Theta functions old and new

Arnaud Beauville

Description

Theta functions, old and new Arnaud Beauville Universite de Nice

• group


• standard tool


• exponential exact


• theta functions


• complex tori


• bundles


• line bundles

Sujets

• Tsinghua University
• Beauville
• Theta function
• Complex torus
• Bundle

Informations

• Publié par: pefav
• Nombre de lectures: 30
• Langue: English

Extrait

Theta functions, old and new
Arnaud Beauville
´Universite de NiceIntroduction
gTheta functions are holomorphic functions onC , quasi-periodic with respect to a lattice. For g = 1
they have been introduced by Jacobi; in the general case they have been thoroughly studied by Rie-
mann and his followers. From a modern point of view they are sections of line bundles on certain
complex tori; in particular, the theta functions associated to an algebraic curve C are viewed as sec-
tions of a natural line bundle (and of its tensor powers) on a complex torus associated to C , the
Jacobian, which parametrizes topologically trivial line bundles on C .
Around 1980, under the impulsion of mathematical physics, the idea emerged gradually that one
could replace in this definition line bundles by higher rank vector bundles. The resulting sections
are called generalized (or non-abelian) theta functions; they turn out to share some (but not all) of the
beautiful properties of classical theta functions.
The goal of these lectures is to develop first the modern theory of classical theta functions (complex
tori, line bundles, Jacobians), then to explain how it can be generalized by considering higher rank
vector bundles. We have tried to make them accessible for students with a minimal background in
complex gometry: Chapter 0 of [G-H] should be more than enough. At a few places, especially in the
last chapters, we had to use some more advanced results. Also we have not tried to be exhaustive:
sometimes we just give a sketch of proof, or we prove a particular case, or we just admit the result.
These notes come from a series of lectures given at Tsinghua University in April 2011. I am grateful
to Tsinghua University and Professor S.-T. Yau for the generous invitation.
1 The cohomology of a torus
1.1 Real tori
Let V be a real vector space, of dimension n. A lattice in V is a Z-module V such that the
induced map
R!V is an isomorphism; equivalently, any basis of overZ is a basis of V . InZ
nparticular Z .=
1 nThe quotient T := V= is a smooth, compact Lie group, isomorphic to (S ) . The quotient ho-
momorphism :V !V= is the universal covering of T . Thus is identified with the fundamental
group (T ).1
We want to consider the cohomology algebra H (T;C). We think of it as being de Rham coho-
mology: recall that a smooth p-form ! on T is closed if d! = 0, exact if ! =d for some (p 1)-form
. Then
fclosedp-formsgpH (T;C) =
fexactpg
Let ‘ be a linear form on V . The 1-form d‘ on V is invariant by translation, hence is the pull
back by of a 1-form on T that we will still denote d‘. Let (x ;:::;x ) be a system of coordinates1 n
on V . The forms (dx ;:::;dx ) form a basis of the cotangent space T (T ) at each point a2T ; thus1 n a
a p-form ! on T can be written in a unique way
X
! = ! (x)dx ^:::^dxi :::i i i1 p 1 p
i <:::<i1 p
where the ! are smooth functions on T (with complex values).i :::i1 p
1An important role in what follows will be played by the translations t :x7!x +a of T . We saya
that ap-form! is constant if it is invariant by translation, that is,t ! =! for alla2T ; in terms of thea
above expression for! , it means that the functions! are constant. Such a form is determined byi :::i1 p
p
its value at 0, which is a skew-symmetric p-linear form on V =T (T ). We will denote by Alt (V;C)0
the space of such forms, and identify it to the space of constant p-forms. A constant form is closed,
p 1p phence we have a linear map : Alt (V;C)!H (T;C). Note that Alt (V;C) is simply Hom (V;C),R
1and maps a linear form ‘ to d‘.
pp pProposition 1.1. The map : Alt (V;C)!H (T;C) is an isomorphism.
Proof : There are various elementary proofs of this, see for instance [D], III.4. To save time we will
n nuse the Kunneth¨ formula. We choose our coordinates (x ;:::;x ) so that V = R , = Z . Then1 n
1 1T = T :::T , with T =S for each i, and dx is a 1-form on T , which generates H (T ;C).1 n i i i i
N
The Kunneth¨ formula gives an isomorphism of graded algebras H (T;C)! H (T ;C). Thisii
means that H (T;C) is the exterior algebra on the vector space with basis (dx ;:::;dx ), and this is1 n
equivalent to the assertion of the Proposition.
What about H (T;Z)? The Kunneth¨ isomorphism shows that it is torsion free, so it can be con-
psidered as a subgroup of H (T;C). By definition of the de Rham isomorphism the image of H (T;Z)
R
pin H (T;C) is spanned by the closed p-forms ! such that !2Z for each p-cycle in H (T;Z).p
n nWrite again T =R =Z ; the closed paths : t7! te , for t2 [0; 1], form a basis of H (T;Z), andi i 1
R
1 1we have d‘ = ‘(e ). Thus H (T;Z) is identified with the subgroup of H (T;C) = Hom (V;C)i Ri
consisting of linear forms V !C which take integral values on ; it is isomorphic to Hom ( ;Z).Z
Applying again the Kunneth¨ formula gives:
pp p Proposition 1.2. For each p, the image of H (T;Z) in H (T;C) = Alt (V;C) is the subgroup of forms
p
which take integral values on ; it is isomorphic to Alt ( ;Z).
1.2 Complex tori
From now on we assume that V has a complex structure, that is, V is a complex vector space, of
g 2g dimension g . Thus V C and Z . Then T := V= is a complex manifold, of dimension= =
g , in fact a complex Lie group; the covering map : V ! V= is holomorphic. We say that T is a
1 ncomplex torus. Beware : while all real tori of dimension n are diffeomorphic to (S ) , there are many
non-isomorphic complex tori of dimension g – more about that in section 3.3 below.
The complex structure of V provides a natural decomposition
Hom (V;C) =V V ;R
where V := Hom (V;C) and V = Hom (V;C) are the subspaces of C-linear and C-antilinearC C
1forms respectively. We write the corresponding decomposition of H (T;C)
1 1;0 0;1H (T;C) =H (T )H (T ):
1;0If (z ;:::;z ) is a coordinate system onV ,H (T ) is the subspace spanned by the classes ofdz ;:::;1 g 1
1;0dz , while H (T ) is spanned by the classes of dz ;:::;dz .g 1 g
The decomposition Hom (V;C) =V V gives rise to a decompositionR
p p p 1 pAlt (V;C) ^ V (^ V
V ):::^ V=
2which we write
p p;0 0;pH (T;C) =H (T ):::H (T ):
p p;0 0;pThe forms in Alt (V;C) which belong to H (T ) (resp. H (T )) are those which areC-linear (resp.
q;rC-antilinear) in each variable. It is not immediate to characterize those which belong to H (T ) for
q;r> 0; for p = 2 we have:
22 2;0 0;2Proposition 1.3. Via the identification H (T;C) = Alt (V;C), H is the space ofC-bilinear forms, H
1;1the space ofC-biantilinear forms, and H is the space ofR-bilinear formsE such that E(ix;iy) =E(x;y).
Proof : We have only to prove the last assertion. For " 2 f 1g, let E be the space of forms"
2 2 2;0E 2 Alt (V;C) satisfying E(ix;iy) = "E(x;y). We have Alt (V;C) = E E , and H and1 1
0;2H are contained inE .1
0For ‘2V , ‘ 2V , v;w2V , we have
0 0 0 0(‘^‘ )(iv;iw) =‘(iv)‘ (iw) ‘(iw)‘ (iv) = (‘^‘ )(v;w);
1;1 2;0 0;2 1;1hence H is contained inE ; it follows that H H =E and H =E .1 1 1
2 Line bundles on complex tori
2.1 The Picard group of a manifold
Our next goal is to describe all holomorphic line bundles on our complex torus T . Recall that line
bundles on a complex manifold M form a group, the Picard group Pic(M) (the group structure is
given by the tensor product of line bundles). It is canonically isomorphic to the first cohomology
1 group H (M;O ) of the sheafO of invertible holomorphic functions on M . To compute thisM M
group a standard tool is the exponential exact sequence of sheaves

0!Z !O !O ! 1M M M
where (f) := exp(2if), andZ denotes the sheaf of locally constant functions onM with integralM
values. This gives a long exact sequence in cohomology
c11 1 2 2H (M;Z)! H (M;O )! Pic(M)! H (M;Z)! H (M;O ) (2.1)M M
2For L2 Pic(M), the class c (L)2 H (M;Z) is the first Chern class of L. It is a topological invari-1
ant, which depends only on L as a topological complex line bundle (this is easily seen by replacing
holomorphic functions by continuous ones in the exponential exact sequence).
When M is a projective (or compact Kahler)¨ manifold, Hodge theory provides more information
1 2 2on this exact sequence. The image of c is the kernel of the natural map H (M;Z)! H (M;O ).1 M
2 2 2This map is the composition of the maps H (M;Z)! H (M;C)! H (M;O ) deduced from theM
2 2 0;2injections of sheaves Z ,! C ,!O . Now the map H (M;C)! H (M;O ) = H is theM M M M
projection onto the last summand of the Hodge decomposition
2 2;0 1;1 0;2H (M;C) =H H H
1In this section and the following we use standard Hodge theory, as explained in [G-H], 0.6. Note that Hodge theory is
much easier in the two cases of interest for us, namely complex tori and algebraic curves.
3
ee(for the experts: this can be seen by comparing the de Rham complex with the Dolbeault complex.)
2 0;2 1;1 0;2Thus the image of c consists of classes 2 H (M;Z) whose image = + + in1 C
2 0;2 2 2;0 0;2H (M;C) satisfies = 0. But since comes from H (M;R) we have = = 0: the imageC
2 2 1;1ofc consists of the classes inH (M;Z) whose image inH (M;C) belongs toH (“Lefschetz theorem”).1
oThe kernel of c , denoted Pic (M), is the group of topologically trivial line bundles. The exact1
1 1sequence (2.1) shows that it is isomorph