space

is then

contained in the tangent space of that submanifold, which is a linear space of dimension ding dimension of

and if at any point

hence is certainly lesg than

it follows that the imbedis at most Therefore the singularities of

-90-

2 2 3 which may be contained in W \W are at least not so bad as r r r 3 the singularities contained in W ; and in particular, if r < g

W

W

3

can be characterized intrinsically as the subvariety of W

2

consisting of those singular points at -which the imbedding dimension of W

2

is equal to

g .

Of course this observation can readily be extended; for 2 -¾ χ e W \W , then

whenever

x-cp(p) will be a regular point of W

for all but a finite number of points p space

ρ e Μ , hence the tangent

Τ (W ) is contained in the intersections of the tangent

spaces of a number of germs of analytic submanifolds. At any regular point of the analytic subvariety W

. , the tangent space of W

.

was determined explicitly in Corollary k to Theorem 10, so that something more can be said about the intersections of these tangent spaces also; but it is convenient to insert first a brief digression, to prepare the way for this calculation. 2 3 -1 For any point χ e W \W , the fibre ψ (χ) of the analytic (r) mapping ψ: Μ > J(M) is a complex analytic submanifold of (r) 1 Μ

analytically homeomorphic to

IP

, as a consequence of Corol­

lary 1 to Theorem 10; thus the points of divisors t € TP

ψ" (χ) can be viewed as

ρ (t) +...+ ρ (t) depending analytically on a parameter .

It may very well happen that some of the points of these

divisors remain fixed as

t

varies, so that these divisors can

actually be written in the form for some index

p. (t) +...+ ρ (t) + Ρ,,,-, +...+ ρ

s < r . The divisors

ρ (t) +... + ρ (t) are obvi­

ously all linearly equivalent, and evidently

-91-

thus and

where

If the index

s

is chosen to

he as small as possible, this decomposition is obviously unique; and the condition that

s be as small as possible is clearly that

Using the terminology introduced in the notes to the subset of of

is the subvariety of gap points

and its complement is the open subset Thus any point

form

of nongap points

can be written uniquely in the

where

and

for some index

for some uniquely determined divisor

so that

In

terms of this decomposition it is clear that any two divisors in the fibre

of the analytic mapping

either

coincide or have only the divisor points

in common.

The

of this divisor are precisely those points such that

so for any point

other than one of these, the translate

p eM

is a regular point

on the subvariety Lemma 5.

Consider a point

which

can be written in the form

where

and correspondingly,

is the line bundle associ-

_

ated to any divisor in the fibre write

of the analytic mapping where

Then for any two points that

and

and

such

are regular points of the subvariety

-92-

the tangent spaces

and

viewed as subspaces to the tangent space of

either coincide

or intersect in a linear subspace of dimension Proof.

Since the tangent bundle to the Jacobi variety is

trivial, the tangent spaces to

at various points can all be

identified with one another canonically; that can be accomplished by translating all these tangent spaces to the same point of using the group operation on j(M)

The dual cotangent spaces to

at various points can correspondingly be identified with one

another canonically, and all can be identified with before.

For each point

such that

point of the subvariety divisor

as is a regular

there will exist a unique positive such that

space to

i , and the tangent can be identified with the

image of the differential

of the analytic mapping

at the point

as a consequence

of Corollary U to Theorem 10, the dual space to that tangent space is the linear subspace

Note that

defined b y

dim

, and that as a consequence of the

Riemann-Roch theorem

din

as wellj so since

these two spaces really coincide. section of the tangent spaces to the subvariety

-93-

Now the interat the two

regular points

and

is just the dual space to the dimension of that intersection

is consequentlydim sors

dim and

If the divi-

are distinct, their common terms

are determined b y the decomposition thus the divisors

as discussed above;

and and

where

have no common points and where

and

Then

and

so that applying the Riemann-Roch theorem again

Substituting this result into the preceding formula, it follows that the dimension of the intersection of the two tangent spaces is as desired, thus concluding the proof of the lemma.

-9U-

If the two tangent spaces

and

in the preceding lemma intersect properly, that is to say, intersect in a linear subspace of dimension

in

germs of manifolds

then the two

and

also intersect properly, in a complex analytic submanifold of of dimension

containing the germ

dimension of any irreducible component of

Since the is not less than

as will shortly be demonstrated, then that intersection must coincide with quently

in an open neighborhood of

and conse-

must be an analytic manifold at the point

It

follows from Lemma 5 that such a proper intersection occurs precisely when

with the notation as in the state-

ment of the lemma; and since

this is just the con-

dition that the dimension possible value.

have the least

In other cases than this, the manifold germs and

need not inter-

sect properly, so their intersection need not be a submanifold of near

and the germ

the intersection.

may be properly contained in

At any rate, the following does generally hold.

Theorem l£(a). write

For any point

where

and

and let

be any positive divisors such that and the analytic variety

Then the imbedding dimension of ; at the point

-95-

is not greater than

Proof. Choosing any two points ρ , ρ x- r+2-v

at each point of Μ

fr) , then

is nonempty.

Recall from Theorem 10 that the fibres of the Jacobi mapping > J(M)

are complex analytic submanifolds of M^ r '

ana­

lytically homeomorphic to complex projective spaces of various dimen­ sions.

To round out the description of this mapping, something fur­

ther should be added to indicate the extent to which the Jacobi mapping exhibits

(r) M v ' as a local product of J(M)

projective space.

-105-

and the appropriate

Theorem 15. subset

V

For any sufficiently small relatively open

of the analytic subvariety

there exists

a one-one analytic mapping

such that Proof.

is the natural projection. The proof of the desired result is a rather straight-

forward modification of the proof of Theorem 10(a). disjoint open subsets

of the Riemann surface

First choose

M

such that the

restricted Jacobi mapping

is a

complex analytic homeomorphism; that there exist such sets is an immediate consequence of the observation that

where

is a branched analytic covering and is a complex analytic homeamorphism, and

being proper analytic subvarieties.

Fixing points

and introducing the divisor

, it follows

that for any fixed line bundle

the mapping is a complex analytic hameamor-

phism from r| in the manifold

onto an open neighborhood of the point P(M) ; this provides a very convenient system

of local coordinates in a neighborhood of any fixed point of

P(M) ,

or equivalently of course, in a neighborhood of any fixed point of In terms of this parametrization, the fibre of the Jacobi mapping the parameters

over the point of

J(M)

described by

consists precisely of the divisors of all

-106-

the holomorphic sections

. Recall-

ing the canonical isomorphism

then upon choosing a basis

for the vector space

it follows that the fibre of the Jacobi mapping over the point

can be described

equivalently as the set of divisors of the form

where

are any complex constants such that

for

Note that the sections

are here viewed as

ordinary complex analytic functions in each coordinate neighborhood

the

Now if

so that

nx g matrix

has rank

, it follows that and near

the subset

can be described as the analytic subvariety consisting of those points

such that

It can be assumed, renumbering the sections neighborhoods

and shrinking the

, if necessary, that the

matrix

sub-

is nonsingular for and hence for all

The linear

equations tions

rank

then have unique solufor arbitrary constant values

and Cramer's formula shows that these solutions

-107-

are

complex analytic functions of the points It should be remarked again that the sections

are viewed as

ordinary complex analytic functions in each coordinate neighborhood , by choosing any fixed trivialization of the line bundle over each neighborhood.

Taking any

v

linearly independ-

ent constant vectors exist

v

, there thus

vectors of holomorphic functions

such that for any fixed point

the sections are linearly

independent and

for

and moreover whenever for

then also ,

Introducing then the mapping

defined by

it is evident from the proof of Theorem 10(a) and from the construction that this is a complex analytic mapping which induces a one-one complex analytic mapping from the analytic variety the subset the theorem.

onto

. That suffices to conclude the proof of -108-

If the subvariety

has rather bad singularities,

a one-one analytic mapping from another analytic variety onto need not have an analytic inverse, so need not be an analytic homeomorphismj but ignoring these complications, the following simple consequences of this theorem easily arise. Corollary 1 to Theorem 15. For any sufficiently small relatively open subset the set

such that

is analytically homeomorphic to the product Proof.

If

so that the set

is

itself a complex manifold, the one-one analytic mapping is an analytic homeomorphism, hence the corollary follows trivially. Corollary 2 to Theorem 15ping

If

the analytic map-

exhibits the manifold

trivial analytic fibration over the manifold Proof.

If

as a locally

J(M) with fibre

then

and

it then follows from Corollary 1 that for any sufficiently small open subset

the inverse image

cally homeomorphic to the product the desired assertion, that

is analyti. This is precisely is a locally trivial

analytic fibration over J(m) with fibre It should be noted that the natural complement to Theorem l M a ) also follows readily from these observations.

-109-

Theorem l^(b).

If V

is an irreducible component of the

analytic subvariety W C j(M) and if V within W r

is not entirely contained

then dim V > rv - (v-l)(g+v)

Proof.

If V C V v+1

is a sufficiently small open subset of

J

V . and V η W = 0r , then it follows from Theorem 15 that there ' ο r ' ' is a one-one analytic mapping λ : V χ 3? > ψ" (V ) C Μ' ' ; and consequently dim ψ" (V ) = dim V + v-l . On the other hand, -1/ \ V V+1 ψ

(V ) is an open subset of G

not contained in G

, hence

it follows from Theorem lU(a) that dim ψ" (V ) > rv - (v-l)(g+v-l) . The desired result follows immediately from these two formulas, and the proof is thereby concluded. As a brief digression, it is of some interest to examine more closely the Jacobi mapping ψ: Μ

s

> j(M) , to secure a

more detailed analysis of the solution of the Jacobi inversion problem; this can be done rather completely for surfaces of suffi­ ciently small genus. Note first that as a consequence of Corol­ lary 3 to Theorem 10* the Jacobi mapping induces a complex analytic homeomorphi sm ψ: M ' S \ G 2

> J(M)\W2 .

T

g g It follows from the Riemann-Roch theorem as rewritten in the form of equation (13) of §2 that VT = k -W _2 , where k e j(M) is the 2 canonical point; thus W is an irreducible complex analytic sub-

-110-

variety of

J(M)

of dimension

g-2 .

It then further follows from

Theorem 11(a) that the singular locus of the subvariety

is

; hence applying the same form of the Riemann-Roch theorem as above,

Over the regular locus

the Jacobi mapping

is a locally trivial analytic f'ibration with fibre sequence of Corollary 1 to Theorem 15. dim

, hence that

Theorem 8 that

Recall from Theorem 7 that

when

when

g = U

, as a con-

j and recall from

and the surface

is not

hyperelliptic, while

when

hyperelliptic, where

is the hyperelliptic point.

Thus if M M

and the surface

is a Riemann surface of genus

is hyperelliptic and

M

is

g = 2 , so that

is the hyperelliptic point of

the Jacobi mapping

j(M) ,

has the properties that

is an analytic submanifold of morphic to

g = U

M

analytically homeo-

, and that

is a com-

plex analytic homeomorphism; the symmetric product

is

obtained from the Jacobi variety merely by blowing up the hyperelliptic point

e e j(M)

to a projective line

, using the

picturesque terminology that has been introduced in complex analysis and algebraic geometry to describe such mappings. mann surface of genus manifold of

J(M)

then

If M

is a Rie-

is an analytic sub-

analytically homeomorphic to the surface

-111-

M

itself; the Jacobi mapping that the manifold

has the properties

is a locally trivial analytic fibration over with fibre

, and that

is a complex analytic homeomorphism.

The symmetric product

is obtained from the Jacobi variety by blowing up each point of the analytic submanifold

to a projective line.

a Riemann surface of genus subvariety of

J(M)

g = ^

then

If M

is

is an analytic

which is the image of an analytic mapping j if

M

is not hyperelliptic this

mapping is a complex analytic homeomorphism and its image is a regularly imbedded analytic submanifold of hyperelliptic the image point manifold

J(M) , while if

M

is

has an isolated singularity at the and the analytic mapping

exhibits the

as being obtained from the analytic variety

by blowing this singular point up into a projective line. Jacobi mapping

The

has the properties that is a locally trivial analytic fibration over

the regular locus of the subvariety that the isolated singular point of inverse image

with fibre

IP

,

(if it exists) has as

an analytic submanifold of

analytically homeomorphic to

, and that

is a complex analytic homeomorphism.

The symmetric product

is obtained from the Jacobi variety by blowing up each point of the analytic submanifold

to a projective line, and

-112-

blowing up the isolated singular point of

(if it exists) to a

projective plane; the singularity occurs only when

M

is hyper-

elliptic. (g)

The proof of Theorem 15 was really accomplished by demon-

strating the existence of an analytic mapping over any sufficiently small relatively open subset that factors through the natural mapping the desired analytic mapping the mapping

to yield ; indeed

constructed in that proof can evidently be viewed

as the composite

which for each

of a one-one mapping

is a linear isomorphism

and the divisor mapping

which The mapping for , or anyequivalently, canthe be and thought structure as -113determining ofof asadetermining complex on the has analytic set on thethe value variety set

the structure of a complex analytic vector bundle over the analytic variety

V ; and the mapping

μ"

is then a one-one analytic map­

ping from the complex analytic projective bundle naturally associ­ ated to this vector bundle onto the analytic variety

ψ

(V) . It

is quite natural to expect that this local construction can be extended to a global construction, that is, that the set U ., Γ(Μ, Θ-(ξζ )) can be given the natural structure of a P 5 e W V \W V o b r r ν v+1 complex analytic vector bundle over the variety W \W such that the divisor mapping is a one-one analytic mapping from the complex analytic projective bundle naturally associated to this vector ν bundle onto the analytic variety

V+1

G \G

; this is of course equiv­

alent to asserting merely that the local mappings ping open sets

V

μ'

for overlap­

are compatible, that is, that whenever ν

V^, V

v+1

are intersecting open subsets of W \W

for which this construc­

tion has been carried out there exists a complex analytic homeomorphism μν

v

: (v1 η v 2 ) χ c v

which for each fixed point and which is such that mappings

ξeV

> (ν χ η v 2 ) χ c v Π V_

is linear on the space

μ* μ = μ' on 1 12 2

(V Π V ) χ C V .

•υ

C

The

μ

„ are then the coordinate transition functions for 12 ν v+1 the vector bundle so defined on W \W In demonstrating that

this global assertion is indeed true it is convenient, and for some other applications quite useful, to show something rather more precise. -11U-

Any analytic line bundle

can be

represented by a flat line bundle

, as observed in

of the earlier lecture notes. bundles over a Riemann surface M group isomorphic to

Indeed the set of flat line of genus

g

form a complex Lie

, the subset of analytically trivial

flat line bundles form a Lie subgroup isomorphic to quotient group is isomorphic to

, and the

p(m) , so that the manifold

is a complex analytic principal bundle over the manifold P(m) with group and fibre equal to

j this is of course merely

an interpretation of the exact cohomology sequence

derived on page 132 of the earlier lecture notes.

It is further

evident from the proof of Lemma 27 of the earlier lecture notes that the flat line bundles over M cycles

can all be represented by co-

for a fixed open covering

of the Riemann surface, in such a manner that these cocycles axe complex analytic functions

of the points

It is then an immediate consequence of these observations that for any sufficiently small open subset the line bundles cycles

for all

and any index r can be represented by co-

for a fixed open covering

of the Riemann surface, and in such a manner that the cocycles are complex analytic functions of and of

; for the analytic fibration

-115-

admits local sections.

This provides a very

convenient explicit representation of the local bundle structure on the sets

Theorem l6(a). subset

V

For any sufficiently small relatively open

of the analytic subvariety

and any index

r , there exist an open covering M

of the Riemann surface

and complex analytic mappings

that for each fixed

the functions

form a cocycle representing the line bundle

; and furthermore,

there exist complex analytic mappings so that for each fixed

the functions form a basis for the space of holomor-

phic sections of the line bundle

, when that

bundle is represented by the cocycle Proof.

Although the first assertion of the theorem was

proved just above, it is convenient to prove both assertions simultaneously; but the preceding proof may well serve as an enlightening motivation for the constructions in this proof.

The proof is

really a natural continuation of the proof of Theorem 15, so the notation and terminology introduced in the course of that proof will be presupposed here. sal covering space

To continue, then, introduce the univer-

of the Riemann surface

that the open sets select open sets and let

M ; and assuming

are simply connected, covering each of the sets

denote the point lying over

-116-

simply, and

denote the point lying over point

ρ €Μ ο

p. . It can be assumed that the "base J is not contained in any of the sets U. . In terms j

of the prime function of the marked Riemann surface

Μ , define

g g(z-,,...,ζ ;z) = Π p(z,z ;z°z ) . g j=l ° J 0 This is then a meromorphic function on identically 1 when

ζ = ζ

U_ x... χ U χ Μ , which is 1 g ' , and which as a function of ζ e Μ

has simple zeros at the points

Γζ . and simple poles at the points J

Γζ. , and is otherwise holomorphic and nonvanishing on Μ ; of J course when

ζ . = ζ . this function remains holomorphic and non-

vanishing at that point also. Furthermore, for any transformation Τ

belonging to the covering translation group Γ

}

this function

has the property that g(z l 5 ---5 z g J T z ) =X(Tjz1,...,zg)g(z1,...,z jz) , where

x(T;z.,...,z ) is a holomorphic nonvanishing function on

U, x. ..x U 1 g

for each

Τ e Γ ; the explicit form of this function '

can readily be determined also, referring back to Theorem k, but will not be needed here. The functions describe flat line bundles over Μ parameters

(z. ,...,ζ ) e U

χ

depending analytically on the

χ U

, or equivalently of course,

depending analytically on the parameters and the functions

X(T;z.,...,ζ ) really

(p ,...,ρ ) e IL. χ. ..x U

g(z..,...,z ;z) correspondingly describe a mero­

morphic family of meromorphic sections of these bundles.

In more

detail, for any covering \!~l = {U } of the Riemann surface

-117-

Μ

by

simply connected open subsets sections, select for each set simply covering

with connected pairwise interan open subset

; and note that for each nonempty intersection

there is a unique covering translation

such that

namely, that transformation tion of

lying over

taking the por-

to the portion of

Then to each nonempty intersection the function

lying over associate

; these are complex

analytic mappings

which for each fixed

point

are easily seen to form a cocycle

and hence to define a flat line bundle over

M . Also to each set

associate the function where

lies over

j these are meromorphic functions which for each fixed point are easily seen to form a meromorphic

section

of the flat line bundle

Note that by construction and consequently

as complex analytic line bundles, With this auxiliary construction out of the way, the remainder of the proof then follows readily from the proof of Theorem 15-

-118-

Suppose that the line bundles

, and

by analytic cocycles in

are also defined

, for the same covering

of

the Riemann surface M ; the products of the appropriate cocycles then clearly yield complex analytic mappings

which for each fixed point

form a co-

cycle

representing the com-

plex analytic line bundle , where

.

The restric-

tions of these functions to the subvariety senting

repre-

in a neighborhood of the point

can be

viwed as analytic mappings

having the

properties required by the first assertion of the theorem. The sections

, forming a basis for

the space of sections of the line bundle sented by local complex analytic maps bundle

can be repre, when the

is represented by a cocycle in

and there are complex analytic mappings ing a section

\ represent-

of the line bundle

that bundle is represented by a cocycle in Then using the analytic mappings

when

, such that intro-

duced in the proof of Theorem 15, define the meramorphic functions

-119-

Note that for each point

it follows from the con-

struction that the functions

are linearly-

independent meromorphic sections of the line bundle , and since in addition

and

, these sections are really holomorphic, hence

form a basis for the space of holomorphic sections of the line bundle That serves to conclude the proof of the theorem. Corollary 1 to Theorem 16.

In the conclusions of Theorem

16(a) the mappings

can be taken to be

of the form

are

analytic mappings which for each fixed point representing a flat line bundle represents the complex analytic line bundle

form a cocycle which in turn , and

are analytic mappings forming a cocycle representing the complex analytic line bundle Proof.

This is merely an observation of what was actually

proved in the course of the proof of Theorem 16(a). Applying this theorem, for any sufficiently small relatively open subset index

V

of the analytic subvariety

r , introduce the mapping

-120-

and any

defined by

where the functions

are as in the

statement of Theorem l6(a).

This mapping is clearly one-to-one and

linear for each fixed

, hence establishes the local product

structure or equivalently the vector bundle structure desired on the set

, in a quite explicit form; indeed the

form is sufficiently explicit that this bundle structure is easily extended to the entire set

Let

be a covering of the analytic subvariety by sufficiently small relatively open subsets such

that the constructions of Theorem l6(a) and of its Corollary 1 can be carried out on each of the sets

. Note that these construc-

tions can be carried out in terms of the same open covering of the Riemann surface

M

for all the sets

simultaneously,

even if there are not necessarily only finitely many of these sets ; for all the flat line bundles over and

M

as well as the bundles

can be described in terms of one covering

, which

can be assumed to consist of simply connected open subsets with connected pairwise intersections.

Thus there are complex ana-

lytic mappings

and

that for any fixed point

the functions

such

form a cocycle representing the complex

-121-

analytic line bundle

; and there are complex analytic mappings , so that for any fixed point

the functions

are a basis for

the space of holomorphic sections of the line bundle

The

condition that these functions are sections of the line bundle means explicitly that (9) whenever

and

. Then over each of these sets

the family

can be given a local product

structure by the mapping

defined by

(10)

the constants

are thus the fibre coordinates of the

resulting vector bundle over the coordinate neighborhood , and it merely remains to compare these coordinatizations over intersections

of coordinate neighborhoods.

For any point

, the cocycles

and over

determine flat line bundles M

which represent the same complex analytic line bundle

; the functions sections over

and

can therefore be viewed as two

of the complex analytic bundle

-122-

described by the exact sequence (8).

Thus there

exists a complex analytic mapping

so

that

(11) To be more explicit, in terms of the basis Abelian differentials over

for the space of

M , there are complex analytic mappings

such that open set

. For each

select analytic functions

in

such that

; and recalling the explicit form for the coboundary mapping

in the exact sequence (8), it follows that (ll) can

be rewritten

(12) for all

Then introducing the com-

plex analytic mappings

defined by , condition (12) can finally

be rewritten in the form

(13)

for all

and

. Now for any

the functions

represent a basis for the

space of holomorphic sections of the line bundle functions

as do the

but these represent holomorphic

-123-

sections in terms of two different cocycles describing the line bundle

, in the sense that the functions

equation (9) while the functions equation over

satisfy

satisfy the corresponding

. However in view of (13) the functions also satisfy (9), hence represent with the

functions line bundle

two bases for the space of sections of the , expressed in terms of the same cocycle repre-

sentation for that line bundlej thus for any are constants

there

determining a nonsingular matrix such that

(HO

whenever

and

. Since the sections

are linearly independent, then for any point are points

there

such that the matrix

... ,v , is nonsingular at

£

and hence in an open neighborhood of

i ; and it follows immediately from (lH) that the functions are holomorphic in equations (lU) at the

, since they are uniquely determined by the v

points

.

Thus considering merely the

sections

represented by the

functions

, it follows from (lU) that

(15)

whenever

for the complex analytic mappings

-12k-

the cocycle

then describes the coor-

dinate transformations relating the coordinatizations (10) over intersecting neighborhoods

and

, in the sense that if and

only if (16) These observations taken together with the preceding results then yield immediately the proof of the following assertion.

Theorem l6(b).

The set

can be

given the structure of a complex analytic vector bundle over the analytic variety

, in such a manner that the fibre over

is the vector space

j and the divisor

mapping

induces a one-one complex analytic mapping from the associated complex analytic projective bundle over onto the subvariety

with fibre , so that

is the bundle

projection. In the special case

there follows immediately

the following natural complement of Corollary 2 to Theorem 10 and Corollary 2 to Theorem 15-

-125-

Corollary 2 to Theorem l6. set

For any index

can be given the structure of a complex

analytic vector bundle over the manifold that the fibre over

the

J(M) , in such a manner

is the vector space

and the divisor mapping

induces an analytic fibre bundle homeomorphism between the associated complex analytic projective bundle and the bundle

-126-

Notes for §3. (a)

An analytic mapping

> Ν between two complex ana­

τ: Μ

lytic manifolds, with the property that the inverse image of any point of Ν

is a nonempty finite set of points of Μ , is locally

a branched analytic covering, in the technical sense customary in the study of complex analytic varieties; that is to say, the map­ ping

τ is proper and light, and there exists a proper complex

analytic subvariety D C Ν τ: Μ- τ" (D)

^ Ν-D

for which the restriction

is a complex analytic covering projection.

A discussion of the properties of such mappings can be found in most texts on functions of several complex variables; the terminol­ ogy used here is that of E. C. Gunning, Lectures on Complex Analytic Varieties (Princeton University Press, 1970).

In particular, when­

ever τ is a one-one mapping it is an analytic homeomorphism. Similar mappings can also be considered where both domain and range are complex analytic varieties with possible singularities, although then they need not be analytic homeomorphisms even when one-one; but such mappings do always preserve dimension. (b)

The subvarieties W - W

C J ( M ) have a definite interest,

as evidenced for instance in the course of the proof of Theorem 11. They are irreducible analytic subvarieties of U ) + Σ η ε c. , x

i=l

where

*

1

1 = 1

I is t h e identity transformation.

1

T h e left hand side is a

constant, while t h e right hand side is a function o f «9- ; and since the images

g .

(5)

i=l

"

It is now quite easy to demonstrate that necessarily some

r. > g-2 for

i j for if r. < g-2 for all i then it follows immediately

from Lemma 7(d) that

d. = r. and that d. < v. for all i , and

'

1

1

1

1

hence from (3) and (5) that g = Σ. ε.ν. > Σ. v. > Σ . r. > g , 1

1

1

an evident contradiction.

=

1

1

1

1

=

'

It can therefore be assumed that

r. > g-2 , indeed from Lemma 7(d) that (6)

d 1 = g-2 , Γ χ > g-2 , v± > g-2 .

-15*+-

Considering then a second standard irreducible component of the analytic subvariety

M , it follows immedi-

ately from (3) and (6) that 7(d) it follows that

1

2 . If

1

and that

then from Lemma 1

only when

g = 3 j which case has been excluded by hypothesis; furthermore if 0 =

= min

, then

that u 2

= 0

and from Lemma 7(c) it follows

, which is one of the desired consequences. then again from Lemma 7(d) it follows that

that

2

only when

If

2 and

g = 1+ , which case has also been

excluded by hypothesis; and as before if

d^ = 0

which is one of the desired consequences.

then

u

,

Therefore the only case

left to consider is that in which (7) and in this case

, where

subgroup of the Abelian Lie group

J(M)

a point of J(m) ; note also that

n = 2 , so that V

two irreducible components, and that

is a Lie

of dimension 1 and

Cg is

has precisely

r^ = g, V^ = g-2, and

1 . To examine this last case more closely, note that for any divisor

for which , the unique divisor in

containing

can be written as

(8)

-155-

with the notation introduced above; but this divisor can also be written in the form (9) so that as a consequence of Lemma 7(a) the point necessarily lies on the subvariety

and therefore

Now on the other hand from (l), (2), and the preceding observations it follows that

so that

for all divisors

Since

; that is to say,

as a consequence of (!+) and since

C^

is non-

singular as a consequence of the observation that follows that variety of

J(M)

is an irreducible analytic subof dimension

g-3 ; and noting that

-156-

since otherwise

and this is clearly

impossible, it further follows that lytic subvariety of

is an irreducible ana-

of dimension

Therefore

must be an irreducible component of the intersection of the desired form, and the proof of the theorem is thereby concluded. There are a number of questions about additional properties of these intersections that come to mind almost at once, even for the case that

but rather than pursuing these matters

further here, let it suffice merely to observe that the reducibility of the intersection

seems to imply that

rather special point of

is a

, as indicated in the following.

Corollary 1 to Theorem 18. intersection

If for some genus

the

has 2 irreducible components, then

the intersection Proof.

u

is of dimension If

then it follows from Theorem 17(a)

that one irreducible component of the intersection is a translate of

W

_ , and hence is of dimension

g—.J

other case in which the intersection

) g-3 ; the only has

ducible components is that considered in detail in the last part of the proof of Theorem 18.

Continuing with the notation introduced

in that proof, for any divisor

where consider the divisor (8) in

, and note that it can be rewritten in the form

-157-

(10) thus setting

it follows from

Lemma J (a.) that the points

and

are all contained in the subvariety

If for same

divisor

the associated divisor

also belongs to

then clearly

is a proper

analytic subvariety of

; and in this case it is easy to

see that the points component

V^

I

are all contained in the irreducible

of the subvariety

V .

(For note that for all

outside another proper analytic subvariety the points of the divisor (8) are distinct on since

; then a divisor

moved along a closed loop in i

M , can be

\

so that the point

| is deformed into any other point

by the natural

analytic continuation, and since the same motion can be viewed as deforming the point

(

into the point

all of

these points necessarily belong to the same irreducible standard component of

V , indeed to the component

V^

since

Then from (l) and (2) it follows that

hence since

C^

is nonsingular that

lent ly that

-158-

or equiva-

tut this implies that

, which is clearly impossible.

This contradiction then shows that necessarily divisors

for all

or equivalently that

and consequently

, so that has dimension

as desired.

That suffices to

conclude the proof of the Corollary.

(c)

Finally something should also be said about these intersec-

tions for Riemann surfaces of genus hypothesis

, Recall that the

was used in Theorem 18, in the discussion of the

possible values for the various parameters the standard irreducible components

V^

associated to

of the analytic subvariety

, to rule out some exceptional cases; but the presence of these exceptional cases reflects the possibility of the occurrence of some special symmetries in these intersections in the cases

and merits some further examination.

Referring to

the proof of Theorem 18, the exceptional cases were those in which the parameters associated to the second standard component had the values

in case that

in case that

and the values

. Note also that the discussion of the last

case considered in the course of the proof of Theorem 18 must also be modified when that

, since from

unless

-159-

it cannot be concluded

Corollary 2 to Theorem 18.

For a Riemaxm surface of genus

and. any point

, the intersection

has at most 3 irreducible components; if it has precisely 2 irreducible components, then either

and

these components are as described in Theorem 17(b), or one of these components is of the form and

x

is some point of

where

'

is a Lie subgroup

J(M) ; and if it has precisely 3 irre-

ducible components, then these all are of the form T^ C 5 with Jacobi

varieties

J(M)

W

and W* C j(M') respectively, and if there is an ana­

C j(M)

and

If Μ

and subvarieties of positive divisors

lytic homeomorphism between the manifolds forming the subvariety W

.. C J(M)

then the Riemann surfaces

Μ

and M 1

j(M)

and J(M') trans­

to the subvariety W

η

C J(M')

are also analytically homeo-

morphic. Proof.

To prove the desired result it is of course suffi­

cient to show that the Riemann surface Μ

itself can be recon­

structed from knowledge only of the complex torus subvariety W

j(M)

and of the

. C j(M) . For this purpose consider those points

u e J(M) such that the intersection W . Π (W . + u) has preg-1 g-1 cisely 2 irreducible componentsj it follows from Theorem 18 that

-169-

there exist such points

u , and that for any such point

the 2 irreducible components are of the form for some points is of the fonn of dimension

where

u

either

and

, or one of the components X C j(M)

and

is an irreducible subvariety

is a Lie subgroup of dimension 1.

These two cases are readily distinguished, for a subvariety of the form

is translated into itself b y adding to it any one of

the infinitely many points of the subgroup variety of the fonn

while a sub-

or

cannot be translated

into itself by any nonzero point of

J(M) , as noted on page

thus it is possible to construct a subvariety either

V C J(M)

such that

for some points

Now note that

and that

; and consequently either

or

In any case the analytic subvariety is analytically homeamorphic to the Riemann surface

M

itself; and since the construction involved only the complex torus J(M)

and the subvariety

, the proof of the theorem is

completed. Corollary 1 to Theorem 19.

The theorem also holds as stated

for compact Riemann surfaces of genus case

the Jacobi variety

of period 3 with no real eigenvalues.

-170-

provided that in the does not admit an endomorphism

Proof.

There is only one compact Riemann surface of genus

g = 0 , while a surface of genus

g = 1

is analytically homeomor-

phic to its Jacobi variety and a surface of genus lytically homeomorphic to the subvariety W

g = 2

is ana-

.. C J(M) ; the theorem

is thus completely trivial in these cases. That the proof of theorem 19 goes through for compact Riemann surfaces of genus and

g = 3

g = k , with the exception as noted, follows immediately from

Corollaries 2 and 3 to Theorem 18, respectively.

That suffices for

the proof of the desired result. The difficulty in carrying the proof through in the exceptional case when

g = h

is merely that of intrinsically distin-

guishing between subvarieties of the form the form J(M)

+ (C.W

- CTW + v) , where

C

+ (W_+v)

and those of

is an endomorphism of

of period 3 with no real eigenvaluesj and the difficulty lies

in the method of proof rather than the theorem itself, which holds in this case as well.

Rather than giving a separate argument in

this special case, though, the reader will be referred to the other proofs of the theorem listed in the references, or left to conclude the proof on his own. As interesting as it may be on its own, this theorem gains immeasurably greater significance if one is also aware that the subvariety W

.. C J(M) can be constructed quite explicitly, at least

up to a translation in J(M) , from the period matrix of the marked Riemann surface alone; hence the Torelli theorem really implies that two marked Riemann surfaces are analytically equivalent if

-171-

they have the same period matrices.

These period matrices are an

interesting set of moduli for describing Riemann surfaces, and their properties have not yet been altogether sorted out.

It

should be remarked that the period matrix of a marked Riemann sur­ face determines not only just the Jacobi variety of that surface, but also the additional structure embodied in the naturally associ­ ated Riemann matrix pair; and this additional structure, sometimes called a polarization of the complex torus, is essential in describ­ ing the hypersurface

W η . Alternatively it is sometimes the g-i

pair consisting of the complex torus variety W

J(M) and the analytic sub-

. C J(M) that is called a polarized complex torus.

The continuation of this tale must be left for another episode in the serial, though.

-172-

Notes for §1+. (a)

It is a familiar result in the theory of functions of

several complex variables that for any nonempty irreducible compo­ nent V

of the intersection of two analytic subvarieties of dimen­

sion

in a complex manifold of dimension g , necessarily

r

dim V > 2r-g ; hence an immediate consequence of Theorem 17(a) is that for any nonempty irreducible component subvariety

«.-,

for any

V

r < g , necessarily

This result is of course just the special case l^(b).

of the analytic dim V > 2r-g . V = 2

of Theorem

Actually a rather straightforward extension of Theorem 17(a)

can be used to give another proof of the general case of Theorem ll)-(b) as well; this proof is due to H. Martens, and can be found in [1)-]. (b)

The various properties of complex analytic subvarieties

used more freely in this section are treated in most of the stand­ ard texts on functions of several complex variables.

It should per­

haps particularly be noted that subvarieties of codimension one in a complex manifold are characterized by the property that they are locally the sets of zeros of a single holomorphic function. Intersections of the form W in a quite similar manner for any

Π (w

+u)

can be described

r = 1,...,g-l ; but for

r < g-1

there are many more cases to consider than in the proof of Theorem l8, and it seems rather doubtful that the results are worth the effort involved in disentangling all the possibilities that arise.

-173-

A comment on the notation is perhaps in order here.

It

seems pointless to maintain separate notations for the Jacobi homomorphism and Μ

r(M, &• )

> j(M)

and the Jacobi mappings Μ

— > J(M)

(r) ' — > j(M) , once the properties of these various mappings

have been established; so contrary to the practice adopted in §3 the same letter

φ has been used for all of these mappings,

leaving them to be distinguished either by context or explicitly as necessary. (c)

The existence of endomorphisms

C: j(M) — > J(M) other

than the trivial ones defined by matrices of the form trary integers plex torus

η

nl

for arbi­

imposes rather severe restrictions on the com­

j(M) ; thus the exceptional cases considered in this

section, especially that described in Corollary 3 to Theorem 18, can only occur for Riemann surfaces whose Jacobi varieties are of quite special forms.

It would be of some interest to see whether

these cases can indeed occur at all. The endomorphisms of nontrivial sort are called complex multiplications of the torus

J(M) ;

their investigation is an interesting subject in its own right, and has been the subject of an extensive literature. (d)

(See [26].)

The proof of Torelli's theorem given here, based on the

analysis of intersections in Jacobi varieties in the preceding sections (b) and (c), is essentially that given by A. Weil in [10], translated from the algebro-geometric to the analytic point of view; separate proofs of Torelli's theorem in the cases of genus

-17^-

g = 3

and

g = k , avoiding the complications discussed in section

(c) and the remaining gap in the proof in these notes, can also be found in [10].

A rather shorter and more direct proof of Torelli's

theorem, based on an extension of the techniques discussed in sec­ tion (a) rather than on the general discussion of intersections in section (b), was given by Η. H. Martens in [2] and [3]. proofs can be found in the references listed next.

-175-

Other

References.

The main source for the material discussed in

this chapter is [10]. Other treatments of Torelli*s theorem and discussions of related results can be found in the following, and in [2], [3], [7]· [23] Andreotti, Aldo, Recherches sur les surfaces algebriques irregulieres, Mem. Acad. Belg. 2j(l952), fasc. 7. [2½]

f

on a theorem of Torelli, Amer. Jour, of Math. 80

(1958), 801-828. [25] Torelli, R., Sulle varieta di Jacobi, Rend. Accademia Lincei 22(19110, 98-IO3. [26] Weil, Andre, On the theory of complex multiplication, Proc. International Symposium on Algebraic Number Theory, Tokyo (1955), 9-22.

-176-

Appendix.

On conditions ensuring that

One special case of the topological argument outlined on page 10U is of particular interest; and since it involves only quite well known topological techniques in a very straightforward manner, it is perhaps worthwhile appending a more detailed discussion of that case. If for some index

r

the subvariety

then the Jacobi mapping

is a nonsingular com-

plex analytic homeamorphism between submanifold

is empty,

and the complex analytic

; thus

can be viewed as a regularly

imbedded analytic submanifold of

J(M) . As such there is a well

defined analytic normal bundle

, which is a complex analytic (r)

vector bundle of rank

g-r

over the manifold

sum of the tangent bundle

M

' ; and the direct

and this normal bundle

is topological1y equivalent to the restriction to the submanifold of the tangent bundle to

J(M) , hence that direct sum

is topological!y equivalent to a trivial vector bundle of rank over the manifold

.

g

Therefore introducing the total Chern

classes

where

, it follows that

(1)

(r) The topological properties of the symmetric products

-177-

M

have been described very conveniently by I. G. Macdonald in [18],

and can be summarized briefly as follows.

The cohomology ring

is generated by elements degree 1 and an element

of

t) of degree 2; these are subject to the

usual commutativity relations, namely that elements of degree 1 anticommute with one another and commute with elements of degree 2 , and in addition to the relations (2) where

are any distinct integers

and

are any integers such that

particular

In

is generated by the single element

and

is generated by the

independent ele-

ment

where ; as usual

(denotes

£

a

n

,

a binomial coefficient. d

,

Chern class of the tangent bundle to the manifold

Setting

the total M

(r)

v

is

(3) These assertions are demonstrated in [l8], with the same notation. An additional useful observation is the following. Lemma 8. integers

and for any index

For any index in

and any distinct

[l,g] ,

and any distinct integers

-178-

in

[l,g] ,

a±

r-l-a / ... σ 1 η = (σ1 l a

Proof.

, , \ r-2 , /, \ r-1 +...+ σ.^ )η + (1-&)τ\ l a

Both relations hold trivially for

second relation also holds trivially for strated by induction on the index

a = 0 , and the

a = 1 ; and will be demon­

a . To prove the first relation

note that as a consequence of (2) it follows that φ. ... φ. η l a then

" = 0 whenever

φ. ... φ. η ~ Χ 1 a

1 < a < ^-(r+l) ; and evidently

= 0 whenever

1 < a < r . Assuming that the

first relation holds for all indices less than the expansion of the product

a

and considering

(σ. -η) ... (σ. -η)η " 1 a

= 0 , all of

the terms in the expansion of this product except for the term σ. ... σ. η 1 1 a

coincide with the corresponding terms in the expan-

sion of the product therefore that

(η-η) ... (η-η)η "

σ. . . . σ . η " 1 a

=η

; and it follows immediately

as desired.

To prove the

second relation note as above that as a consequence of (2) it follows that

φ. •""l

... φ. η

= 0 whenever

1 < a < r . Assuming then

"""a

that the second relation holds for all indices less than lows that

-179-

a

it fol­

which yields the desired result and concludes the proof of the lemma. Combining the preceding observations leads rather directly to the following result. Lemma 9-

If for some index

r

the subvariety

is empty, then the Chern classes of the normal bundle have the form

and

with the usual notation for the binomial coefficients.

-180-

Proof.

If the subvariety

is empty then it

follows from (l) and (3) that

2

recalling that

and that )

CK = 0 .

The class

consists of those terms of degree

2r

in the

above summation; and applying the first relation in Lemma 8 and expanding the summation, it follows that

-l8l-

Note that

CO

for any indices

C

if not well known, this is at

least easily demonstrated by induction, and the details will be omitted.

Applying (1+) it then follows that

and this can be simplified by applying a standard form of the Vandermonde convolution formula [see J. Riordan, Combinatorial Identities (Wiley, New York, 1968), formula (5) on page 8], yielding the result that

as desired.

The class

terms of degree

:onsists of those in the original summation; and expanding the

resultant summation and applying the second relation in Lemma 8 , it follows that

-182-

-183-

Applying (U) once more this can be rewritten

and after further simplification by an application of the Vandermonde convolution formula as before it follows that

which serves to complete the proof of the lemma. The formulas of Lemma 9 are already simple enough for the present purposes, even though further simplifications are still possible.

Thus for the special case that is nonzero only for

note that so the sum-

mation in the first formula of Lemma 9 can be restricted to these two indices and the result easily calculated; and the calculations are equally simple for the special cases that

and

g = 2r-3 , and for both the first and second formulas of Lemma 9The relevant results are the following: (5)

if

then

(6)

if

then

-l8U-

(γ)

if g = 2r-3 then η = 0 and _ r-k r

r-1

„ r-3 /2r-lw , ^ r(r-l)v r-2'v 1

/2r-3\ r-1 r-1' '

^

λ r-2 g' '

These observations then lead almost immediately to the principal result of this appendix. Theorem 20.

On a Riemann surface of genus

g > 2 if

? 2 2r - (g+2) > 0 then necessarily W j= 0 and G ^ φ . ρ

Proof.

It clearly suffices merely to show that G j=- φ

for the least value of the index

r such that 2r - (g+2) > 0 ,

hence for the index r such that for the index r such that

g = 2r-2 when g is even and

g = 2r-3 when

g is odd. If

ρ

G = φ when

g = 2r-2

then since

g-r = r-2

necessarily

η

=0 ,

(r) n^ ε Η (Μ ' ,Z) i s a Chern class for the normal where as Ν(NT before bundle ') , a vector bundle of rank g-r over the manifold undj ; but (6) shows that η φ 0 , a contradiction.

Μ when η

g = 2r-3 then since g-r = r-3 necessarily

. = 0 ; but (7) shows that η

If G = φ

η = 0 and

/ 0 , a contradiction again,

and the proof is thereby completed. This result completes Theorem Ik in a natural manner, at least in the case that

ν = 2 . For Theorem 1Mb) implies that for

any nonempty irreducible component V

of the analytic subvariety

p

W C J(M) necessarily dim V > 2r - (g+2) , recalling Corollary 1 to that theorem; and Theorem 20 shows that whenever 2r - (g+2) > 0 there necessarily exist some nonempty components of the subvariety p

W C j(M) . Combining these results with Theorem J it follows that -I85-

2r - (g+2) < dim W 2 < r-2 r =

(8)

A superficial glance at Theorem lh(a.) might lead one to expect a somewhat better result than that given by Theorem 20, until one recalls that

G r

is generally fibred over

w with fibre dimenr 2 sion 1 and hence that any irreducible component of G must be of

dimension at least 1; indeed it is apparent from (5) that the argu­ ment in Theorem 20 fails for the case that

g = 2r-l .

Corollary 1 to Theorem 20. Any compact Riemann surface of genus

g

admits a representation as a branched analytic cover­

ing of the Riemann sphere

IP

of at most

[**=—] + 1 sheets.

Proof.

The assertion is trivial for g = 0,1 . If g > 2 ο it follows from Theorem 20 that W / 0 whenever 2r - (g+2) > 0 , hence in particular that W fi 0 that value of ξ

with

r

c(|) = r

for

r = [ g ? ]+ 1 . Thus for

there must exist a complex analytic line bundle and

y{t)

> 2 ; and the quotient of any two

linearly independent holomorphic sections of

ξ is a meromorphic

function which represents the given Riemann surface as a branched analytic covering of the Riemann sphere of at most

r sheets,

thereby completing the proof of the Corollary. This Corollary indicates the particular interest associated to the problem of determining whether \T jL φ ·} other proofs of the Corollary have been given by T. Meis [2k],

G. Kempf [12], and S.

Kleiman and D. Laksov [lU], and rather incomplete proofs appeared

-186-

much earlier in the literature. whenever

rv - (v-l)(g+v) > 0

The analogous proof that W

/ 0

requires rather more topological

machinery, along the lines indicated in [12] and [14].

-I87-

Index of symbols

M , compact Riemann surface of genus (r) M

', symmetric product of r copies of

j(M), Jacobi variety of P(M), Picard variety of [39 subvarieties of positive divisors,