Workshop on Curves and Jacobians pp

Arnaud Beauville

Description

Niveau: Supérieur, Doctorat, Bac+8
Workshop on Curves and Jacobians, pp. 1–25 Classical theta functions and their generalization Arnaud Beauville Abstract. We first recall the modern theory of classical theta functions, viewed as sections of line bundles on complex tori. We emphasize the case of theta functions associated to an algebraic curve C : they are sections of a natural line bundle (and of its tensor powers) on the Jacobian of C , a complex torus which parametrizes topologically trivial line bundles on C . Then we ex- plain how replacing the Jacobian by the moduli space of higher rank vector bundles leads to a natural generalization (“non-abelian theta functions”). We present some of the main results and open problems about these new theta functions. Introduction Theta functions are holomorphic functions on Cg , 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 Riemann 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 sections 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.

• bundles leads


• group


• standard tool


• complex tori


• bundle


• e?


• functions


• line bundles

Sujets

• Beauville
• Complex torus
• Bundle
• Function

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 12
• Langue: English

Extrait

Workshop on Curves and Jacobians, pp. 1{25
Classical theta functions and their generalization
Arnaud Beauville
Abstract. We rst recall the modern theory of classical theta functions,
viewed as sections of line bundles on complex tori. We emphasize the case
of theta functions associated to an algebraic curve C : they are sections of a
natural line bundle (and of its tensor powers) on the Jacobian of C , a complex
torus which parametrizes topologically trivial line bundles on C . Then we ex-
plain how replacing the Jacobian by the moduli space of higher rank vector
bundles leads to a natural generalization (\non-abelian theta functions"). We
present some of the main results and open problems about these new theta
functions.
Introduction
gTheta functions are holomorphic functions on C , 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 Riemann 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 sections 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 de nition 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.
These notes follow closely my lectures in the Duksan workshop on algebraic
curves and Jacobians. I will rst develop the modern theory of classical theta func-
tions (complex tori, line bundles, Jacobians), then explain how it can be generalized
by considering higher rank vector bundles. A more detailed version can be found
in [B5].
1991 Mathematics Subject Classi cation. Primary 14K25, 14H60; Secondary 14H40, 14H42.
Key words and phrases. Complex torus, abelian variety, theta functions, polarization, theta
divisor, stable vector bundles, generalized theta functions, theta map.
I wish to thank the organizers for the generous invitation and the warm atmosphere of the
workshop.
12 ARNAUD BEAUVILLE
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;Z
nequivalently, any basis of over Z is a basis of V . In particular =Z .
1 nThe quotient T :=V= is a smooth, compact Lie group, isomorphic to ( S ) .
The quotient homomorphism :V !V= is the universal covering of T . Thus
is identi ed with the fundamental group (T ).1
We want to consider the cohomology algebra H (T;C). We think of it as being
de Rham cohomology: recall that a smooth p-form ! on T is closed if d! = 0,
exact if ! =d for some (p 1)-form . Then
fclosed p-formsgpH (T;C) =
fexact pg
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 coordinates on V . The forms (dx ;:::;dx ) form a1 n 1 n
basis of the cotangent space T (T ) at each point a2 T ; thus a p-form ! on Ta
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
An important role in what follows will be played by the translations t :x7!x+aa
of T . We say that a p-form ! is constant if it is invariant by translation, that is,
t ! = ! for all a2 T ; in terms of the above expression for !, it means that thea
functions ! are constant. Such a form is determined by its value at 0, which isi :::i1 p
p
a skew-symmetric p-linear form on V =T (T ). We will denote by Alt (V;C) the0
space of such forms, and identify it to the space of constant p-forms. A constant
pp pform is closed, hence we have a linear map : Alt (V;C)!H (T;C). Note that
1 1Alt (V;C) is simply Hom (V;C), and maps a linear form ‘ to d‘.R
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 use the Kunneth formula. We choose our coordinates (x ;:::;x )1 n
n n 1so that V =R , = Z . Then T =T :::T , with T =S for each i, and1 n i
1dx is a 1-form on T , which generates H (T ;C). The Kunneth formula givesi i i
N
an isomorphism of graded algebras H (T;C)! H (T ;C). This means thatii
H (T;C) is the exterior algebra on the vector space with basis (dx ;:::;dx ), and1 n
this is 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 considered as a subgroup of H (T;C). By de nition of the de
p pRham isomorphism the image of H (T;Z) in H (T;C) is spanned by the closed
R
p-forms ! such that ! 2 Z for each p-cycle in H (T;Z). Write againpCLASSICAL THETA FUNCTIONS AND THEIR GENERALIZATION 3
n nT =R =Z ; the closed paths :t7!te , for t2 [0; 1], form a basis of H (T;Z),i i 1R
1and we have d‘ = ‘(e ). Thus H (T;Z) is identi ed with the subgroup ofii
1H (T;C) = Hom (V;C) consisting of linear forms V ! C which take integralR
values on ; it is isomorphic to Hom ( ;Z). Applying again the Kunneth formulaZ
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 which take integral values on ; it is isomorphic to
pAlt ( ;Z).
1.2. Complextori. From now on we assume that V has a complex structure,
g 2g that is, V is a complex vector space, of dimension g. Thus V =C and =Z .
Then T :=V= is a complex manifold, of g, in fact a complex Lie group;
the covering map :V !V= is holomorphic. We say that T is a complex torus.
1 nBeware : while all real tori of dimension n are di eomorphic 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-linearC C
and C-antilinear forms respectively. We write the corresponding decomposition of
1H (T;C)
1 1;0 0;1H (T;C) =H (T )H (T ) :
1;0If (z ;:::;z ) is a coordinate system on V , H (T ) is the subspace spanned by1 g
1;0the classes of dz ;:::;dz , while H (T ) is spanned by the classes of dz ;:::;dz .1 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=
which 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
are C-linear (resp. C-antilinear) in each variable. It is not immediate to charac-
q;rterize those which belong to H (T ) for q;r> 0; for p = 2 we have:
22 2;0Proposition 1.3. Via the identi cation H (T;C) = Alt (V;C), H is the space
0;2 1;1of C-bilinear forms, H the space of C-biantilinear forms, and H is the space
of R-bilinear forms E such that E(ix;iy) =E(x;y).
Proof : We have only to prove the last assertion. For " 2 f 1g, let E be"
2the space of forms E 2 Alt (V;C) satisfying E(ix;iy) = "E(x;y). We have
2 2;0 0;2Alt (V;C) =E E , and H and H are contained inE .1 1 1
0For ‘2V , ‘ 2V , v;w2V , we have
0 0 0 0(‘^‘ )(iv;iw) =‘(iv)‘ (iw) ‘(iw)‘ (iv) = (‘^‘ )(v;w) ;4 ARNAUD BEAUVILLE
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
1 the rst cohomology group H (M;O ) of the sheafO of invertible holomorphicM M
functions on M . To compute this group a standard tool is the exponential exact
sequence of sheaves
0!Z !O !O ! 1M M M
where (f) := exp(2if), and Z denotes the sheaf of locally constant functionsM
on M with integral values. This gives a long exact sequence in cohomology
c1 1 1 2 2(1) H (M;Z)! H (M;O )! Pic(M)! H (M;Z)! H (M;O )M M
2For L2 Pic(M), the class c (L)2 H (M;Z) is the rst Chern class of L. It1
is a topological invariant, 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 K ahler) manifold, Hodge theory provides
1more information on this exact sequence. The image of c is the kernel of the1
2 2natural map H (M;Z)!H (M;O ). This map is the composition of the mapsM
2 2 2H (M;Z) ! H (M;C) ! H (M;O ) deduced from the injections of sheavesM
2 2 0;2Z ,! 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
(for the experts: this can be seen by comparing the de Rham complex with the
Dolbeault complex.)
2Thus the i