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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 image of c consists of classes 2 H (M;Z) whose image =1 C
0;2 1;1 0;2 2 0;2 + + in H (M;C) satis es = 0. But since comes fromC
2 2;0 0;2H (M;R) we have = = 0: the image of c consists of the classes in1
2 2 1;1H (M;Z) whose image in H (M;C) belongs to H (\Lefschetz theorem").
oThe kernel of c , denoted Pic (M), is the group of topologically trivial line1
bundles. The exact sequence (1) shows that it is isomorphic to the quotient of
1 1H (M;O ) by the image of H (M;Z). We claim that this image is a latticeM
1 1in H (M;O ) : this is equivalent to saying that the natural map H (M;R)!M
1H (M;O ) is bijective. By Hodge theory, this map is identi ed with the re-M
1 1 1;0 0;1 0;1striction to H (M;R) of the projection of H (M;C) = H H onto H .
1
In 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.
eeCLASSICAL THETA FUNCTIONS AND THEIR GENERALIZATION 5
1 1Since H (M;R) is the subspace of classes + in H (M;C), the projection
1 0;1 oH (M;R)! H is indeed bijective. Thus Pic (M) is naturally identi ed with
1 1the complex torus H (M;O )=H (M;Z).M
2.2. Systems of multipliers. We go back to our complex torus T =V=.
Lemma 2.1. Every line bundle on V is trivial.
2 1Proof : We have H (V;Z) = 0 and H (V;O ) = 0 (see [G-H], p. 46), henceV
Pic(V ) = 0 by the exact sequence (1).
Let L be a line bundle on T . We consider the diagram
L L
V T
The action of on V lifts to an action on L = V L. We know that LT
is trivial; we choose a trivialization L! VC. We obtain an action of on
VC, so that L is the quotient of VC by this action. An element of acts
linearly on the bers, hence by
(z;t) = (z +;e (z)t) for z2V; t2C
where e is a holomorphic invertible function on V . This formula de nes a group
action of on VC if and only if the functions e satisfy
e (z) =e (z +)e (z) (\cocycle condition"):+
A family (e ) of holomorphic invertible functions on V satisfying this con- 2
dition is called a system of multipliers. Every line bundle on T is de ned by such
a system.
A theta function for the system (e ) is a holomorphic function V ! C 2
satisfying
(z +) =e (z)(z) for all 2 ;z2V :
Proposition 2.2. Let (e ) be a system of multipliers, and L the associated 2
0line bundle. The space H (T;L) is canonically identi ed with the space of theta
functions for (e ) . 2
Proof : Any global section s of L lifts to a section s^ = s of L =V L overT
V , de ned by ^s(z) = (z;s(z)); it is -invariant in the sense that ^s(z+) =s^(z).
Conversely, a -invariant section of L comes from a section of L. Now a section
of L =VC is of the form z7! (z;(z)), where :V !C is holomorphic. It
is -invariant if and only if is a theta function for (e ) . 2
0Let (e ) and (e ) be two systems of multipliers, de ning line bundles 2 2
0 0L and L . The line bundle L
L is the quotient of the trivial line bundle
0 0 0V (C
C) by the tensor product action (z;t
t ) = (z +;e (z)t
e (z)t );
////6 ARNAUD BEAUVILLE
0therefore it is de ned by the system of multipliers ( e e ) . In other words, 2
multiplication de nes a group structure on the set of systems of multipliers, and
we have a surjective group homomorphism
fsystems of multipliersg! Pic(T ) :
A system of multipliers (e ) lies in the kernel if and only if the associated 2
line bundle admits a section which is everywhere = 0; in view of Proposition
2.2, this means that there exists a holomorphic function h : V ! C such that
h(z+
e (z) = . We will call such systems of multipliers trivial. h(z)
Remark 2.3. (only for the readers who know group cohomology) Put H :=
0 H (V;O ). The system of multipliers are exactly the 1-cocycles of with values inV
H , and the trivial systems are the coboundaries. Thus we get a group isomorphism
1 H ( ;H )! Pic(T ) (see [M1], x2 for a more conceptual explanation of this
isomorphism).
2.3. Interlude: hermitian forms. There are many holomorphic invertible
functions on V , hence many systems of multipliers giving rise to the same line
bundle. Our next goal will be to nd a subset of such systems such that each line
bundle corresponds exactly to one system of multipliers in that subset. This will
involve hermitian forms on V , so let us x our conventions.
A form H on V will beC-linear in the second variable, C-antilinear
in the rst. We put S(x;y) = ReH(x;y) and E(x;y) = ImH(x;y). S and E are
R-bilinear forms on V , S is symmetric, E is skew-symmetric; they satisfy:
S(x;y) =S(ix;iy) ; E(x;y) =E(ix;iy) ; S(x;y) =E(x;iy)
Using these relations one checks easily that the following data are equivalent:
The hermitian form H ;
The symmetric R-bilinear form S with S(x;y) =S(ix;iy);
The skew-symmetric R-bilinear form E with E(x;y) =E(ix;iy).
Moreover,
H non-degenerate () E non-degenerate () S non-degenerate.
2.4. Systems of multipliers associated to hermitian forms. We denote
byP the set of pairs (H;), where H is a hermitian form on V , a map from
1to S , satisfying:
E(; )
E := Im(H) takes integral values on ; ( +) =()()( 1) :
(We will say that is a semi-character of with respect to E ).
0 0 0 0The law (H;) (H ; ) = (H +H ; ) de nes a group structure on P . For
(H;)2P , we put
1[H(;z)+ H(; )]
2e (z) =()e :
6CLASSICAL THETA FUNCTIONS AND THEIR GENERALIZATION 7
We leave as an (easy) exercise to check that this de nes a system of multipliers. The
corresponding line bundle will be denoted L(H;). The map (H;)7! L(H;)
from P onto Pic(T ) is a group homomorphism; we want to prove that it is an
isomorphism.
Theorem2.4. The map (H;)7!L(H;) de nes a group isomorphism P! Pic(T ).
Sketch of proof : One proves rst that the rst Chern class c (L(H;)) is equal to1
2 2E2 Alt ( ;Z) = H (T;Z). This can be done by using the di erential-geometric
de nition of the Chern classes ([ G-H], p. 141), or in terms of group cohomology
([M1],x2).
LetQ be the group of hermitian forms H on V such that Im(H) is integral on
. The projection p :P!Q is surjective, because a semi-character is determined
by its values on the elements of a basis of , and these values can be chosen
1arbitrarily. The kernel of p is the group of characters Hom( ;S ). Consider the
diagram
p
1Hom( ;S )0 P Q 0
o L L
c1o 2Pic (T ) Pic(T )0 H (T;Z)
2o 2where L () :=L(0;), and (H) = Im(H)2 Alt ( ;Z) =H (T;Z). The equality
c (L(H;)) =E =(H) implies that the diagram is commutative.1
Now we claim that is bijective onto Im(c ): indeed we have seen in section1
2 22.1 that a form E 2 Alt ( ;Z) H (T;Z) belongs to Im(c ) if and only if it= 1
1;1belongs to H , that is satis es E(ix;iy) =E(x;y) (Proposition 1.3). By section
2.3 this is equivalent to E = Im(H) for a hermitian form H2Q; moreover H is
uniquely determined by E , hence our assertion.
oFinally one proves using Hodge theory that any line bundle M in Pic (T )
admits a unique at unitary structure, that is, M =L(0;) for a unique character
o of . In other words L is bijective, hence L is bijective.
2.5. The theorem of the square. This section is devoted to an important
result, Theorem 2.6 below, which is actually an easy consequence of our description
of line bundles on T (we encourage the reader to have a look at the much more
elaborate proof in [M1],x6, valid over any algebraically closed eld).
0 0Lemma 2.5. Let a 2 V . We have t L(H;) = L(H; ) with () =(a)
() (E(;a).
Proof : In general, let L be a line bundle on T de ned by a system of multipliers
(e ) . Then (e (z +a)) is a system of multipliers, de ning a line bundle 2 2
0L ; the self-map (z;t)7! (z +a;t) of VC is equivariant w.r.t. the actions of
////////////e//8 ARNAUD BEAUVILLE
de ned by ( e (z +a)) on the source and (e (z)) on the target, so it induces an
0 isomorphism L! t L.
(a)
1[H(;z)+ H(; )]
2We apply this to the multiplier e (z) = a()e ; we nd
h(z+)H(;a)e (z +a) =e (z)e . Recall that we are free to multiply e (z) by for h(z)
H(a;z)some holomorphic invertible function h; taking h(z) = e , our multiplier
[H(;a) H(a; )] 2iE(;a)becomes e e =e e .
Theorem 2.6. Let L be a line bundle on T .
1) (Theorem of the square) The map
o 1
:T! Pic (T ) ; (a) =t L
LL L a
is a group homomorphism.
22) Let E2 Alt ( ;Z) be the rst Chern class of L. We have
? ?Ker = = ; with :=fz2V j E(z;)2Z for all 2 g:L
3) If E is non-degenerate, is surjective and has nite kernel.L
4) If E is unimodular, is a group isomorphism.L
Proof : By the Lemma, is the compositionL
o" L1 oT! Hom( ;S )! Pic (T ) ;
owhere "(a), for a = (a~)2 T , is the map 7! (E(;a~), and L is the isomor-
phism 7!L(0;) (Theorem 2.4). Therefore we can replace by " in the proof.L
Then 1) and 2) become obvious.
1Assume that E is non-degenerate. Let 2 Hom( ;S ). Since is a free
Z-module, we can nd a homomorphism u : !R such that () = (u()) for
each 2 . Extend u to aR-linear form V !R; since E is non-degenerate, there
exists a2V such that u(z) =E(z;a), hence "((a)) =. Thus " is surjective.
Let us denote by e : V ! Hom (V;R) the R-linear isomorphism associatedR
?to E . The dual := Hom ( ;Z) embeds naturally in Hom (V;R), and isZ R
1 ? ?by de nition e ( ); then e identi es with , so that the inclusion
corresponds to the map ! associated to E . This map has nite cokernel,j
and it is bijective if E is unimodular; this achieves the proof.
oRemark 2.7. We have seen in section 2.1 that Pic (T ) has a natural structure
of complex torus; it is not di cult to prove that the map is holomorphic. InL
particular, when E is unimodular, is an isomorphism of complex tori.L
0Corollary 2.8. Assume that c (L) is non-degenerate. Any line bundle L with1
0 c (L ) =c (L) is isomorphic to t L for some a in T .1 1 a
0 1 o 1Proof : L
L belongs to Pic (T ), hence is isomorphic to t L
L for somea
a in T by 3).
The following immediate consequence of 1) will be very useful:
eeCLASSICAL THETA FUNCTIONS AND THEIR GENERALIZATION 9
P
rCorollary 2.9. Let a ;:::;a in T with a = 0. Then t L
:::
t L =L .1 r i a a1 r
3. Polarizations
In this section we will consider a line bundle L = L(H;) on our complex
torus T such that the hermitian form H is positive de nite . We will rst look for
a concrete expression of the situation using an appropriate basis.
3.1. Frobenius lemma. The following easy result goes back to Frobenius:
Proposition 3.1. Let be a free nitely generated Z-module, and E : !Z
a skew-symmetric, non-degenerate form. There exists positive integers d ;:::;d1 g
with d jd j :::jd and a basis ( ;:::; ; ;:::; ) of such that the matrix1 2 g 1 g 1 g !
0 d
of E in this basis is , where d is the diagonal matrix with entries
d 0
(d ;:::;d ).1 g
As a consequence we see that the determinant of E is the square of the integer
d :::d , called the Pfa an of E and denoted Pf(E). The most important case1 g
for us will be when d = =d = 1, or equivalently det(E) = 1; in that case one1 g
says that E is unimodular, and that ( ;:::; ; ;:::; ) is a symplectic basis of1 g 1 g
.
Proof : Let d be the minimum of the numbers E(;) for ;
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