Complex Abelian Varieties and Theta Functions (Universitext)

Proof. We need to find a complex linear mapping P: V ~ HomlR(V, ill) which induces ,I)+1r/2

fl(l,1)

(a(l)e1rH(V,')+1r/2

H(I,I)) -1 =

e1rH (x,1)

where x in V is a lifting x. Thus B,( v) are constant in v and

R(k)(v) = Re (;iH(x, v)) - 1m (;iH(X, iV)) =

~ImH(x,v)+~lmH(x,v) =

E(x,v).

So R(k) is a IR-linear function of x. To check that it is a complex function we need to verify that E(ix,v) = -E(x,iv) but this follows because E(ix,iv) = E(x, v). 0 The proof shows that
8

Chapter 1. Complex Tori

The kernel of
x

Lemma 1.10. If H is non-degenerate, K(fi') is a finite group of order detL E. Proof. By the above we have K(fi') ~ {x E VIE(x,l) E 7l}/L. Thus the result follows from standard linear algebra. 0 Another result along the general lines of discussion of this section is the theorem of the cube.

Theorem 1.11. If fi' is an invertible sheaf on a complex torus X, a) the sheaf (71'1 +71'2 + 71'3)*fi' 0 (71'1 +7I'2)*fi'0- 1 0(7I'1 +7I'3)*fi'0- 1 0 (71'2 + 71'3)* fi'0- 1 071'; fi' 0 71' 2fi' 0 71';fi' on X X X X X is trivial. b) For any integer m we have an isomorphism m* fi' ~ fi'0 m(m+I)/2 0 (-1)* fi'0 m(m-I)/2 .

Proof. Let (a, H) be A.-H. data for fi'. Then A.-H. data for the sheaf in part a) are (a(ll + 12 + 13)a- I (l1 + 12)a- I (l1 + 13)a- I (l2 + 13)a(lt)a(l2)a(l3), H(ZI + Z2 + Z3, WI + W2 + W3) - H(ZI + Z2, WI + W2) - H(ZI + Z3, WI + W3) - H(Z2 + Z3, W2 +W3) +H(ZI' WI) +H(Z2, W2) +H(Z3, W3))' As H is bilinear the H-part of this data is O. Thus the a-part is a homomorphism L X Lx L ~ U(l), whose value is clearly one when restricted to the component subgroups. Therefore the a-part is 1 and, hence, a) is tJ:ue. The part b) can be formally deduced from a) or more directly by reproducing the above type of argument. I leave this detail to the reader. 0

Exercise 1. If (a,H) are A.-H. data on a complex torus X = V/L and H is nondegenerate, prove that for some point x of X, T; fi' is a symmetric sheaf. Exercise 2. Let f : X ~ Y be a homomorphism between two complex tori and fi' is an invertible sheaf on Y. Then we have a commutative diagram

xLy t/>j!e 1

Pico(X)

1t/>!e

L

Pico(y) .

Chapter 2. The Existence of Sections of Sheaves

§ 2.1 The Sections of Invertible Sheaves (Part I) Let (a,H) be A.-H. data for a complex torus V/L. Our objective is Theorem 2.1. The space r(v/ L,!l'( a, H)) of holomorphic sections of !l'( a, H) is non-zero if and only if both a) H is positive semi-definite and b) a is identically one on L n Ker H. The proof will be given in two sections. Here we will reduce to the most interesting case where H is positive definite.

Step 1. r( V / L, !l'( a, H)) = 0 ifthere exists v in V such that H ( v, v) < O. The best idea here is to appeal implicitly to the curvative property of the natural Hermitian metric in !l'( a, H). This metric is interesting for other reasons. To define it let f( v) = c- 1rH (v,tI). Then f is positive IR-analytic function on V and it satisfies the functional equation f(v

+ I) = e- 21rRe H(v,/)-1rH(/,/) f(v)

for alII and v in V. Let .2c where 0 < C = -H(v, v). Therefore k(A) = O. In particular
Step 2. r(v/ L, !l'(a, H))

= 0 if a

is not one on L n Ker H.

10

Chapter 2. The Existence of Sections of Sheaves

Let
Step 3. r(V/L, f£(a,H))

=1=

0 if H is positive definite.

This step is postponed because we want to give an explicit construction of all these sections and will prepare the notation for this in the next section. Exercise 1. Compute the curvature 2~i8810gf of our metric on f£(a, H). Show that it is an invariant differential form. FUrthermore this is the unique metric with this properly such that 111110 = 1. Exercise 2. Let X be a non-zero complex torus. Show that the Poincare sheaf on X x Pico has no non-zero sections.

§ 2.2

The Sections of Invertible Sheaves (Part II)

Let (a,H) be A.-H. data for a complex torus V/L where diml[: V = g. We will assume that H is positive definite. As usual E = ImH. We intend to write the sections of f£( a, H) using theta series. There is some necessary notation. Let A be a subgroup of the lattice L such that A = L n IR A, E is zero on A x A and rank A = g. The existence of such subgroups is easy. If A' satisfies the last two conditions, then A = L n IR . A' will work. To find A' choose linearly independent VI, ••• ,Vg in L inductively such that E( vfo Vj) = 0 for all j < k. The set A' = ZlVI + ... +Zlv g • Thus A is a direct summand of L. Let W = IR . A ~ IR ®71 A. Then E is zero on W x W. The first point is that the canonical mapping A ®71 C ~ V is an isomorphism because W n iW is zero as the intersection is a complex subspace on which E (and hence the positive definite form H) is zero. <

§ 2.2 The Sections ofInvertible Sheaves (Part 11)

11

Because E is zero on A x A the mapping a : A ~ U (1) is a homomorphism. Hence there is a complex linear transformation A on V such that A is real on W and a(a) = e 21ri -X(a) for all a in A. Furthermore as H is real on W X W there is a unique complex symmetric bilinear form S on V x V such that H( WI, W2) = S( WI, W2) for all WI and W2 in W x W. By complex linearity S = H on V x W. Also if W is in Wand v in V we have t;(H - S)(w, v) = E(w, v) because the left side is t;(H - S)(v,w) = t;(H - H) = -ImH(v,w) which is the right side. For v in V let v : V ~ C be the complex linear mapping such that v( w) = E(w, v) for all w in W. Thus t;(H - S)(w,v) = v(w) for all w in Wand hence by complex linearity for all w in V. Finally, let A~ = Hom71(A, Zl) which will be identified with a subgroup of Homa::(V, C). Let f( v) be a global section of !l'( a, H); i.e., f is a holomorphic fWlction on V such that f(v + I) = a(l)e1rIl(tI,IHtH(I,I) f(v) for all I in L. Consider g(v) = e-1- S (tI,tI)-21ri-X(tI) f(v). Then 9 is still holomorphic but now

g( v + I) = a(l)e -21ri-X(I) e 1r(H -S)(tI,IH1-(1I-S)(I,I) g( v) = a(l)e -21ri-X(IH1rii(l) e 21rii (tI) g( v) .

Thus by construction 9 is periodic with respect to A. Hence we may expand it in a Fourier series g( v) = EXEA ~ cxe21rix(tI) where the C x are complex constants. Our fWlctional equation gives

L

(xxe 21riX (I)) e 21rix (tI) =

xEA~

L

(cxa( I)e -21ri-X(I H

1rii(I») e 21ri (x+i)(tI) •

xEA~

Setting coefficients equal we get the equations

(*)

C

x

=

a(l)e- 21ri (-X(IHx(I)H1rii(I)C

x

_i

for alII in L and X in A~ .

Conversely once we show that the Fourier series of such Cx converges reasonably we may reverse the process to find all possible sections f. Let T( a, H, A, U) be the space of all complex valued functions Cx of X in U~ such that (*) holds. The main result is

Theorem 2.2. For any c in T( a, H, A, U) the series

tPu,-X(c) =

c1- S (tI,tlH21ri-X(1l) (

L

cxe21riX(V»)

xEU~

converges uniformly on compact subsets to a global section of !l'( a, H) and tPu,-x : T( a, H, A, U) ~ rev/ L, IR( a, H)) is an isomorphism.

12

Chapter 2. The Existence of Sections of Sheaves

Proof. (Actually the convergence is very fast but the notation is confusing). Let Ilxll be a nonn on Homv(V, C). We claim that

(t)

lex I :5 c-cllxll

2

for all X in A~ for some positive constant C .

First we will show that this implies convergence. Assume that v lies in a given compact subset of V. Then It/Ju,.x(c) I :5 E EXEU~ ICxl eDllxll for some constants E and D. Thus the series is majorized by C EXEU~ e-cllxIl2+Dllxll whose ratios compare with C Ex e- CIIxIl2 . This last series converges similarly to E(n;)EZn e-l>~ = [1; (E";EZ e-n~). Anyway the convergence is very fast. We will take time proving (t) because we want to develop the structure of T( a, H, A, U) so as to compute its dimension. As A is a direct summand of L we can find a complementary subgroup B with L = A ffi B. The first remark is that in the conditions (*) we need only check them with 1 in B. This follows because the multipliers a(a)e-2"-i(.x(a)+x(a»+"-ia~a are 1 by construction when a is in A. The conditions (*) mean that Cx is determined by the arbitrary constants cx. where x* are a complete set of cosets representatives of B in A~. Explicitly

(to

c

. = a(b')e -2,,-i(.x(h)+x(b»-,,-ib ~(b) c x.+b x.

for all bin B and all x*. Next we want to see that the number of x* (i.e. Coke: B ~ A~}) is finite. Write E as a matrix in tenns of the decomposition L = A ffi B. Thus E is given by [_OF

~]

where F and G are integral matrices. Here F represents

~ : B ~ A ~. Now det E = (det F)2 which is non-zero as E is nondegenerate. Hence Coke) is a finite group with (det E)I/2 number of elements. As lal = 1 and (*) is finite then (tt) will imply the estimate (t) if we can show that Imb~(b) is a negative definite fonn on B. This is easy. Just write b = m+ in where m and n are in W. If b -I then n -I (otherwise u' would be in A = WnL). Now 1m b(b) = Im(b(m) + ib(n)) = Im(E(m, b) +iE(n, b)) = E(n, b) = E(n,m) + E(n,in) = E(n,in) = ImH(n,in) = -H(n,n) which is negative as H is positive definite. Thus (t) is true as the series converge. 0

°

°

vVe get more infonnation from the proof. Theorem 2.3. dimv r(V/L, f£(a,H)) = VdetE. As this number is positive we have finished the proof of Theorem 2.1.

§ 2.3 Abelian Varieties and Divisors

13

Exercise 1. IT H is positive definite,

dimr(VIL, fi'(a,H))

= V#K(fi') =

V

deg4>!£(OI,II) .

Exercise 2. Let fi' be an invertible sheaf of the form fi'(a,H) on VIL such that fi' has a non-zero section. Show for positive integers m that dim r (VI L, fi'®m)

= (dimr(VI L, fi')) m(dim V-dim Ker 1I)

•

Exercise 3. Let f : X ~ Y be an isogeny and fi' be an invertible sheaf on Y given by A.-H. data with a positive definite Hermitian form. Then dim r( X, j* fi') = (dim r(Y, fi')) ( deg J) .

§ 2.3 Abelian Varieties and Divisors A complex torus VILis an abelian variety if there exists a positive definite Hermitian form H on V such that its imaginary part is integral on L x L. Such a pair (VI L, H) is called a polarized a helian variety. Lemma 2.4. Let X be a complex torus. There is quotient abelian variety Y of X such that the projection 7r : X ~ Y is a universal homomorph.i.~m from X into an abelian variety. Proof. Let X = VI L. Let ft' be the set of all positive semi-definite Hermitian forms on V with integral imaginary part. Let K = nHEJt' Ker H. For dimension reasons we may find a finite number HI,'" ,Hr of elements of ft' such that K = Ker Hi. Let H = HI + ... +Hr • Then by construction Ker H = K. Let Y = VI K I 11m L. Thus H gives a polarization of Y and hence Y is an abelian 0 quotient. A little thought should give the universal property of Y.

ni

Corollary 2.5. Let fi' be an invertible sheaf on X such that fi' has a non-zero section. Then fi' = 7r* J{ for an invertible sheaf J( on Y and 7r*:

r(Y,J()

~

r(X,fi')

is an isomorphism. Proof. Consult Step 2 of the proof of Theorem 2.1.

o

14

Chapter 2. The Existence of Sections of Sheaves

Next we will explain our results in tenns of divisors. Recall that a divisor on a complex manifold X is an element of the free abelian group on irreducible divisors which are the irreducible closed subvarieties of X of co dimension 1. A divisor D is effective if all its multiplicities are non-negative. We have a homomorphism {Divisor on X} ~ Pic(X) which sends D to the sheaf ~Jx(D). Two divisors DI and D2 are linearly equivalent (written DI '" D2) if and only if by definition ~Jx(Dl) ~ ~Jx(D2) if and only if there exists a non-zero meromorphic function f on X such that the divisor (f) of zeros and poles of f equals DI - D2. The complete linear system IDI of a divisor D consist::; of all effective divisors linearly equivalent to D. If X is compact there is a natural bijection between IDI and the projective space of lines in r (X, ex(D)). Let X be a complex torus and 7r ; X ~ Y be its abelianization. Corollary 2.6. The homomorphism 7r induces a bijection between a) {divisors on Y} and {divisors on X} which respects linear equivalence, and b) {meromorphic functions on Y} and {meromorphic functions on X}. Proof. This is a rcfonnulation of Corollary 2.5 in classical language. I leave it as an exercise for those who want to speak both languages. 0

Therefore if you are just interested in the geometry of divisors or the algebra of meromorphic functions on the complex torus you need only work with abelian varieties. A useful fact about divisors is the theorem of t he square which is a special case of Proposition 1.8. Proposition 2.7. For any divisor D on a complex torus X and point x and y of X, (D+x+y)+(D)",(D+x)+(D+y). Exercise 1. Show that any complex torus of dimension one is an abelian variety. In fact there is a canonical choice of a polarization. Exercise 2. Let II = (1,0), 12 = (0.1), 13• 14 be a basis for a lattice L(l3, 14) in (;2. For fixed 13 • 14 let H be a polarization of (;2/ L( 13,14)' Vary 13 and 14 in (;2 to 13 and 1 4 , Show that there is one non-trivial (;-analytic condition on 13• 14 such that there is a polarization jj on (;2/ L(13. 14) with imaginary part = the analytic continuation of the imaginary part of H.

§ 2.4 Projective Embeddings of Abelian Varieties

15

§ 2.4 Projective Embeddings of Abelian Varieties We will begin with two simple consequences of the theorem of the square. Let D be an effective divisor on a complex torus.

Lemma 2.8. If n is an integer

~

2 then the linear system InDI has no base

points. Proof. Let y be a given point of X. We want to find a divisor E in InDI which does not pass through y. Now y is contained in D + x if and only if x is contained in the codimension one subvariety y - D. Thus for general choices of Xl, ... ,xn-b Y is contained in (D + xd + (D + X2)+ .. . +(D + xn-d + (D - El:5:j:5: n -1 Xi) = E. By the theorem of the square 2.7 E is contained in InDI. 0

Lemma 2.9. The linear system

IDI

contains reduced divisors.

Proof. Let D = EnjDj where D, are the components. If nj > 1 replace njDj by a divisor Ej formed from Dj as in the last lemma. Thus D ~ E Ej = E. We want to choose the x's for Ej general so that E is reduced. This follows by the following maneuver. Given two non-empty effective divisors FI and F2 then the set of X such that FI + x = F2 is contained in the divisor F2 - II where II is any point of H. 0

Let (a,H) be A.-H. for the sheaf ex(D) on X. We begin the serious discussion with

IDI IE + x = E, for some non-zero x in X} of IDI is the union of a finite number of proper linear subspaces if H is positive definite.

Proposition 2.10. The subset {E E

Proof. Assume that E = x + E for some E in IDI. Thus T~xex(E) ~ ex(E). Hence x is contained in the finite group I«ex(D)). In particular x generates a finite subgroup S of X. Let 7r : X ~ XIS = Y be the quotient homomorphism. There is a divisor F OIl Y such that E = 7r- 1 F. We write X = VIL and Y = VIM. Let (a,H) be A.-H. data for the sheaf ey(F). As (al",H) is A.-H. data for ex(E) ~ ex(D), H and aiL are fixed. Clearly a is determined by the value a( I') where I' is a lifting of x. Furthermore nI' is contained in L for some integer n (= order x) and Q(l')n = a( nl'). Therefore there are only a finite number of choices of a. By construction the cone over {E = x + E} in IDI is union of the images of all r(Y, f£( a, H)) in rex, f£( aiL, H)). The remaining

16

Chapter 2. The Existence of Sections of Sheaves

point is that these finitely many subspaces are proper if x (see exercise on § 2.2).

:f.

0 as deg 7r > 1 0

Now we are in a position to prove the classical theorem of Lefschetz. Theorem 2.11. If H is positive definite and n InDI defines a projective embedding

~

3 then the linear system

Proof. By Lemma 2.8

:r

We have much control of the existence of abelian fWlctions now. Corollary 2.12. Let X be complex torus of dimension g. The following are equivalent:

§ 2.4 Projective Em beddings of Abelian Varieties

17

a) X is an abelian variety, b) there are g algebraically independent meromorphic functions on X, and c) X is complex projective variety. Proof. The Theorem 2.10 proves that a) implies c). By standard algebraic geometry c) implies b). In fact the transcendence degree of the field of algebraic functions on a projective variety equals its dimension. To prove that b) implies a). Let 7r : X ~ Y be the abclianization of X. By Corollary 2.6, X and Y has the same meromorphic functions but by the above we know that the transcendence degree of the function field on Y = dim Y. This equals dim X if 0 and only if X = Y. Thus a) is true if b) is true.

Exercise 1. Show that Lemma 2.7 and Theorem 2.10 are sharp when X has dimension one.

Chapter 3. The Cohomology of Complex Tori

§ 3.1

The Cohomology of a Real Torus

Let L be a lattice in a real vector space W. Then the quotient ltV/L is a real torus. Using a basis I!, ... ,12 of L we have an isomorphism

rt

J: Hi(W/L,Zl)

x HheRharn(W/L, C)

~ C.

ily DeRham's theorem the pairing identifies HbeRham(W/ L, C) with

Hom71(H i (W/L,Zl),C) ~ Hi(W/L,C). The integral cohomology Hi(W/ L, Zl) is identified with the subset of classes in HhcRham(W/L, C) which have integral periods. Consider the invariant (Wlder translation) differential i-form w on V/L. Then w is determined by its value at zero which is an arbitrary element of Ai(Homm.(W, C)). Using coordinates one can easily check the truth of Lemma 3.1. a) Any invariant form is closed and each DeRham class contains a unique invariant form. b) We have a natural isomorphism Ai(Homm.(W, C)) ~ HhcRharn(W/ L, C) such that Hi(W/ L, Zl) is identified with Ai(L~) where

20

Chapter 3. The Cohomology of Complex Tori

Next we consider the Riemannian geometry of WIL. Assume that we are given an invmiant Riemannian metric on WIL or what is the same as the Euclidean metric on W. We have the usual * -operator on differential forms. As the metric is invariant * of an invariant form is invariant. Recall that a form w is called coclosed if d*w = O. As WIL is compact, a form is harmonic if it is closed and coclosed. Thus the invariant forms are harmonic and conversely! Lemma 3.2. On the real torus WI L the harmonic forms are exactly the in· variant forms. Proof. By standard Riemannian geometry on compact manifolds any DeRahm class contains a unique harmonic form. 0 Exercise 1. Let w be a one-form on WI L. Then w is invariant if and only if p.*w = 7l"rW + 7l"2W where p. : WIL x WIL ~ WIL in the group law.

§ 3.2 A

Complex Torus as a Kahler Manifold

A Riemannian metric on a complex manifold X i" Hermitian if it has the form H( v, v) where H is a Hermitian form on the tangent space of X with respect to its given complex structure. The metric is called Kahler if the 2-form (Kahler form) ImH(v,w) is closed. Let VI L be a complex torus. Let H be a Hermitian form on the tangent space V. Then H extends uniquely to an invariant Hermitian metric on VIL. As the Kiihler form is automatically invariant it is closed. Thus we have a Kahler metric on the compact complex manifold VI L and we may use the properties of the Hodge decomposition. We need to write a given invariant (harmonic) differential form into its (p, q)-components. This is very easy. As a C-valued IR-linear function on V can be written uniquely in the form 11 + 12 where the l's are complex linear we have an isomorphism of C-vector spaces

HomlR(V, C) Therefore Ai (HomlR(V, C))

= Homa::(V, C) ffi Homv(V, C)

= EB p+9 =i AP (Homa::(V, C)) 0

.

Aq (Homa::(V, C)).

In terms of cohomology this is the Hodge decomposition HheRharn(VI L, C) = EBi=C-tq Hp,q. As this is independent of H we may forget it. As Hi(VI L, (f)v/ d = H ,l Hodge theory gives

§ 3.3 The Proof of the Appel-Humbert Theorem

21

Theorem 3.3. Hi(V/ L, lfJ v / d ~ Ai (Homc(V, C)) where we a.9sociate the Dolbeault cohomology class to the appropriate invariant (0, i)-form. We will need simple calculation. Let H( v, w) be a Hermitian fonn on V. Then ~88H(v, v) is an invariant (1, l)-fonn on V and hence we may regard it as one on V / L. We simply want to compute the element of A2 (HomlR (v, C)) corresponding to it. The result is Lemma 3.4. a) t88H( v, v) is the invariant form corresponding to the skewsymmetric form ImH(v,w). b) A real skew form E( e, w) on V has type (1,1) if and only if E( iv, iw) = E(v,w). c) All rcal invariant (1,1) forms on V/L may be written uniquely in the form t88H'(v,v) where H' is a Hermitian form on V. Proof. b) is easy. It says that the space of (1, l)-fonn is the I-eigenspace for the operator E(v,w) ~ E(iv,iw). The fonn of type (0,2) or (2,0) are -1eigenvector for the operator. Thus the statement is clear. For a) we compute in coordinates (Zb'" ,Zg) on V. We may assume that H is diagonal, i.e. H(z, z) = Eakzkzk where the ak's are real. The ImH(z,w) = Eaj(Yjuj - XjV,) and 88H(z, z) = Eajdzj 1\ dZj = i Eaj(dYj 1\ dXj - dXj 1\ dYj). Thus the result is clear. The point c) is a combination of a) and b) by Proposition 1.2. 0

§ 3.3

The Proof of the Appel-Humbert Theorem

Let X = V / L be a complex torus. Let A.-H. be the group of all Appel-Humbert data and Pic(X) the group of isomorphism classes of invertible sheaves on X. Then we have a homomorphism p : A.-H ~ Pic(X) which sends (a, H) to the isomorphism class of !l'( a, H). The Appel-Humbert theorem says that p is an isomorphism. Step 1. P is injective.

If (a, H) is in the kernel, !l'( a, H) is trivial. Thus !l'( a, H) and its inverse !l'(a-1,-H) have a non-zero section. Thus by Theorem 2.1 Hand -H are both positive semi-definite. Therefore H = 0 and L n Ker H = L. Hence by the theorem again a is identically one. This completes Step 1. We first recall some generalities. The Picard group Pic(X) is isomorphic to Hl(X, lfJx ) where lfJx is the sheaf of holomorphic units. We have the exact sequence of sheaves 0 ~ 71. ~ lfJx ~ lfJx ~ 0 which gives an exact sequence of groups

22

Chapter 3. The Cohomology of Complex Tori

If f£ is an invertible sheaf 6[f£] = cl(f£) is the (first) Chern class of f£ and it determines f£ topologically. As H2(X, Z) C H2(X, C) we have computed cl(f£) differentiably as below. Choose a metric on f£. Then Cl (f£) is represented by 2~; of the curvature • of the metric, which is a 2-form. Explicitly if A is the square length of a local holomorphic section of f£ then. = 881ogA. Therefore a Chern class is an integral (1, I)-form. Step 2. Im6 = Im6p.

vVe will show that any integral (1, I)-form w is cohomologous to the first Chern class of f£( a, H) for some (a, H) in A.-H. Such a two-form w is cohomologous to the invariant differential of a skew-symmetric form E on V such that E(iv,iw) = E(v,w) by Lemma 3.4b). Let H be the Hermitian form on V with imaginary part E. As w is integral, E is integral on L x L. Thus by Lemma 1.6 there exist a such that (a, H) is in A.-H. This step will be finished by applying. Lemma 3.5. If f£ = f£( a, H), then 2~i of the curvature of the canonical metric on f£ is the invariant (1,1) form corresponding to E = 1m H. Proof. The square length of the Section 1 of f£ is e- 1rH (v,,,). Thus the Chern class 6[f£] is 2~i881og (c- 1rH (v,,,») t88H(v, v) which is essentially E by Lemma 3.4 a). 0

Vie will be done if we can complete Step 3. Ker 6 C 1m p.

By the sequence (*)Ker6 = Image i. Let fJ be an element of H1(X,e x ). Then by Theorem 3.3, fJ is the cohomology class of an invariant (0, 1)-form w which we may identify with a anti-complex linear function k on V. The general fact is that i(fJ) is represented by the flat sheaf with multipliers exp( k(l)) for I in L. (This is proven as follows. Note that fJ is the image of the cohomology class'Y in Hl(X, C) represented by k. Then i(fJ) is the imag~ of the element exp('Y) of Hl(X, C*).) We have seen in the proof of Lemma 1:7 that any flat sheaf is isomorphic to one of the form f£(a, 0) where (a,O) is in A.-H. Thus i(fJ) is contained in the image of p and we are finished.

§ 3.4 A Vanishing Theorem for the Cohomology of Invertible Sheaves

23

Remark. It is a general theorem about compact Kahler manifolds that the image of Hl(X, l!'J in H2(X, C) is exactly the integral (1,1) classes and the image of Hl(X,l!'JX ) in Hl(X,l!'JX) is represented by unitary flat bundles.

x)

§ 3.4 A Vanishing Theorem for the Cohomology of Invertible Sheaves Let X = V/L be a complex torus. Let (a,H) be A.-H. data. Let AO be the space of all Coo sections of fi'( 0:, H). Thus A ° consists of all Coo -functions f on V such that f( v + 1) = alc'll"II(v,l)+fII(I,I) f( v) for alII in L. We have a inner product (f,g) = e-'II"II(v,tJ) f(v)g(v)dv where dv is the invaricmt measure on

Ix

V. We will consider some differential operators on A 0 • Let (Zl, ••• ,Zg) be coordinates on V. Then we may assume that H is diagonal; i.e., H (z, w) = El:5:,:5:9 h' ZjWj for some real numbers hI, ... ,M. The first operator is just (8~., ). As the multipliers of fi'( a, H) are complex analytic in v, the product

formula shows that 8~z, maps A ° into A 0 • We have adjoint operators.

(8t

Lemma 3.6. a) The operator b)

((8~)*f,g)

=

(f'(8~)g)

r = - 8~j + 7rhiz and

((8~)f,g)

j

=

maps AO into AO.

(f'(8~,)*g)·

_ C) (...!L)(...L)* 8%. 8z· - (...L)*(...L) 8%· 8%. - trh j . J

J

J

J

Proof. For a) let f be an clement of A 0 • Then

Thus a) is true. To prove b) it will be enough by Stokes' theorem to show that

24

Chapter 3. The Cohomology of Complex Tori

and

where the inner product is the pointwise version. Now

~(J,g) = ~ (e- 1rH (Z,Z) J(Z)g(Z)) f)Zj f)Zj = _7rh j zje- 1rH (z,z) J(z)g(z) + e- 1rH (z,z) (f)~j )J(Z)g(Z)

+ e- 1rH (z,z) J(Z) ~

f)Zj

= - ((

9

f)~j ) * J,g) + (J, (f)~j )g)

This proves the first equation. The second equation is conjugate to the first. This proves b). For c) just apply the operators to a function. 0 Next we let A * = EBo
As with (a~. )

) the operator 8 on differential forms

sends An to An+l and

2

8 =-- O. We want an expression for the adjoint (8)* of 8. Consider the linear operator defined by (8)*(hdz I ) = El:s;d:5:#I( _l)dH ( a:' )* hdzI-{id}. d Lemma 3.7. a) (8*) takes An into An-I. b) (8)* is adjoint to (8). c) ((8)*8 + 8(8)* )(/IdZI)

= (E1 9

:5:g( a~, )*( a~,)

+ 7r EjEI hi)hdzI

.

Proof. a) is clear from Lemma 3.6. Also b) follows from the lemma and the explicit formula for 8 given by 8(hdzI) = El
at

The cohomology Hi(X, !l'(a, H)) on X = Cq /L which we may consider as a Kiihler manifold is isomorphic to Ker Ll : Ai ~ Ai where Ll = (8)*8 + 8(8)* is the Laplacian. Thus by Lemma 3.7 c) as Ll respects the decomposition of forms KerLl = EBIEInd HI where #I=i

§ 3.5 The Final Determination of the Cohomology of an Invertible Sheaf

25

Lemma 3.8. Let N = {I ~ j ~ glh i < a} and SN = {I ~ j ~ glh i ~ a}. HI = a if either I ~ SN or I ~ N. Proof. If we apply Serre duality we sec that HI(fi'(a,H)) is dual to H[I, ... ,gl-I (fi'( a-I, -H)). Thus the first alternative fer fi'( a-I, -H) implies the second alternative for fi'( a, H). Thus we need to see that if we have k in I such that

hk

> a then HI = a.

Let w

=

jdzI be an element of HI

= Ker D. Then

Thus 'L. iEl hi < a if w =1= a. Thus we are done if hk is bigger than 'L.iEl-k hi. VIle may make a diagonal change of coordinates such that this is true and not change the cohomology upto isomorphism. Thus HI = a. 0

Corollary 3.8. Hi(X,fi') = EBN~/~S'N HI . #1=;

§ 3.5 The Final Determination of the Cohomology of an Invertible Sheaf Let (a, H) be A.-H. data for an invertible sheaf fi' on a complex torus X = V / L. Let z = dim Ker H. Let Y = V/Ker(H) + L = (V/KerH)/L'. Then H is induced from a Hermitian definite form H' on V / Ker H. Let n be the number of negati ve eigenvalues of H. Let KO (fi') = Ker H / L n Ker H as usual. Then we have our objective

Theorem 3.9. a) Hi(X, fi') = a if i < n or i > n + z. b) If a ~ i ~ z, Hn+i(x, fi') ~ Hn(x, fi') 0 Hi (KO(fi') , 19K O(!C»). c) If alLnKer H ¢. 1, then Hn(x, fi') = a and, otherwise, dim Hn(x, fi') = JdetL' E' where E' = .~imH'. Proof. Part a) follows directly from Corollary 3.8. We may choose our coordinates such that hi = a if 1 ~ i ~ z and hi < a if z + 1 ~ i ~ z + n. To

26

Chapter 3. The Cohomology of Complex Tori

prove b) we will show that for any J C [1, ... ,zl multiplication by dZ J gives an isomorphism H[Z+l, .. ,z+nj with HJU[z+l, ... ,z+nj' This will prove b) by Corollary 3.8 and Theorem 3.3. On the other hand the mapping is an isomorphism by the differential equation of the HI as hi = 0 if 1 ::; i ::; z. For c) we introduce a new complex structure on the real vector space V = (;g. In the new complex structure Zj, ... ,Zz, Zz+l, . .. ,zz+n, Zz+n+l' ... ,Zg are the complex coordinates. Let V be this new complex space. Let H(z, w) be the Hermitian fonn EZ+l< 'z+n hi ZjWj.~Then H ~and _1_ ~ H have the same imaginary part. Now (a, H) is A.-H. data for V / Land H is semi-positive. We claim

If we prove this claim then part c) will follow from Theorem 2.1, its proof and Theorem 2.3 where we determined the sections of sheaves. Now for the claim let N = [z+l, ... ,z+nl. Thus Hn(x,fi') = HN and its elements correspond to forms W = fdzN in AOdz such that Llw = O. By the usual reasoning in Kiihler geometry this differential equation is equivalent to 8w = (8)*w = 0 ¢:} =0

H. )

if j

rt Nand (8~')* f )

= 0 (or, rather, - 8~')

f + 7rh f j

= 0) if j EN.

The isomorphism Hn(x,fi') ~ HO(V /L,fi'(a,H)) will send W = f dZN to g( z) = exp( -7r EiEN h j Zj Zj )f( z). The differential equations for 9 are = 0 if j

rt N

and

*)

If;

= 0 if j EN; i.e., 9 is holomorphic on

V.

It only remains to

check that f is in AO if and only if 9 is in AO(a,H). This is routine and we will do it one way. Assume that f is in AO. Then g(z

+ 1) =

exp ( - 7r

L hizjzj -7r( L hj(z;lj + zili + liTi)) )al iEN

jEN

x exp(7rH(z, 1)

7r

+ 2H(l, l))f(z)

= a, exp (7r H( Z, 1)

+ iH(l, 1) )g( z)

.

So 9 is in AO(a,H).

o

§ 3.6 Examples Let fi' be an invertible sheaf on a complex torus X = V / L. Theorem 3.10 (Riemann-Roch). The Euler characteristic x(fi') of fi' E(-l)idimHi(X,fi') is the intersection number Cl(g~)9 where 9 = dimX.

Proof. Let fi' = fi'(a,H) be some Appel-Humbert data. Then Cl(fi') is the invariant two-form on X corresponding to the skew-symmetric fonn 1m H = E.

§ 3.6 Examples

~~'

27

~,,{ = gfcl(f£)I\(g 1 times)l\cl(f£).

Then by linear algebra ±vuetL E~.f£ll\ ... l\d':&2g Thus H is non-degenerate if and only if

C!

(~). 9

=I o.

Assume that H is degenerate we need to see that X(f£) = O. This case follows from Theorem 3.9a) and b). Because E(-l)iH i (KO(f£),l!'JKO(.2'») = 0 as the cohomology of the structure sheaf is an exterior algebra. If H is nondegenerate then z = 0 and Hi(X,f£) is non-zero when i = n and its dimension is ±JdetLE. Thus we need only check that ± = (_l)n. This is a question in linear alg;ebra which we don't do. (Hint: if H' is a pOf>itive definite form on V when (ImH,)g is positive). 0 A frequently used special case is Corollary 3.11. If D is an ample divisor on an abelian variety X then a) dimT(X,l!'Jx(D)) = the intersection number ~Dg and g. b) Hi(X,l!'Jx(D)) = 0 ifi

> O.

Another special case of Theorem 3.9 is Corollary 3.12. If f£ is in Pico(X) but f£ =1= l!'Jx, then Hi(X,f£) all i.

=0

for

Let ~ be the Poincare sheaf on X x X~ where X~ is dual complex torus. Corollary 3.13. The only non-zero cohomology group of ~ is Hg(X x X~,~) which has dimension one. Proof. We left this as an exercise (Hint: prove that X(~) = ±l and that the tangent space of X(dimg) is an isotropic subspace of H(~)). 0

Next we want to make some applications to families of cohomology groups. Let f : X ~ S be a smooth proper morphism of connected analytic spaces. Assume that the fibers X. = f-l(S) are abelian varieties. Let f£ be an invertible sheaf of X. Theorem 3.14. If for all s in S the sheaf f£lx. is ample on X., then a) f*f£ is a locally free sheaf on S and for each s, (f*f£). ~ T(X., f£IX.) is an isomorphism. b) Rif*f£ = 0 ifi

> O.

28

Chapter 3. The Cohomology of Complex Tori

Proof. Follows from Corollary 3.11 by proper flat base extension. (Actually a) can be proven directly for using the Fourier expansions of sections of !l' along the fibers.) 0

We will need to know one calculation of higher direct images for the Poincare sheaf ~. Theorem 3.15. R g 7r x- o (~) is the one-dimensional sky-scraper sheaf situated at the zero point 0 of X~ and the other higher direct images are zero.

Proof. By proper flat base extension for 7r x~ : X X X~ ~ X~ the Corollary 3.12 implies that support (R i 7r x- o (~)) C {O} for all i. Hence the Leray spectral sequence gives an isomorphism

Thus this result follows from 3.13.

o

Chapter 4. Groups Acting on Complete Linear Systems

§ 4.1 Geometric Background Let !l' be a very ample invertible sheaf on an abelian variety X. Then we have a projective embedding of X in lP n by the complete linear system IDI of sections of !l'.

Lemma 4.1. Translation Tx : X ~ X by a point x of X extends to a projective transformation of IP n if and only if x is contained in the finite group K(!l') = {x E XIT;!l' ~ !l'}.

Proof. We have already seen in Lemma 1.10 that K(!l') is finite. Assume that Tx extends to a projective transformation. Then for any E in IDI the divisor E - x is linearly equivalent to E. As!l' ~ ~Jx(E) and T;!l' ~ 19x (E - x) we must have x in K(!l'). Conversely if a: T;!l' ~ !l'is an isomorphism, then the global effect of a defines an isomorphism IP n ~ IP n which extends the action of Tx on X. 0 As the above extension is clearly unique because X spans lPn, we have a projective representation of the finite abelian group K(!l') on IPn. It will shortly turn out that IP n is a very simple irreducible representation of K(!l') but the purpose of this section is to lay the groundwork for this presentation. Let p: G ~ PGL(n) be a projective representation of a group G. Then we have an exact commutative diagram

1

-----+

C*

-----+

1

-----+

C*

-----+

H

G

!p'

!p

GL(n + 1) ~ PGL(n)

-----+

0

-----+

1

where H is the fiber product of a and p. Thus our projective representation p is determined by the ordinary representation p of the central extension H of G by C*.

30

Chapter 4. Groups Acting on Complete Linear Systems

Now let ft' be an invertible sheaf on X, we want to construct the natural central extension H(ft') of K(ft') by C* together with a representation of H(ft') on T(X, ft') which will give above projective representation of K(ft') on ]pn when ft' is very ample. This group H(ft') is called the theta (or Heisenberg) group of ft'. By definition an clement of H(ft') is a pair (x, a) where x is a point of K(ft') and a is an isomorphism a : T;ft' ~ ft'. The product (y, (3) * (x, a) =

(x

+ y, (30 T;a)

where (30 T;( a) is the composition T;+yft'

= T;(T;ft')

TO(<»

~

T; ft' ~ ft'. One checks that H(ft') is a group with identiy (0,1). Furthermore C* is a subgroup of H(ft') if we identify a complex number k with (0, k) and we have a central extension 1 ~ C* ~ H(ft') ~ K(ft') ~ O. To get the corresponding representation on T(X, ft') for (x, a) in H(ft') and a section a of ft', define (x, a)· a to be a(T;( a)). From the definition this gives a representation of H(ft') in which C* acts by multiplication. To study the non-commutativity of H(ft'), we introduce the form e.!l' : K(ft') x K(ft') ~ C* which is defined by the commutators in H(ft'). Explicitly e.!l'(x, y) = (x, a)(y, (3)(x, a)-ley, (3)-1 where (x, a) and (y, (3) are any elements of H(ft') lying over x and y in K(ft'). Next we compute the form e.!l'. Let (a,H) be A.-H. data for ft' and E = ImH where X = V/L as usual. Recall that K(ft') = L.l./L where L.l. = {v E VIE(v,l) E 71. for alil in L}. Thus if m and n are clements of L.l., we want Lemma 4.2. e.!l'(m,n) = e- 21riE(m,n).

Proof. We will first describe the transformation (m, A) in H(ft') lifting the element in in K(ft'). It is convenient to think of the T;' : homomorphism A' : ft' ~ ft' associated to A, which is gotten by A 0 Tfi.. Recall that ft' is the sheaf which assigns to each L invariant open subset U of V, the holomorphic funtions f on U such that feu + l) = a,e 1rH (n,Il+{H(I,I) feu). We claim that A'(J)(u) = e- 1rH (u,m)-t H (m,m) feu + m) is such an operation. The point is that A' takes ft' into ft' and is a T,';.-homomorphism almost by definition. To show that A' takes ft' into ft' note that if f is a section of ft' over u, then (after some calculations) A'(f)(u + l) = e21riE(m,l)ale1rIl(u,Il+{-Il(I,I)A'(f)(u). Hence as E(m, l) is always an integer A'(J) is a section of ft' over U - m. Next let B' be an operator with n instead ofm. We need to find the constant

A = e.!l'(m,n) such that A' . B' = AB'· A'. Now A' . B'(J)(u) (terms symmetric in m and n). Therefore A = e- 21riE(m,n).

= e- 1rH (m,n)

Corollary 4.3. If ft' is non-degenerate then the center of H(ft') is C*.

0

§ 4.2 Representations of the Theta Group

31

Proof. Given a non-zero element in of K(!l') we need to find another element =1= 1. In other words such that E( m, n) is not an

n of K(!l') such that e.!l'( in, n)

integer. By assumption we know that the form E is non-degenerate. We need to see that if E(m, n) is an integer for all n in MLihen is in L. This is easy. Let I!, ... ,hg be a basis for L. Then the dual basis is a basis for M. Hence M is contained in the span of the original basis (double dual basis) which is L. 0

itfl

Exercise 1. Define a natural structure of complex analytic group on H (!l') such that the sequence 1

--+

C*

--+

H(!l')

--+

K(!l')

--+

0

is an exact sequence of complex analytic group.

§ 4.2

Representations of the Theta Group

The theory of projective representation of finite abelian groups is an exercise for a college student. We will present a special case where we can be very explicit. Let H(!l') be the theta group for a nondegenerate sheaf !l' on an abelian variety. Let 7r : H(!l') ~ K(!l') be the projection. Take a subgroup A(!l') of K(!l') which is maximal with respect to the requirement that 7r- 1 A(!l') is abelian. Then the form e.!l' defines a homomorphism

l: K(!l')

--+

Hom71 (A(!l'), C*)

by l(k)(a) = e.!l'( a, k). The kernel oU is exactly A(!l') by maximality of A(!l'). By Corollary 4.3 the similar mapping l' : K(!l') ~ Hom71(K(!l'), C*) is injective. Hence it is an isomorphism because the dual group has the same order. Therefore l is a surjection because Hom71(K(!l'), C*) ~ Hom71(A(!l'), C*) is surjective. In summary we have an exact sequence

0--+ A(!l')

--+

K(!l')

--+

Homil (A(!l'), C*)

--+

0

and hence #A(!l')2 = #K(!l'). We want to consider an irreducible representation V of H(!l') on which C* acts by multiplication. As 7r- 1 A(!l') abelian we may find a non-zero eigenvector Vx with eigenvalue a character X of 7r- 1 A(!l') which is good in sense that it is one on C. Two such good characters defer by multiplication 7r*'of a character of A(!l'). Now let h be an element of H(!l'). So h . Vx is also an eigenvector with eigenvalue X7r*l(7rh) because ahv x = e.!l'( a, h)h. av x = X( a)e.!l'(7ra, 7rh)h. vx . Therefore X could be an arbitrary good character of 7r- 1 A(!l'). As the

32

Chapter 4. Groups Acting on Complete Linear Systems

line spanned by hv x depends only on the coset of kin H(!l')/A(!l'), we have EBkEIl(.2')/ A(.2') Ck . Vx is a H( !l')-invariant subspace of V. By reducibility we have V = EBkEH(.2')/".-lA(.2') Ck· v x ' Clearly the action V is simply determined by our implicit choice of coset representatives but nothing else because if I . k = k'a where k and k' are coset representatives and a is in 71"-1 A(!l'), then l(k· v x ) = x(a). k' . v x ' Theorem 4.4. The theta group H(!l') has a unique upto isomorphism representation T on which C* acts by multiplication. Furthermore dim T =

J#K(!l'). Proof. We have seen the uniqueness. Conversely if one defines V as above, one need only know that there is a good character X of 71"-1 A : i.e. a splitting 1

---+

C* ~

71"-1

A(!l')

---+

A(!l')

of this sequence of abelian groups (as an extension of a finite group by a divisible group splits!). For the dimension count note that dim V = #H(!l')/7I"-1 A(!l') =

#K(!l')/A(!l') = J#K(!l').

0

Theorem 4.5. Let i be the index of!l'. Then the natural representation of

H(!l') on Hi(X,!l') is isomorphic to T. Proof. By Theorem 3.9 dimHi(X,!l') = detLE = JK(!l') by Lemma 1.10. The natural action of (x,a) in H(!l') on Hi(X,!l') is (x,a). a = ao T;(a) and so C* acts by multiplication. Therefore by Theorem 4.4 this theorem is true. 0 In the rest of this section we will describe the standard form of the representation T. First of all having a good character X of 71"-1 A(!l') is the same as having a subgroup A'(!l') of 71"-1 A(!l') (= Ker X) such that 71" induces an isomorphism A'(!l') ~ A(!l'). Such a subgroup is called a level subgroup. Now assume that we have another subgroup B(!l') with the same properties as A(!l') and lifting A'(!l') and B'(!l') of A(!l') and B(!l') and K(!l') = A(!l')ffiB(!l'). The choice of such subgroups A'(!l') and B'(!l') is called a decomposition of

H(!l'). We can be very explicit about the theta group H(!l'). Any of its elements can be written u,niquely as A . a . b where A E C,' a E A' (!l') and b E B' (!l'). Thus H(!l') ~ C X A'(!l') X B'(!l') ~ C X A(!l') X B(!l') as set. The product in H(!l') is given by

(A,a,b)'(A',a',b') =(AA'e.2'(b,a'),a+a', b+b') .

§ 4.3 The Hermitian Structure on r(X,fi')

33

We can represent T as functions on B(fi'). The group action is given by «A,a, b). f) (c) = Ae~(a,c)-l fCc

+ b)

.

It is trivial to check that this is such an irreducible representation. Call this the standard method for T. Lastly we need in this situation B(fi') is identified with the character group of A(fi') by the form e~. The analytic theory of theta functions involve the construction of an explicit isomorphism of H(fi')-modules p~ : T ~ T(X,fi')

where fi' is an ample sheaf on an abelian variety. The algebraic theory says that p~ is determined upto constant multiple (apply Schur's lemma to these irreducible representation). The analytic theory allows a determination of this multiple. Next we will note a lemma of Mumford which uses the irreducibility.

Lemma 4.6. Let fi' ~ Y 0 Jt be an am.ple sheaf on an abelian variety such that Y and Jt have non-zero sections. Then

T(X,fi') = ET(X,Y')' T(X,Jt') where fi' = Y' 0 Jt' and Y' (I) is topologically equivalent to Y (Jt). Proof. The left side of the inequality is a non-zero subspace. By Theorem 4.5 it will suffice to show that it is invariant under the action of H(fi') because T(X, fi') is irreducible. Let a : T; fi' ~ fi' be an isomorphism in H(fi'). By definition a(T(X,Y')T(X,Jt')) = a(T(X, T;Y')).T(X, T*Jt') = T(X,Y").T(X,Jt") where yll = T;.¥' and I ' = T;I. As yll 0Jt" ~ fi' and .¥" (Jt") is equivalent to Y (Jt), we are done. 0

§ 4.3 The Hermitian Structure on r(X,fi') Let fi' be an invertible sheaf on an abelian variety X. Let X = V / Land fi' = fi'( a, H) for some A.-H. data (a, H). In the proof of Theorem 2.1 we have introduced a canonical Hermitian inner product in fi'( a, H) by defining

(a, T)x = e- 1rH (x,x)a(x)T(x) where x is a point of V over a point x in X and a and T arc sections of fi'( a, H) defined around x. Thus fi'( a, H) has a natural Hermitian invertible sheaf structure. Such sheaves have a remarkable property.

34

Chapter 4. Groups Acting on Complete Linear Systems

Lemma 4.7. Let a and fJ be two Hermitian isomorphisms !l'l ::::::t!l'2 between two invertible sheaves with Hermitian structure over a connected complex manifold Y. Then a and fJ differ by multiplication by a unitary complex number. Proof. Consider 'Y = fJ-1a. It is a Hennitian endomorphism of !l'l which is nowhere zero. Let (7 be a nowhere vanishing local section of !l'l. Then 'Y( (7) has the same property and the same length as (7. Therefore the ratio 'Y( (7) / (7 is a holomorphic function which always has absolute value 1. Hence it is a constant u (locally) where lui = 1. Thus u = fJ-1a or rather ufJ = a as Y is connected. 0

This gives us a compact fonn Hc(!l') of the theta group H(!l'). The element (x, a) of H(!l') are in Hc(!l') if the isomorphism a : T;!l' ~!l' is Hermitian when we get T;!l' the Hennitian structure gotten by pulling back the natural one on !l'. One can easily check that Hc(!l') is a subgroup of H(!l'). Clearly we have a complex 1

--t

U(l) ---+ Hc(!l') ---+ K(!l') ---+ 0 .

This is exact on the left by Lemma 4.6. To check exactness on the right we need to know that for any x in K(!l') , we can find (x, a) with a Hennitian. We will do this explicitly by noting the transfonnation A' of Lemma 4.2 as Hennitian by verifying of the definitions. If !l' is ample then we have a Hermitian inner product of T(X,!l') given by

((7, T) =

J

(7(v)f(v)f(v)dv

V/L

where the invariant measure dv on V/L is normalized such that IV/I, dv = 1 where f(v) = e- 1rH (tJ,tJ) is the metric in !l'(a, H) introduced in the proof of Theorem 2.1. By definition the action of the compact theta group Hc(!l') preserves these inner products on T( X, !l'). Let A'(!l') and B'(!l') be a decomposition of H(!l'). The decomposition is unitary if A'(!l') and B'(!l') are contained in Hc(!l'). In the last section we have introduced an identification of the representation T with the fWlctions on B'(!l') ~ B(!l') in K(!l'). We also may introduce a standard Hermitian structure on T such that the action of Hc(!l') is unitary. The inner product (f, g) of two functions f and 9 on B(!l') is EbEB(~) f(b)g(a). Thus the basis of delta functions {Ob} is a unitary basis of T. It is automatic from the definition that this inner product is Hc(!l') = U(l) X A'(!l') X B'(!l') invariant. In the above situation we may measure the length of an analytic theory p~ : C(B(!l')) ~ T(X,!l') of the theta functions. As Po< is H(!l')-invariant.

§ 4.4 The Isogeny Theorem up to a Constant

35

as we are dealing with irreducible representation PIe multiplies the length of vectors by a constant. Thus if we define IIp!£1I = V(P!£OO,P!£Oo). Hence for any functions f and 9 on B(ft') we have (p!£f,P!£g) = IIp!£11 2 (f,g).

§ 4.4 The Isogeny Theorem up to a Constant Let f : X ~ Y be an isogeny of abelian varieties. Let M be an ample invertible sheaf on Y. Then ft' = !* Jt is ample on X. We intend to describe the pull-back mapping !* : T(Y, Jt) ~ T( X, ft') using a theory of theta functions of Jt and ft'. If we have a compatible (to be defined) decompositions (A'(ft'), B'(ft')) of H(ft') and A'(Jt), B'(Jt)) of H(Jt). We will state the objective first. Assume that we have compatibility and isomorphisms PIe : ClB(ft')] ~ T(X,ft') and PJt : C[B(Jt)] ~ T(Y,Jt) which are H(ft') and H(Jt)-module homomorphisms.

Theorem 4.7. p'f/ !*PJt(Ob)

=

constant

O:b'EB(!£) Ob')

for all bin B(Jt).

I(b')=b TIns remarkably simple formula is the reason that the theory of isogenies is simple. Basically it says that the algebra of !* is determined upto constant by the geometry of f. The analytic theory of the next chapter will determine the constant in the formula. To understand the meaning of compatibility we need to understand the relationship between the two theta groups H(ft') and H(Jt). We may assume that X = V j Land Y = V j M where the lattice M contains Land f(v + L) = v + M. Also assume that Jt is given by A.-H. data (a, H). Then ft' is given by (aIL,H). We have inclusions L.l.:J M.l.:J M:J L. Hence we have a diagram

K(ft')

= L.l.jL :JM.l.jL

11 M.l.jM = K(Jt)

n.

where M.l.jL = {x E L.l.jLle 27riE(x,m) = 1 for all m in MjL = Ker Now we have a natural homomorprusm Ker(f) '-+ H(ft') which sends a coset m of M into the transformation represented by

f(v)

---+

a(m)-le- 7rH (tf,m)tH(m,m) f(v

+ m)

.

Trivially this is a homomorphism and T(Y, Jt) is the subspace of Ker(f)invariants in T(X, ft'). By Lemma 4.2 the centralizer N of Ker(f) is the inverse image of M.l.jL under H(ft') ~ K(ft'). There is a natural surjective homomorphism f3 : N ~ H(Jt) with kernel = Ker(f) given by sending a

36

Chapter 4. Groups Acting on Complete Linear Systems

transformation into the same formula by thinking of it operating on different sheaves .!l'( aiL, H) and .!l'( a, H). Compatibility means that a) j3(A'(.!l') n N) = A'(Jt), b) j3(B'(.!l') n N) = B'(Jt), and c) Ker(f) = (Ker(f) n A'(.!l')) ffi (Ker(f) n B'(.!l')). Now we are ready for the Proof of Theorem 4.7. By Theorem 4.5, T(Y,Jt) is an irreducible representation of H(Jt). Now f* : T(Y,Jt) ~ T(X,.!l') is j3-linear (i.e. f*(j3(x) . 0") = X· f*(0") for x in N). As j3 is surjective it follows that f* is uniquely determined upto to constant multiple as an j3-linear injection. To prove this theorem just note that T/;,l f*'1.6 is also a j3-linear injection by an exercise using the actions on C[B(.!l')] and C[B(Jt)] and the compatibility conditions. 0

Exercise 1. Finish the proof of the theorem with complete details.

Chapter 5. Theta Functions

§ 5.1

Canonical Decompositions and Bases

Let L be a free abelian group of finite rank with a non-degenerate integralvalued skew-symmetric form E : L X L ~ 71. In this section we will develop the structure of the symplectic lattice L. An isotropic subgroup is a subgroup A of L such that E( aI, a2) = 0 for all al and a2 in A. Lemma 5.1. There exists two isotropic subgroups A and B of L such that

AEBB

= L.

Proof. If L =I 0 then there are elements a and b of L such that E(a,b) == m is positive and minimal. Thus the subgroup of values E( a, 1) (or E(l, b)) is generated by m. For any 1 in L we may write 1 = l' +(E(a, l)/m)b-(E(b, l)/m)a where E(l,a) = O. If (a,b).1. denote the subspace of all such 1, we have 71a EB 71b EB (a, b).1.. Here (a, b).1. is non-degenerate for E and we may repeat the constnlction if necessary. Therefore we have a basis aI, b1 , a2, b2, ... ,a g , bg of L such that all these elements are perpendicular except for all aj and bj • For this lemma we may take A = EB71aj and B = EB71bj. 0

So the rank L is even. We can even find a "canonical" basis al , b1 , a2, b2, ... , a g , bg such that they are all perpendicular except aj and bj and E( al , b1 ) IE(a2, b2 )1 ... IE(a g , bg ) as positive integers. Lemma 5.2. Given a decomposition A EB B = L as in Lemma 1 we may find bases aI, ... ,a g of A and b1, ... ,b g of B which form a canonical basis. Proof. The form E gives an homomorphism

38

Chapter 5. Theta Functions

coefficients. Now let al, ... ,a g be the dual basis of A. Then we have solved the problem. 0 Exercise 1. Find a canonical basis of 71 4 where E is given by the matrix 0 [

~2

! ~5 ~41 2

-3

-2

.

o

Exercise 2. Using the existence of a canonical basis show that the proof of Lemma 5.1 actually produces one. Exercise 3. Show that the integers {E( ai, bi )} depend only on Land E. (Hint: What are the elementary divisors of matrix representing E?).

§ 5.2 The Theta Function Let X be an abelian variety V f L with polarization H. Let A EB B be a decomposition of the lattice L with respect to the skew-symmetric form E = ImH. By the proof of Lemma 1.6 we have a unique multiplier a on L such that alA = alB = 1 such that (a, H) is A.-H. data. The sheaf f£ = f£( a, H) is called excellent with respect to the decomposition A EB B = L, or just excellent if there is no confusion. The whole theory of theta function revolves around the special properties of excellent sheaves. Consider L~ = {v E VIE(v,l) E 71 for all 1 in L}. We have L~ = A ~ EB B~ where A = L ~ n A 0 1R and B = L ~ n B 0 IR. Thus K( f£) = L ~ f L is the direct sum A ~ fA EB B~ f B. Let A(f£) = A ~ fA and B(f£) = B~ f B. One may check that A(f£) and B(f£) are maximal isotropic subgroups of K(f£). Lemma 5.3. There is a canonical decomposition A'(f£) and B'(f£) of H(f£) which refines A(f£) and B(f£). Furthermore A'(f£) and B'(f£) lie in Hc(f£). Proof. We need to define section 0" A and O"B of Hc(f£) ~ K(f£) over A(f£) and B(f£). If a is in A(f£) then let O"A(a) be given by the transformation f(v) ~ exp(-7rH(v,a)tH(ii,a))f(v + a) where a in V lies over a. We have already noted that 0" A( a) E Hc(f£). One must check that 0" A is a homomorphism. This follows from a routine calculation as E is zero on A~. Furthermore it does not depend on the choice of a. Then section O"B is defined in the same way. 0

§ 5.3 The Isogeny Theorem Absolutely

39

The central result is Lemma 5.4. There is a canonical H(fi')-module isomorphism

p!£: C(B(fi'» ---+ T(X,fi') .

Proof. P!£(oo) must be a A'(fi')-invariant section of fi'. Then P!£(Ob) = O'B(b)(p!£(oo». Thus it will suffice to determine P!£(oo) explicitly. We return to the proof of Theorem 2.13. Set U = A, U' = B and A = 0. Let P!£(oo) = e 1r / 2B (tI,tI)O:bEB e- 1rib -(b)+21rib-(tI». By the theorem this defines a non-zero section of fi'. We must check that it is A'(fi')-invariant. In other words that it is a section of the excellent sheaf on VIA ~ ffi B with polarization H, but this follows from the theorem ill that case. 0 The expression EbEB e- 1rib -(b)+21rib-(v) is called the theta function. It is useful to have an explicit expression for P!£(Ob). Note that the constant term of this Fourier series is 1. Corollary 5.5. For c in B(fi') represented by an element

P!£( oc)

c of B~

= e 1r / 2S(v,tI) (2: e -1ri(b-c)-(b-c)+21ri(b+c) -(V»)

we

have

.

bEB Proof. P!£( oc)( v) = P!£( (0,0, -c )*00) = O'B( -c) . P!£( 00)( v) = e- 1rH(tI,-c)-1r/2H( -c,-c) e1r/2S(v-c,tI-c) e- 1rib -(b)+21rib -(tI-C»)

(2:

bEB

= e 1r/2S( tI,tI) (2: e 1rib -(b)-21ric -(c)-1r/2H(c,c)+1r/2S(c,c) -1r(S( v,c)- H( tI,c»+21rib - (tI) ) bEB

t(

but H - S)( v, c) = c~( v) by the proof of Theorem 2.13. Thus the exponent equals -7rib ~(b )+27rib ~(c)-7ric~(c)+27ric~( v )+27rib ~(v) which equals -7ri(b+ c)~(b + c) + 27ri(b + c)~(v) as b~(b) is symmetric quadraticform on B because H is real on our B x B. Thus we have the result. 0

§ 5.3 The Isogeny Theorem Absolutely Let f ; X ~ Y be an isogeny of abelian varieties where X = VI Land Y = VIM for lattice M :J L and f is induced by the identity. Let A ffi B = M be a decomposition of M. If A n L ffi B n L = L then we have a decomposition of

40

Chapter 5. Theta Functions

L. In this case we will say that f is compatible with the decomposition of M. Assume that this happens. Let Jt be an excellent sheaf on Y. Then !l' = !* Jt is an excellent sheaf on X. Thus we have canonical decomposition A'(!l') and B'(!l') of H(!l') and A'(Jt) and B'(Jt) of H(Jt). Lemma 5.6. Thesc decompositions are compatible in the sense of Section

4.4.

Proof. We consider f : M.l.IL ~ M.l.IM

= K(Jt). By assumption M.l.IL = M.l.IM = A~/AffiB~/B where f preserves the decompositions. Clearly for a E A~/A n L the transformation O"A(a) of !l' induces the transformation O"A(f(a)) of Jt. Thus a(A'(!l') n N) ~ A'(Jt) and similarly with B. We need only see that Ker(f) = (Ker(f)nA'(!l'))EB(Ker(f)n B' (!l')) but Ker(f) = MIL = AlAn L ffi BIB n L. This proves the result. 0 A~AnLffiB~/BnL and

Finally we have a definitive isogeny theorem. Theorem 5.7. In the above situation p-;/ !*P.6(Ob) B( Jt).

= Eb'EB(.2')

o~ for all bin

!(b')=b

Proof. By Theorem 4.7 we know this formula upto constant. Thus we only need to know C where p-;/!*P.6(oo) = C(E!(b')=oObl). By Corollary 5.5 in the Fourier series expansion of Ob' the term e 7r / 2B (fJ,fJ) (constant) occurs only 00 and the constant is 1. Thus Ce 7r / 2B (fJ,fJ) is the term of !*P.it(oo) but this is e 7r / 2B (fJ,fJ). So C = 1. 0

We will later see that this innocent looking theorem is responsible for a meriad of relations between theta functions.

§ 5.4

The Classical Notation

Let X = VI L be an abelian variety wi th polarization H and A ffi B = L be a decomposition with respect to E = ImH. Then we have the form u(v) on B~. This form has marvelous properties: a) u ~(v) is symmetric and b) Imu~(v) is negative definite. The part a) was used in the proof of Corollary 5.5 and b) was used in the proof of Theorem 2.13. We will consider the classical description of polarized abelian varieties in terms of a canonical basis al, •.• ,a g and b1 , ••. ,bg in A and B. We may write E( aJ, ~ b.)bj = - El
§ 5.4 The Classical Notation

complex basis of V. We want to relate the 9 x 9 matrix

T = (Tn

41

to the form

u(v). Let E(a~,bi)bj = bi. Then (bi) is a basis on B~. Lemma 5.8. (bD~(bi)

= -Tf.

Proof. v( w) is complex linear Wand equals E( w, v) when w is in A. Thus (bD~(bi)

= (b~ ~)CE-T}a/) = - 'ETJE(ak,bD = -Tf.

0

Therefore we have proven the first part of Theorem 5.8. The matrix T is a complex symmetric 9 X 9 matrix and the imaginary part 1m T is positive definite. Conversely, any such matrix arises from the above situation with fixed elementary divisors {Cj = E( aj, bj)}. Proof. Let V = (;g with unit basis al, ... ,ago Let bj = Cj('E19~g Tfak) where cil .. ·Ic g is a successively divisible sequence of positive integers. Then as 1m Tf is non-singular L = ffiZla; ffi ffiZlb i is a lattice. We define a skew-symmetric form E on the real space of V by setting E( aj, ak) = E(bj , bk ) = 0 = E( a"~ bk) if 1 =I k and E(a/, b/) = c/. We want to write E as a matrix for the basis all ... ,a g , iall ... ,ia g • By the linear algebra this is t

[1

o

Re-T]-l [0-1

1m - T

1] [1 0 0

Re_T]-l

1m - T

.

Using this the matrix is

As

T

is symmetric this reduced to

(Im-r)-l]

o

.

Thus this matrix is of type (1,1) and the corresponding Hermitian form H has matrix ((1m T )-1) with respect to the complex basis al, ... ,a g which is positive definite. Thus we have constructed the required example. 0 We may now write our theta functions in terms of the matrix B(fi') be given by 'El~j~g njbj for some integer nj. Then

(*)

1J~(ljb)(Z)

= c+i'z(ImT)-l z (

2:::

Let bin

c+ 1ri '(im+n)T(im+n)+21ri (im+n)z)

mEZ9

where

t

T.

e is the diagonal matrix with entrees C], . .. ,Cg.

42

Chapter 5. Theta Functions

It will turn out that the r's are natural analytic parameters for the family of polarized abelian varieties. The matrices H and S are not complex analytic functions of r. So they did not appear in the classical theory which ignored the complex differential metric geometry. We will eventually eliminate them when we do moduli.

§ 5.5 The Length of the Theta Functions Let A EB B = L be a decomposition of the lattice of a polarized abelian variety X = VjL with polarization H. Let !l' be the excellent sheaf given by H. We have constructed a natural H(!l')-isomorphism PIe : C(B(!l'» ~ T(X,!l'). We intend to compute the length IIp!£11 of our theory of theta functions PIe as introduced at the end of Section 4.3. We need to have an answer for this length. As we have used many times we have an isomorphism A ®71 C ~ V. Thus ,VA ®z C:::: N V where 9 is the dimension of X. The free Zl-module ,VA has two generators ±k. We may identify the dual of ±k with an invariant holomorphic g-form ±w on X. The

JI Ix

I.

needed invariant is w /\ w We will call this the geometric height of the decomposed polarized abelian variety (X, H) with respect to A and denote it by h(X,H,A).

Theorem 5.9.

IIp!£11 2

=

2g j(detL E)1/4 . heX, H,A).

Proof. We will choose a canonical basis aI, ... ,a g and bl , ... ,bg of A and B. Let e be the diagonal matrix with coefficients E(a;,b j }. So we may assume that al, ... ,a g are the unit basis for V = cg = {(z;)} and bj = -ej El::;k::;g rlak' The height h( X, H, A) is just the square root of absolu te value of dZ l /\ ••• /\ dZg /\ dZl /\ ... /\ dZg = (±1 )2g volume of a fundamental domain for Zln + reZl n g in cg = ±2 ITl::;j::;nejdet(Imr) = ±2 g detedet(Imr). Thus h(X,H,A) = (2g det e det(Im r»l /2. By definition IIp!£11 2 = e- 7rH (tJ,tJ) P!£(oo)P!£(oo)dv where dv is the invariant measure on V normalized such that dv = 1. In coordinates this integral is

Ix

Ix

J

e7r'z(Imr)-'Ze7rRe('z(Imr)-'z)(

Ix

2:::

e7riI'merem-'PHepl+27ri['mcz-'pezl)dv.

m,pE71'

X

Write Z = x + iy where x and y are real. Thus as r is a symmetric matrix the integral becomes

J

e 27r ' y (Im r)-'y (

X

2::: m,pE71 9

e7ri['merem-'pcreple27ri[('m-'p)ex+i'(m+p)eYl)dv.

§ 5.5 The Length of the Theta Functions

43

Consider the homomorphism (7 : X ~ IRg / 1m T·e71 g == Y defined by (7(z) = y. Then Ker((7) = IRg /71 g and we may compute the integral by integration over 1 l the fibers of (7. As e27rirxdx = 0 if -lOis an integer and dx = 1. Our integral becomes e- 27r 'y(Im r)-'y( EmEZ' e- 27r 'me(Im r)em-47r'me Y )dy where dy is the invariant measure on Y such that dy = 1. Let y = (1m T ) eA:. Changing variables in our integral we may write it as

Jo Jy

J

Jo

r

Jy

e- 27r '.oe(Imr)e",(

m.. /z.

2:::

e- 27r 'me(Imr)cm+47r'me(Imr)e"')dA:

mEZ9

where dA: is the Euclidian volume clement in ]I{9. Now we may write this as

J 2:::

e- 27r '(.o+m)c(Imr)e(",+m)dA:=

IIV /z. mEZ9

J

e- 27r '",e(Imr)e"'dA:

m.9

because T is symmetric. By the following theorem this is det (27r~m r) -1/2 / det e 2 = 2 g / 2 det(lm T )1/2/ det e (det ;),/2 h(X,IH ,A)' The result follows from (det e)2 = IdetL EI. 0 We need Theorem 5.10. If A is symmetric 9 X 9 matrix with ReA positive definite and b is a 9 vector, then JJRn e-('xAx+2'bx)dx = det(7rA- 1 )1/2 e 'bA-'b where the square root is positive if A is real. Proof. By analytic continuation we may assume that A and b are real. Clearly the problem is independent of the choice of coordinates in IRg. So we may assume that A = In. Thus our integral is

J

e-(L;xJ+2L;bjxj)

IR n

II 1 $.; $.g

dx; =

II

J

2

e-(x +2bjx)dx .

1 $.; $.g IR

A little reflexion verfies that we need only treat the case 9 = 1. Thus we want 2 2 to show that Jm. e-(ax +2bx)dx = (7r/a)I/2 e b /a when a> O. By completing the square in the numerator and making a linear change of variable, one is reduced to the case where b = 0 and a = 1. This last case is done by Gauss's well-known polar coordinate trick. 0

Chapter 6. The Algebra of the Theta Functions

§ 6.1

The Addition Formula

Let X = V / L be an abelian variety. We will fix a decomposition L = A ffi B with respect to an ample invertible sheaf !l' = !l'( Ct, H) which is excellent. In this section we will consider a special case of the isogeny Theorem 5.7. Let n be a positive integer. Let Y be the product xn of X with itself n-times. Then we have the sheaf !l'(c) = Q91:5:;:5:n 7rJ!l'0 C; on Y where all C; are positive integers. Clearly !l'(c) is ample and excellent with respect to the product decomposition An ffi Bn of the lattice L n of Y = vn / Ln. Now let d = (d1 , ••• ,dn ) be another such sequence. Let C (resp. D) be the diagonal n x n matrix of integers with entrees Cl, ••. ,C n (resp. d l , ••• ,dn ). Let F be an n x n matrix of integers. Then F define a homomorphism F : Y ~ Y where F(xl> ... ,x n ) = CEI:5:;:5: n FIx;). Lemma 6.1. !l'(c) is isomorphic to F* !l'(d) if and only if t F· D . F = C. In which case F is an isogeny.

Proof. To check that two excellent sheaves are isomorphic it is enough to see that they have the same Hermitian form or what is the same as the same
46

Chapter 6. The Algebra of the Theta Functions

Proof. The theory of theta functions are explicit expressions given in Chap. 5. To check this equation one simply applies the definitions. 0

Thus we have in terms of these product decompositions.

Theorem 6.3 (addition). P!£(c) -] F*P!£(d)Ob

= Eb'EB(!£(c» Ob'

for all b in F(b')=b B(f£(d)) where Ob = Q9]::;:;::;:n obi where (bj) = b with bj in B(f£0 ci).

o

Proof. Just apply Theorem 5.7.

Thus F* is simply 00bj goes to Eb'=B(!£(c» 0obi' We will give some special J(b')=b cases but we regard the theories of theta functions P* as identifications. Let m be a non-zero integer. We take n = 1 = d and F = m. Thus d = m 2 • m2 Consequently (m)* f£ ~ f£0 . Our formula became the classical m-plication formula

Theorem 6.4. P;~m2

0

m*

0

P!£(Ob)

=

Eb'EB(!£®m2)

Ob' for all bin B(f£).

mb'=b Let (d], d2 ) be two positive integers. Let F be the matrix

tF· D· F = C where c = (d] + d2,d1 d2(d] which is just F* in coordinates we have

+ d2)). If '1

[~ ~~:]. Then

denotes p;~c)F*P!£(d)

Theorem 6.5. Then for b] in B(f£0 d,) and b2 in B(f£0 d2)

b; +b;=b, d2b; -d, b; =b 2

Now we will assume that f£ is (d] + d2 )-power of an invertible sheaf. Thus B(f£) contains all the (d] + d2 )-torsion in B(f£0d,d 2(d,+d 2»). In this situation we will give a basis of ClB(f£0 d,) X B(f£0 d2)] and ClB(f£0d,+d 2) X B(f£0 d,d 2(d,+d 2»)] such that '1 has a simple diagonal form. Let J be the group of (d] +d 2 )-torsion in B(f£). For b] in B(f£0 dl) and b2 in B(f£0 d2) and X a character of J let X(b 1 , b2, X) = EiEJ X(j)Ob, +i0ob2+d2;. As (bl,b 2,X) runs through [B(f£0d,) X B(f£0d 2)]/{U,d2j)} x J* the X(bbb 2,X) form a basis of C[B(f£0 d,) X B(f£0d 2)].

§ 6.2 Multiplication

47

For b~ in B(f£0 dl+d 2) let Y(b~, X) = 'EiEJ X(j)Ob~+i' Also for b~ in B(f£0d , d2(d , +d 2») let Z(b~, X) = 'EiEJ X(j)Ob~+i' Then as (bt, X) runs through B(f£0 dl+ d2)/J X J* then Y(b~,X) forms a basis of ClB(f£0 dl+ d2)] and !>"imilady with {Z(b~, X)}. Our addition formula gives Theorem 6.6. 7J(X(bt,~, X) = Y(b~, X) 0 Z(b~, X)

if bl

= b~

+ b~

and b2 =

d2b~ - dlb~.

Proof. By Theorem 6.5 7J(X(bt, b2, X)) = 'E X(j)OCl 0oC2 where CI +C2 = bl + j and b2 + d2j = d2cI - d l c2 for some j in J and CI in B(f£®d 1 +d 2) and C2 in B(f£0d , d2(d , +d 2»). Thus (CI, C2) runs through (b~ + j + j', b~ - j') with j and j' in J as J is the (d l + d2 )-torsion in B(f£0dld2(dl+d2»). Thus

7J(X(bl ,b2,X)) =

2:::

xU +j')X(-j')Ob~+i+1' 00b~-j'

j,j'EJ =

(2::: X( k )Ob~

+k )

(2::: X(l)Ob~+I) IEJ

kEJ

o

= Y(b~, X) 0 Z(b~, X) .

§ 6.2 Multiplication Continuing with the notation of the last section we will assume that d l = d2 = 1. Thus f£ is a square and we have an isomorphism F*(7I"if£071"2f£) ~ 7I"i f£ 02 0 7I"2f£02 where F(XI,X2) = (Xl + X2,XI - X2). In this case B(f£0d 1 +d 2) = B(f£0d 1 d2(d 1 +d 2») = B(f£02) and Z = Y and Theorem 6.6 gives the global effect of F* as 7J(X(b l ,b2,X)) = Y(b~,X) 0 Y(b~,X) where b~ are in B(f£0 2) and F(b~, b~) = (b l , b2). The next idea is to consider the global effect of the inclusion i of X as X x 0 in X x X. As Foi: X ~ X 2 is the diagonal, (Foi)*: r(X,f£) 0 r(X,f£) ~ rex, f£02) is just multiplication. On the other hand (Foi)* is just (evaluation of the second factor at zero)o7J. Before we write the formula for multiplication we will make a more or less trivial generalization of these ideas. The problem is to compute the multiplication M x" x2 ; rex, T;, f£) 0 rex, T;2f£) ~ r(x, T;, f£ 0 T;2f£) for arbitrary points XI and X2 of X. As F is surjective we have points YI and Y2 of X such that F(YJ,Y2) = (XJ,X2). Thus we have a commutative diagram T(Y',Y2) :

X x X

--+

X

!F T(X"X2) ;

X

X

X

X

X

!F --+

X

X

X .

48

Chapter 6. The Algebra of the Theta Functions

We have a covering diagram of sheaves

7r;(T;, .!l'02) 0 7r2(T;2.!l'0 2) ~ 7r2.!l'02 07r2.!l'02

i

i

Taking global section we may consider the horizontal arrows as identification via translations and the two vertical arrows contain the same information. Now composing with i we see that the multiplication has the form

2

M x"x2",,'2 : T(X, T;,.!l') 0 T(X, T;2.!l') ---+ T(X, T;,.!l'0 )

which can be computed by (evaluation of the second factor at -Y2) the above identifications we have

0

'1. With

Let ~ be the Poincare sheaf on X x X~ where X~ is the dual abelian variety. Let ~'" == ~Ixx", for some point a in X~. Then ~Q' is equivalent to zero. With this notation we have Theorem 6.8. Let J{ be an ample invertible sheaf on X. a) For a in a non-empty subset of X~ the multiplication

is surjective for fixed f3 and'Y in

X~.

b) For a in a nonempty subset of X~ the multiplication

is surjective for fixed f3 and'Y in X~. c) If n ~ 2 and m ~ 3 then the multiplication

is surjective for arbitrary f3 and'Y in

X~.

Corollary 6.9. The graded ring EBn>o T(X, J(0 71 ) is generated as a C-algebra by T( X, JI) if J{ is an m-power with m ~ 3. Proof. Without loss of generality we may assume that JI is excellent. In the previous theory let .!l' = J{0 2. For a) we take Xl and X2 such that 2
§ 6.3 Some Bilinear Relations

49

each b~ in B(fi'0 2) and X in J*, we have some b~ such that Y(b~,x)I-Y2 =1= 0 where F(b~, b~) is in B(fi')02. Given b~ we can take b~ = b~. Then Y(b~, X) is a non-zero section of fi'0 2. Hence its value is non-zero in nonempty open subset of Y2 in X~. We need to note that such a Y2 is possible. Now 2Y2 = Xl - X2. Thus 4
T (X,J{0 5 0 9l'/H')') =

2:::

T (X, J{ 0 9l'(l+0) T (X, J{04 09l'')'-0)

OEX

= 2::: T (X, J{ 0

9l'/H0) T (X, ",,02 0 9l'-0)

o x

T (X, ",,02 0

9l'')')

[6 general]

by b) and the fact that linear spaces are closed and the symbols depend continuously on 6. Now this is contained in the subspace T(X,J{ 03 09l'(l)T(X, ",, 02 0 9l'')'). Thus c) is true. 0

§ 6.3 Some Bilinear Relations Let fi' be excellent with respect to the decomposition L = A ffi B. The problem is to describe the kerncI R(fi',fi') of the multiplication m : T(X,fi') 0

T(X, fi') ~ T(X, fi'02). In this section we will assume that fi' is n-power of an invertible sheaf where n is an even integer ~ 4. By Theorem 6.8 c) we know that m is surjective. On the other hand in Theorem 6.7 we have seen that m is diagonal with respect to the correct bases in the two spaces. This means that for any X and a given b~ in B(fi'0 2) there exists a b~ in B(f£0 2) such that Y(b~,x)lo =1= 0 where b~ +b~ and b~ - b~ are in B(fi'). Clearly the set of these b~ are the coset b~ + B(fi'). Therefore we have proven Lemma 6.10. For each X, any B(fi'). coset in B(fi'02) contain an element b such that Y(b, x)lo =1= o. Furthermore consider R(fi',fi') n _-I(CY(bi,x)) == S(b~,X)' As __ is diagonal, R(fi',fi') is spanned by the S(bi, X)· Also let b be an element of b~ +B(fi') such that Y(b, x)lo =1= O. Then by Theorem 6. 7 S(b~, X) is spanned by Y(b, x)loX(c+ b~, c- b~, X) - Y(c, X)loX(b + b~, b- b~, X) when c runs through

50

Chapter 6. The Algebra of the Theta Functions

b~ + B(fi'). As a the above expression is a relation whenever Y(b, X) have proven

=I 0 we

Proposition 6.11. R(fi',fi') is the span of the expressions Yea, x)loX(c + b, c - b, X) - Y(b, x)loX(c + a, c - a, X) where a

== b == c((B(fi')))

are elements of B(fi'0 2 ).

Now by Theorem 6.7 we have by evaluating at zero X(b + c, b - c, x)lo = Y(b, x)lo . Y(c, x)lo if b == c((B(fi'))) and band c in B(fi'0 2 ). Therefore by Lemma 6.10 and Proposition 6.11 we may conclude Theorem 6.12. R(fi', fi') is the span of the expression X(a + d,a - d)loX(c + b,c- b) - X(b + d, b- d)loX(c+ a,c- a) where a

== b == c == d((B(fi')))

are elements of B(f£02).

This theorem is better than the proposition because the value of the X s are more elementary than the value of the Y s. Next we will see that in the above situation that the bilinear relations R( fi', fi') generate all homogeneous forms vanishing on X when we embed X in projective space via fi'. Let Jt be an ample sheaf on X. If Jt is an m-power with m ~ 3. By Theorem 6.9 we have a surjection

SymdT(X,Jt)] ~

ED T(X,Jt0 n )

.

n~O

Let I be the kernel of this surjection. In the next section we will prove Theorem 6.13. a) If m = 3 then I is generated as an ideal by its forms of degree 2 and 3. b) If m ~ 4 then I is generated as an ideal by its forms of degree 2. Remark. Of course the form of degree 2 in I is just the image of R(Jt, Jt)( C

T(X,Jt) 0 T(X,Jt)) in Sym2(T(X,Jt)). Exercise 1. Show that the above works for T; Jt if we may replace evaluation at zero by evaluation at any point x of X (Hint: use translation).

§ 6.4 General Relations

51

§ 6.4 General Relations Let .Ali and fi2 be two l!'Jx -modules. Then R(.AIi, fi2) is the kernel of the multiplication T(X,.AIi) 0 T(X,fi2) ~ T(X,.AIi 0fi2).

Let .AI" be a fixed ample sheaf on our abelian variety X. Let !l'I, !l'2 etc be ample invertible sheaves on X which are equivalent to the 11,12 etc-power of .AI".

Theorem 6.14. If l3

~

2 and either II

~

3 and l2

~

4 or II

~

2 and l2

~

5,

then

This will prove Theorem 6.13. For a) note that R(Jt,Jt) and R(Jt,Jt0 2 ) essentially generate I and have degree 2 and 3. For b) I is generated by R( Jt, Jt) which has degree 2. We will prove Theorem 6.14 by a gambit similar to that used to show multiplication is surjective. Explicitly we will use

Proposition 6.15. If II

+ l2

R(!l'I,!l'20!l'3)=

~

5 and Ibl2 ~ 2, then

L

R(!l'1,!l'209l'",)T(X,!l'309l'_",).

"'EX

First we will show that this implies Theorem 6.14. Write

By Proposition 6.15

but by Theorem 6.8

for general

Q.

Thus

R(!l'1,!l'2 0 !l'3)

=(

L '" general

R(!l'1,!l'4 0 9l'",)T(X,!l's 0 9l'_",)) T(X, !l'3)

52

Chapter 6. The Algebra of the Theta Functions

the reverse inclusion is obvious. Note that we implicitly used that R(fi'l, fi'4 0 ~ -a) is a vector bundle on X~, which follows from the exact sequence of vector

bundles

o ~ R(fi'I, fi'4 0

~ -(1) ~ T(fi'l)

® T(fi'4 0

~ -a)

V

~ T(fi'l 0 fi'4 0 ~ -a) ~

0

(Theorem 6.8). Thus it remains to check Proposition 6.15. For that look at the commutative exact diagram

We need to show that any linear functional A on T(X, fi'J)0T(X, fi'20fi'3) which induces zero on R(fi'I, fi'2 0 ~a) 0 T(X,fi'3 0 ~-a) for all a in X~ is induced by a linear functional fL on T(X,fi'1 0 fi'2 0 fi'3)' By assumption A induces a linear functional on T(X,fi'J) 0 T(X,fi'2 0 ~a) 0 T(X,fi'3 0 ~-a) which comes from one say fLa on T(X,fi'1 0 fi'2 0 ~a) 0 T(X, fi'3 0 ~ _a). By Lemma 4.6 fL if it exists is uniquely determined by the family {fLa}. On the other hand, {fLa} clearly depends regularly on a in X~. If we could show that such a family is induced by such a fL we would be done (just apply Lemma 4.6 to the second arrow to check that these fL induces A). Next we write our problem in terms of sheaf theory on X~. For any ample sheaf fi' on X let W±(fi') = 7l'x-*(7l' fi' 0 ~±l) be a locally free sheaf on X~ whose fibers at a are T( X, fi' 0 ~±a). We have multiplication

x

for two ample sheaves fi'l and fi'2 on X. Thus we have the dual homomorphism

Our problem is just to show Proposition 6.16. M is an isomorphism.

Proof. We have seen that Lemma 4.6 is equivalent to the injectivity of M. To show that M is surjective we will show that the two spaces have the same dimension.

§ 6.4 General Relations

53

Now:K == (W+(.!l'I) 0 W-(.!l'2))~ = (W+(.!l'I))~ 0 (W-(.!l'2))~ has fibers (T(.!l'1 0 9l',,))~ 0 (T(.!l'2 0 9l'_,,))~ ~ Hg(.!l'~-1 0 9l'_<» 0 Hg(.!l'~-1 09l',,) by Serre duality where 9 = dim X. Thus if /F = 7I"r( .!l'~-1 0 7I":,39l'0- 1) 0 7I"2(.!l'~-1 0 7I" 23 9l') is a sheaf on X x X x X~, we have an isomorphism R 29 7l"3·/F ~ As .!l'1 and .!l'2 are ample this is the only non-zero direct image of /F via 71"3. Thus by a degenerate Leray spectral sequence we have isomorphism H29+ i (/F) ~ Hi(:K). Hence we need only compute the cohomology of an invertible sheaf on the abelian variety X x X x X~. One method to finish is to show Ix(/F) I = IX(.!l'1 0 .!l'2)1. Hence /F is nondegenerate and H 2g(!F) ~ HO(:K) is its non-zero cohomology group and has dimension = dim T(.!l'1 0 .!l'2). As this calculation is messy we will use a more dramatic way to compute the cohomology of /F. The trick is

r.

Claim. If ..1 is the diagonal in X x X, R g7l"12./F ~ 7I"r .!l'~-1 0 7I"2.!l'~-11.:l ~ (.!l'1 0 .!l'2)0-1 is the only non-zero higher direct image of /F via 71"12. Thus if we prove the claim then a degenerate Leray spectral sequence gives an isomorphism H9+i (/F) ~ Hi ((.!l'1 0 .!l'2)0- 1) which equals T( (.!l'1 0 .!l'2))~ if j = g. Therefore T(:K) ~ H2g(/F) ~ Hg((.!l'1 0 .!l'2)0- 1) ~ T((.!l'1 0 .!l'2))~ and we will be done. To prove the claim write /F as 7I"r2( 7I"r .!l'~-1 0 7I"2~-1) 0 (71"3 + 71"2, 7I"3)*9l'. Thus Ri7l"12./F~ (7I":.!l'~-1 071"2.!l'~-1)

o Ri 7l"12·(-7I"1 +71"2,71"3)*9l'

~ (7I":.!l'~-1071"2.!l'~-1) 0(-71"1 + 71"2)* (Ri 71"1 09l')

but by Theorem 3.15 Ri 71"10 9l' ~ (!'Jo if j = 9 and is otherwise zero and (-71"1 + 71"2)(0) = ..1. So the claim follows by the Kiinneth formula for higher direct images. 0

Chapter 7. Moduli Spaces

§ 7.1 Complex Structures on a Symplectic Space Let W be a real vector space of dimension 2g with a non-degenerate skewsymmetric form E. Recall the symplectic group SymIR(E) consists of all lRlinear transformations A of E such that E( Ax, Ay) = E( x, y) for all x and y in W. We want to consider complex structures on W. Such a complex structure (W, J) is determined by a real linear isomorphism J : W ~ W such that J2 = -1 which gives the effect of multiplication by i in the complex structure. The symplectic group of transformations of W preserving E operates on such complex structures. For A in SymIR(E) and complex structure Jon W

Thus the isomorphism A : (W, J) ~ (W, A * J) given by A is complex linear for the two different complex structures. This is just the equation A(J(W)) = (AJA-I)(A(W)). A complex structure J on W is reasonable if the following two conditions a) and b) hold. a) E(JWI' JW2) = E(WI,W2) for all WI and W2 in W. By Proposition 1.2 there is a unique Hermitian form HJ on (W, J) with imaginary part E. By Corollary 1.4, H J is non-degenerate. The other condition IS

b) H J is positive definite (equivalently, E(J(x), x) > 0 for all x in W -{O}). We shall be very interested in the properties of the space Reas(W) of all reasonable complex structures on W. We will first see that it is a homogeneous space under SymIR(W), Theorem 7.1. The space Reas(W) is invariant under the action ofSymlR(W), Furthermore SymIR(W) acts transitively on Reas(W) and the stabilizer of a reasonable complex structure J on W is the unitary group of the Hermitian form H J on (W, J).

56

Chapter 7. Moduli Spaces

Proof. The first statement is fairly routine. For A in SymlR(W) and J in Reas(W) we want to show that A * J is in Reas(W). Note that E(AJA-IwI,AJA-IW2) = E(JA-IwI, J A-IW2) = E(A-IwI,A-IW2)

= E(WI,W2)

.

Next we will prove that A : (W, J) ~ (w, A * J) is unitary for the matrices H J and HA*J. in other words,

Recall that HJ(z,w) = E(Jz,w) +iE(z,w). Thus HA*J(Az,Aw) = E(AJA- I Az, Aw)

+ iE(Az,Aw)

= E(Jz,Aw) +iE(z,w) = HJ(z,w). Thus the first statement is true. For the transitivity we will establish a natural bijective correspondence between {symplectic bases of W} and {unitary bases in H J with J in Reas(W)}. As SymR(W) acts transitively on {symplectic bases of W}, this will show that it acts transitively on J. Also it will show the second statement because if A is in the stabilizer of J, A is unitary for H J on (W, J) by the first paragraph. Conversely unitary transformations correspond to a change of the unitary basis. So by the bijectivity any unitary transformation comes from a symplectic transformation. Let a l l ' " ,a g , bl , ... ,bg be a symplectic basis of W. Then E( ak, a;) = 0 = E(b k , b;) for all j and k, E(ak' bj) = 15; for all k and j. We have an associated complex structure J such that J( ai) = -bi and J(bj) = +aj for all j. The associated Hermitian form H J is H J(LAja; ;

+ Il;b;, L ;

"rja;

+ t5;b;)

=

+ LAi'Yj + Il;I5; + i LAjt5; ;

1li'Y; .

;

Thus H J is positive definite; i.e. J is in Reas(W). Furthermore al," . ,a g are a unitary basis with respect to HJ. Conversely if al,'" ,a g are a unitary basis for HJ for some reasonable complex structure J, then al,'" ,a g , -J(al),""J( a g ) is a symplectic basis for W. Clearly this bijection is natural with respect to the SymIR(W) action. 0 We have actually proven more. Consider the space B = {( w, J) where w is in W and J is in Reas(W)}. Then projection 7r : B ~ Reas(W) is a complex vector bundle with respect to the complex structure (W, J) on the fiber over J. In fact we have a Hermitian structure on this bundle given by H J on the fiber over J. We have proven

§ 7.1 Complex Structures on a Symplectic Space

57

Corollary 7.2. B is a homogeneous Hermitian bundle over Reas(W) with respect to the group SymIR(W), In turns out that Reas(W) has the structure of a Hermitian symmetric space. In other words Reas(W) has a complex structure, Hermitian metric (all invariant Wlder SymIR(W)) and an holomorphic isometry
_~]

where K and L are real symmetric g x g matrices.

[~ ~]

Proof of Claim. The condition on such real matrices

~ ~] [~ ~] = - [~ ~] [ ~1 ~] =-[~1 ~] [~ ~]

[ 1

and

t

as a')

[~ ~] [~1 ~]

58

Chapter 7. Moduli Spaces

The rest of the proof of this claim is obvious. Now

2 2 L]2=Tr[K +L -K LK-KL = 2 Tr(K2

KL-LK] K2+L2

+ L2) =

2 2)k;:')2

+ (l;:,)2

m,n

as K and L are symmetric. Hence II II is positive definite. It remains to show that II II is Hermitian. Now

. [K L

2

L] -K

=-

[0-1 01] [KL

L] -K

=

[-L K

K] L

Thus we can think of K as the real part and L as the imagina.ry part of

[~ _~ ]. So the matrix is Hermitian.

0

In the next section we will show that the almost complex structure on Reas(W) is a complex structure by introducing coordinates.

§ 7.2 Siegel Upper-half Space Let 9 be a positive integer. The Siegel space Sg is the complex manifold of all symmetric 9 X 9 complex matrices T with positive definite imaginary part. Thus dimSg = (g + 1)g/2. Let al, ... ,a g , b1, ... ,bg be a symplectic basis of our real vector space W. Thus W = $lRaj $ $IRbj with the standard symplectic form E.

Theorem 7.4. We have a bijection R: Reas(W) ~ Sg defined by the above choice. Proof. For J in Reas(W)((W, J)/ L, H J) is a principally polarized abelian variety where L = $Zla; $ $IRb j and (al' .. , ,a g , b1, ... ,bg ) are a canonical basis of L with respect to the skew-symmetric form E = ImH J . Thus we may apply the procedure of the Theorem 5.8 to produce the point R(J) of Sg. Let (x;) and (yj) be the 9 x 9 real matrices such that bj = - E(xj + yjJ)ak' Thus R(J) = (xj + iyj) == T. In that theorem we have seen that R(J) is in Sg and a.ll of Sg occurs this way uniquely. 0

From the point of view of Siegel space the complex vector space (W, J) is isomorphic to the constant complex space cg = $Caj but the Hermitian structure is changing. Thus we have a holomorphic trivialization

§ 7.2 Siegel Upper-half Space

59

but the Hermitian matrix in the fiber is given by Ek,jZk ((ImT)-l); Zj by Section 5.4. We will next work out how SymIR(W) acts on the bundle (Jg x Sg (in particular how it acts on Sg). In terms of our basis al," . ,ag , bl , ... ,bg of W an element of SymIR(W) is a 2g

X

2g matrix

[~ ~]

where

Ct,

(J, 'Y, 0 are 9

Lemma 7.5. The action of SymlR(W) on

X

c g X Sg

9 matrices such that

is given by

[~ ~] *(Z,T)=(ZI,T')=C(CT+d)-lz,(aT+b)(cT+d)-I). Proof. Let A

= [~ :].

Let J be the complex structure R- 1( T) on W. Then

we have a commutative diagram ffiIRal ffi ffiIRbj

=

(I,-r) -------+ (;g

(w, J)

!A ffiIRa; ffi ffiIRb j

!x

= (W, A * J)

(I,-r') -------+ (;g

where the complex linear isomorphism X is given a complex 9 the same name. Therefore X(l, -T) = (1, -T')

[~ ~]

X

9 matrix of

or rather (X, -X . T) =

(a - T'C, b - T'd). So X = a - T'C and T = -(a - T'C)-I(b - T'd). We want to reverse the roles of T and T'. The inverse of A is

The reversed equations are

So

60

Chapter 7. Moduli Spaces

and

We next have Corollary 7.6. a) SymIR(W) acts by holomorphic transformation on Sg. b) R : Reas(W) ~ Sg is biholomorphic.

Proof. a) follows directly from r ~ (ar + b)( cr + d- 1 ) being holomorphic in r. As we are dealing with SymIR(W) homogeneous space by Theorem 7.1, for b) we need only check that R gives a complex isomorphism on tangent space at one point say

R-l(iI) =

[~I ~]

==

J .

First we will compute T(w,J)(V) for any tangent vector V of Reas(W) at any complex structure J. Let f be a variable with f2 = O. Then by definition

R(J + fV)

= R(J) + a(W,J) (V) .

Let

V(aj) = Lu~ak

+ Lw;Jak

k

for 15,j 5,g.

k

Then

R(J + fV) = (rj)

+ f(O'j)

where

(rf) = R(J) and (0';) = T(w,J)(V) . Let

(rf) = (xj)

+ iCy;)

and (O'j) = (rj)

+ i(sj)

be the real and imaginary parts. By definition

bk = -

L [ext + fr{) + (y{ + fs{)(J + fV)] aj j

= - [ L(x{ + y{J) + f(r{ + s{ V + s{ V + s{J)] aj . j

Thus

L(r{ j

or, rather

+ s{ V + s{J)aj =

0 for all 1 ~ j 5, 9 ,

§ 7.2 Siegel Upper-half Space

61

~)r{ +s{J)aj = - Ly{Vaj = - LY{(Lu}ak + LW}Ja k) j

j

j

I

I

In tenns of matrices this gives

R

= -y . U

and S

=-y

.W .

Thus in the particular case (Y

= l)T(w,J)(V) = -U -

iW

where

V(aj) = L ujak k

+L

wjbk .

k

Thus T(w,J) is injective. We need to check that T(w,J) is surjective and complex linear; i.e., T(w,J)(iV) = +W - iU. In the proof of Lemma 7.3 we have seen that in tenns of the basis

aI,'"

,a g , b2 , ••• ,b g it is given by

[+~ ~~]

and

U and Ware symmetric real matrices. Thus T(w,J) is surjective. Furthermore in that lemma we have seen that

.[U +W] = [-W U] +W -U U W

2

Thus T(w,J)(iV)

=

.

-W +iU and hence T(w,J) is complex linear.

o

Thus we have a SymIR(2g) invariant Hermitian metric of Sg corresponding lion Reas(W). The homogeneous space Sg is actually a symmetric space. The symmetry about

II

iI is T ~ _T- 1 = (AT

+ B)(CT + D)-l

where

Exercise 1. If T = x +iy is in Sg, then R-l(T) is the endomorphism given by

in the basis al, ... ,ab,b1, ... ,bg.

62

Chapter 7. Moduli Spaces

§ 7.3 Families of Abelian Varieties and Moduli Spaces Let L be a lattice in W such that the form E has integral values on L x L. For each complex structure J in Reas(W) we have the polarized abelian variety S(J) = ((W, J)I L, H J). We can continuously construct a family of the varieties as follows. Consider 7r' : B I L ~ Reas(W) where B I L is the bundle B modulo L. Explicitly BIL = {(w, J) I w E W} I I {(WI, J) '" (W2, J) if WI - W2 E L}. In the last section we have seen that 7r : B ~ Reas(W) is a trivial holomorphic bundle. So B I L is a complex manifold and 7r' is holomorphic, smooth and proper mapping and the polarization H J depends continuously on the fibers. This is the basic family of polarized abelian varieties. The basic moduli spaces are M(W, L) == Reas(W) modulo the equivalence relation J I ' " h if S(Jd and S(h) are isomorphic polarized abelian varieties.

Lemma 7.7. We have a natural bijection

M(W,L) = Sym71(L) \ Reas(W) and Sym71(L) acts properly and discontinuously on Reas(W). Proof. For the first statement let A he an element of Sym71(L) and J he a point in Reas(W). Then A : (W, J) ~ (w, A * J) takes L in L and H J into HA*J. Conversely if B : X(J) ~ X(J'), then B is given by an isomorphism A : W ~ W which takes L to L and H J to H JI. Thus A preserves E = ImHJ = ImHJI. Hence A is in Sym71(L) and B = A. The second statement means that a) for all J in Reas(W) the stabilizer G J of J in Sym71(L) is finite and there is a Grinvariant open neighborhood UJ of J such that for all A in Sym71(L), A·UJnu J =1=

§ 7.4 Families of Ample Sheaves on a Variable Abelian Variety

For each I in Z let P, and P, be disjoint neighborhoods of J and VJ = K n np, and VJI = K' n 1-1 P, solves the problem.

63

I J'. Then 0

Therefore if we give Sym71(L) \ Reas(W) the quotient topology by b), it is a Hausdorff space. By a) it has a quotient analytic space structure which is locally isomorphic to the quotient of a manifold by a finite group. Thus the moduli M(L, W) is naturally a separated analytic space. A naive mistake is to try to construct a family of polarized abelian varieties over M(L, W). The guess is consider 7r" : Sym71(L) \ BIL ~ M(L, W). The problem is the fiber over X(J) is X(J) \ \ Aut(X(J)) which is not an abelian variety. The remedy is to consider a smaller subgroup r c Sym71(L) such that r has no fixed points in Reas(W). Then 7r" : r\B I L ~ r\Reas(W) is a family of principal polarized abelian variety over a smooth base. In Proposition 9.7 we will see that we make take r = Ker(Sym71(L) ~ Aut71 / n71 (LlnL)) where n is an integer ~ 3. In this case r \ Reas(W) is moduli space of polarized abelian varieties with a kind oflevel n structure and it is a Galois covering of M(L, W) with Galois group Sym71(L)lr as r is a normal subgroup. Next we want to study the action of Sym71(L) on Reas(W) using coordinates. Let al>'" ,a g , b1 , ••• ,bg be a canonical basis of L with elementary divisors el, ... ,ego Let e be the diagonal matrix with coefficients el, ... ,ego Then the skew-form E is given by tegral matrix [;

~]

[~e ~].

such that

An element of Sym71(L) is a 2g

~] [~e ~]

t [;

[;

~]

[

X

2g in-

~ e ~ ]. Let

re denote this group. Let b: = -!; b;. Then ai, ... ,ag , bi, . .. ,b~ is a symplectic basis of Wand as such it defines a bijection R: Reas(W) ~ Sg. We know how SymlR(W) acts on B ~ (;g x Sg in tenus of a block decomposition of SymlR(W) with respect to the second basis. If [; element of SymlR (W) is

[~1 e c

e

~]

is in

re

then the corresponding

!Ied] . Thus the corresponding action of B is e

[~ ~] *(Z,T) = ((cT+de)-l ez , (aT+be)(CT+de)-l e). So M(L, W) is isomorphic to re \ Sg with the action given by the second coordinate. given by

§ 7.4 Families of Ample Sheaves on a Variable Abelian Variety In the situation of Section 7.3 we want to consider families of invertible sheaves on the basic family 7r' : B I L ~ Reas(W). First of all we consider the multiplier A/(z) = o:/e+1rHJ(z,/>+fHJ(/,1) where 0:/+/1 = O:r 0:/ , ( _l)E(/,/') as usual. For fixed

64

Chapter 7. Moduli Spaces

J this defines an ample invertible sheaf J{J on X(J). This family does not depend holomorphically on J but only real analytically (recall in coordinates HJ is given by (ImT)-1 where T = R(J)). What we can do is change the horizontal structure on the family J{J so that it is holomorphic. This can be done in many ways depending on the choice of maximal isotropic subgroup A of L. This idea goes back to the proof of Theorem 2.1 in section 2.2. For J in Reas(W), let SA,J be the symmetric complex bilinear form on (w, J) such that SA,J = HJ on A ®71 IR x A ®71 IR. Let ~(A, J) : (W, J) ~ C be the complex linear mapping such that v~(A,J)(w) = E(w,v) for all w in A IR. Assume that a is identically one on A. Then K/,J( v) = a(l)e+ 1ri / (A,J)(/)+21ri/~(A,J)(tI) is factor of automorphy. Let YA,J be the invertible sheaf on X(J) such that multiplication by e{SA,J(tI,tI) gives an isomorphism If'A,J : J{J ~ YA,J.

®71

Lemma 7.8. YA is an invertible sheaf on B I L. In other words the multipliers K/,J(v) are holomorphic for (v,J) in B.

Proof. Let al, . .. ,a g be a basis of A. Then we choose a canonical basis al, . .. ,a g , b1 , ••• ,bg of L. Let ai, ... ,a g , b~, ... ,b~ be the associated symplectic basis of W. g g Then we have the standard isomorphism If' : B ~ cg x Sg and L ~ tl EB tl = EB71aj EB EBbZ( -bj ). Claim. In terms of this isomorphism K/,J is given by

where I

= II EB 12

and

T

= R( J) and z corresponding to v.

This is clear because K/,Av) is one if I is in A by the calculation in the proof of Theorem 2.1. Thus K/ 1 +I"J(v) = K/2,J(V) which can be computed by Lemma 5.2 which determines ~ in terms of T. 0 Next let A be another maximal isotropic subgroup of L such that alA == 1. Then we have the isomorphism K1 == If' A,J 0 1f'"A:J : YA,J ~ YA,J'

Lemma 7.9. K1 is an isomorphismYA ~ Y A of invertible sheaves on BIL.

A' . b ul . l' . b -{(SProo. A,J (tI,tI)-SA ' J(tI.tI» . We nee d to f K A IS given y m tIp lcatIon y e see that the function S A,i v, v) - SA,J( v, v) == QJ( v) is C-analytic in v and J simultaneously. We will do this in coordinates (z, T) with respect to aI, ... ,a g , b1 , ••• ,bg. Let aI, ... ,ii g, b1 , ••• ,b g be another canonical basis of L such that

§ 7.4 Families of Ample Sheaves on a Variable Abelian Variety

A=

fB71aj. Thus we have a symplectic transformation

[~ ~]

65

in Sym71(L)

o

which takes the ordinary basis to the one with the "'. This result will prove Lemma 7.9. Sublemma 7.10. QJCv) is given by -2i tz t"'{ thT complex basis ai, ... ,a g of (W, J).

+ 6e)-l z

in terms of the

Proof. In this basis SA,J(V,V) is given by tz lm (T)-l z by the remarks in the proof of Theorem 5.8. Let r be point in Sg corresponding to J computed with respect to a1,'" ,a g , b1, ... ,bb. Similarly SAjv,v) is given by tZIm(7')-lzwith respect to the complex basis ai, ... ,a g • We know that z = e t( O'T + 6e ) -1 Z and 7' = (aT + fJe)hT + 6e)-l e. We also have the equation

t-I (-)-1-::z= t z Im( T)-1-Z zmT as H J is independent of the coordinates. Now QJ(v) = tZIm(r)-l z - tZ lm (T)-l z which is given in the ordinary bases by the matrix t{ e thT + 6e )-1} Im( (aT + fJe )hT + 6e )-1 e )-1 {e thT + 6e)-1} - (ImT)-l. By (*) we have t{e thT+6e)-1}Im((aT+fJe)hT+6e)-le)-1{e thr+6e)-le}

= (ImT)-l

.

So QJCv) is given by t{e thT + 6e)-1} t{ e thT + 6e)-1 }-l(lm r)-l(e thr + 6)-1 )-1 {e t X

hT+6e)-le}-(lmT)-1

= (1m T )-1 [thr + 6) thT + 6e)-1 - 1] = (ImT)-l [thr + 6) - thT + 6eWhT + 6e)-1 = (1m T )-1 [-2i t( "'{ 1m T) t( "'{T + 6e )-1] = -2i t"'{ t( "'{T + 6e )-1 .

0

Now .-vA has a Hermitian metric so that
1

K~ .K1=K~ when these isomorphisms are defined. Exercise 1. Give a formula for the metric on .-vA.

66

Chapter 7. Moduli Spaces

§ 7.5 Group Actions on the Families of Sheaves We continue with the situation of the last section. For all J in Reas(W), K(YA,J) is independent of J. In fact it is K = L~ fL. Let A(E) = A0rn.nL~ f L depend on the choice of the maximal isotropic subgroup A of L. For the next result we need a definition. Let X be a complex manifold on which a group r acts by biholomorphic mappings. Let !l' be a coherent sheaf on X. An action of ron!l' is the following data; for each element 'Y of r we have an isomorphism cx'Y : 'Y * !l' ~ !l' such that for any pair 'Yl and 'Y2 of elements we have a commutative diagram

K(J{J) =

CX'Yl ''Y2 :

('Yl . 'Y2)* !l'

-------+

II C'(2)*C'Y1 *!l')

!l'

i "." 'Y; ("'n)

-------+

'Yi!l'

or simple CX'Yl 'Y2 = CX'Y2 0 'Yi( CX'Yl)' If !l' = lfJx, cx'Y is multiplication by a no-where zero holomorphic flllCtion A i x) on X such that the factors of automorphy equation A'Yl''Y2 (x) = A'Y2 (x )A'Yl ('Y2 . x) holds. Now for each J we have an action of A(E) on YA,J given by transcribing via the automorphism
Lemma 7.11. We have a natural action of A(E) on YA via isometries. Proof. Let al, . . . ,a g , b1 , ... ,b g be a canonical basis of L such that A = fB71.aj. Thus an element of A(E) has the form a = E ~aj where the nj are integers. J We need to compute the operation on YA,J given by a where R(J) = T. Let 9 be a section of Y A,J. By definition a takes this to the section leu)

=

e-tSj(u,u)e+tsJ(u+a,u+a)g(u

because SJ( -, a)

= H J( -, a).

+ a)e-1rHJ(a,a)-fHJ(ii,ii) =

g(u

+ a)

Thus this action is independent of J.

We apparently did something trivial. To do something less trivial, let L

0

=

A fB B be a decomposition of L with respect to E. Assume that cxlB == 1. We have a similar group B(E) of K which acts on YB,J for all J. Similarly we have

§ 7.5 Group Actions on the Families of Sheaves

67

Lemma 7.12. We have a natural action of B(E) on YA via isometries. Proof. Now we assume that B = ffilLbj. A typical element B(E) is represented by b = I: ?-bj where the nj are integers. As before we need to see that J -SJ(u, u) + SJ(u + b, u + b) - 2HJ(u, b) - HJ(b, b) is holomorphic in J and u. The above expression equals +2SJ(u, b)

+ SJ(b, b) -

~

2H J(u, b) - HJ(b, b)

So the action of b sends g(u)

f-+

~

= -2i(2b JA , (u) + bJA(b)) , = -2i(2tnu + tnrn ) .

e- 21ri 'nU-1ri'nrn g (u - re-1n).

o

Now let H(A, B) = C* x A(E) x B(E) with the usual multiplication. We have a natural action of H(A,B) on YA. Thus we have an action of H(A,B) on the direct image 7r:YA which is a sheaf on Reas(W) where H(A, B) acts trivially. In other words we have an action of H(A,B) on each fiber 7r:YA,J = T(X(J),YA,J) which varies analytically with J. In fact for each J, H(A,B) acts by the theta group H(YA,J). There is another group acting on YA. Let TO/ be the subgroup of Sym71(L) consisting of clements which preserve a on L.

Lemma 7.13. We have a natural action of TO/ on YA by isometries. Proof. Let (7 be in Tex. Then "(* YA = Y~-l(A). The action is given by K A.-v,.-l(A). To check that this is group action we need to see that

1A

=

but

So the result follows because of the remark at the end of the last section. Everything is an isometry because (7*J{J = M~-lJ. o Taking direct image we have

Corollary 7.14. We have a natural action of Tex on the sheaves 7r:YA == QA on Reas(W) by isometries. Proof. The sheaf 7r:YA is a locally free sheaf on Reas(W) by Theorem 3.14. Its metric is given by integration of the inner product in YA over the fibers

68

Chapter 7. Moduli Spaces

of 7r' with respect to normalized relative Haar measure. The corollary follows directly from the properties of integration. 0 Now we have action of the two groups rex and H(A,B) on .Al'A. We may ask "How are they related"? Lemma 7.15. rex normalizes the action ofH(A,B). Hence we have a scmidirect product H(A,B) X rex acting on .Al'A.

Proof. Let m = (A, a, b) be an element of the compact group He( A, B) where IAI = 1. Let p be an element of rex' We want to show that (pmp-l)* = m'* for some clement m' of Hc(A, B). Now (pmp-l)* = p*-lm*p* is the composition of an isomorphism .Al'A,J --t .Al'A,p-t J with a covering of translation by a + b and the inverse .Al'A,p- t J --t .Al'A, J. Thus (pmp -1)j is a covering of translation by pea + b) and, hence has the form (p(J),c, d)*. As the above isomorphisms are all isometries, we must have Ip(J)1 = 1. Now p is an analytic function of J. Therefore it is constant. 0

Chapter 8. Modular Forms

§ 8.1 The Definition We can write a canonical invertible sheaf on space Reas(W) as a fixed real vector space W of dimension 2g. We recall that we have the homomorphic bundle B ~ Reas(W) on rank g. One simply considers the higher exterior power Ag B. The fiber of Ag B at J is canonically isomorphic to the g-exterior power of the tangent space of the abelian variety X(J) at the origin. As B is homogeneous with respect to the action of SymIR(V) so is Ag B. Recalling that B is Hermitian we see that Ag B is Hermitian in an invariant way but this is not the interesting metric on Ag B. We will define the related sheaves. Let Jf{' be the sheaf on Reas(J) such that a section Jf{' over U is a holomorphic g-form W on 71"-1 U which is invariant on the fibers of 71". Let "( be an element of SymIR(W). Then "{* of differential forms defines an isomorphism of "{* Jf{' ~ Jf{' which gives a ( continuous) action of the group SymIR(W) on Jf{' by the chain rule. The interesting metric on Jf{' assumes a given lattice L in W where E is integral on Lx L. We identify a section of Jf{' with a differential form on B / L in the obvious way. The interesting metric is given by (WI, (2) J = (2~)9 JX( J) WI /\ W2. By the change of variables in integrals, the group Sym 71 ( L) acts on Jf{' by isometries in this metric. Another definition is as follows. A modular form for a group T in SymIR(L) of weight k is a global section W of Jf{'0k which is invariant under T; i.e. "{*w = W for all "{ in T. If 9 = lone normally requires a growth condition at the cusps which I will ignore. Clearly the ring of modular forms of all weights forms a graded ring. Unfortunately these rings have not yet been determined explicitly (this is the first time we encountered abstract mathematics in this book). In this section I want to write the notions in terms of coordinates for comparison with the classical notation. Let al, ... ,a g , b~, .. . ,b~ be a symplectic basis for W. We have the bijection R: Reas(W) ~ Sg. Then 71" : B ~ Reas(W) is just the projection on the second factor 71" : (;g X Sg ~ Sg. Thus a section of H over an open subset 7I"-I(U) is an expression W = I( T )dz i /\ ... /\dz g where I is a holomorphic function of T on

70

Chapter 8. Modular Forms

U. Thus action of [:

~]

in SymIR(W) sends w into "'{*w

= f( (XT + fJ/("'{r +

6)) det("'{r + 6)-ldz 1/\ .. . /\dz g because "'{( z, r) = (("'{r + 6)-1 z, (ar 6)-1). Thus a modular form of weight k is an expression

where f is a holomorphic function of Sg satisfying f(( ar

f( r) det( "'{r + 6)k for all

[~ ~]

in

+ (3)("'{r +

+ (3)("'{r + 6)-1)

=

r. Such a function is called automorphic.

t

Let al,'" ,ag , b1, . .. ,bg be a canonical basis of L. Let bi = b; as usual. Let WI = f( r )dz 1 /\ • • • /\dz g and W2 = g( r )dz 1 /\ ••• /\dz g be two sections of :K. Then we want to compute their inner product.

Lemma 8.1. (Wl,W2)r A = ED,. Zla,..

= 21gf(r)g(r)(h(X(J),HJ,A))2 where R(J) = rand

Proof. By definition h(X(J), H(J), A)

= ..j2g (±dz1/\ .. . /\dz g , ±dz1/\ ... /\dz g ). o

Thus the geometric height gives the metric in :K.

§ 8.2 The Realtionship Between 1r~.AoA and H in the Principally Polarized Case In this section we assume that Idet EI = 1. Thus 7r~YA is an invertible sheaf. In the situation of Section 7.5 we have a group action of rex by isometries on Q(A) == 7r~YA and by the last section we have an action rex C Sym71(L) on :K with respect to metric defined by L. The obvious question is "Is there any relation between these two actions?". The key to understanding this problem is Lemma 8.2. Let !l' be an invertible sheaf with a Hermitian stritc.ture on a connected complex manifold X. Let r be a group acting holomorphically on X. Then any two actions of r on!l' which acts by isometries differ by a character

X:

r

~

U(l).

Proof. Let * and ** be the two actions. Then for each "'{ in r we have two isometries "'{*, "'{** : "'{*!l' ~ !l'. By Lemma 4.7 we have a unitary constant X( "'{) such that X( "'{ h* = "'{**. As * and ** are group actions, X is a character ~~ 0

§ 8.2 The Realtionship Between

7!"~.h'A

and II

71

To use this result in practice, let !l'l and!l'2 be two invertible sheaves with Hermitian structures on X. Assume that both !l'1 and !l'2 are given T-actions. Then if a : !l'1 ~ !l'2 is an isometry, there is a unitary character X of !l' such that a is isomorphism between sheaves with T-action where the action of T on !l'l is twisted by x. We will apply this principle when X = Reas(W), !l'1 = Q(A)0 2 and !l'2 = Jf{'0- 1. To define the isometry a : Q(A)0 2 ~ Jf{'0- 1 we use a canonical basis al, ... ,ab, bI, ... ,bg of L. Define a in coordinates by aU( T)1]) =

f( T) a~l /\ ... /\ a~9 det~~;1/8. By Theorem 5.9, this is an isometry. Thus we get Theorem 8.3. There is a character X : Tcx isometry of sheaves with T"'{-action

~

U(l) such that we have an

where [l denotes twisting the T"'{-action by the character

x.

With this theorem in hand all of our previous questions boil down to the single one, "What is X?". We can give an elementary result. Lemma 8.4. Let'Y be an element of Tcx such that 'Y(A) detC'YIA).

=

A. Then

xCv) =

Proof. In coordinates 'Y is given by [~ ~] where taeo = e. So det( 'YIA) = det( a) = det( 0- 1). Now 'Y* dz 1/\ . .. /\dz g = det( 0- 1)dz1 /\ • •• /\dz g because 'Y x Z = e t(oe)-1 z. Thus 'Y*-aa /\ .. . /\-aa = det( 'YIA)-aa /\ .. . /\-aa . To prove this %1 Z9 %1 Zg lemma it is enough to show that 'Y*1] = 1] or, what is the same as 'Y*1](00) = 1]( 00). One easily checks that 1](00) depends only on A not B. 0 Let T cx,A be the subgroup of all such a as in the lemma. Let T( n) be the kernel of SymIR(L) ~ GL71./n71.(L/nL). Then trivially we have T(4) C T(2) C T cx. In the next section we will prove Lemma 8.5. a) Tcx is generated by Tcx,A together with square roots of elements of Tcx,A. b) T(2) is contained in the group spanned by conjugates of Tcx,A in T cx . c) T(4) is contained in the group spanned by conjugates of Kerxlr",A in

Right now we can conclude

72

Chapter 8. Modular Forms

Theorem 8.6. a) X4 == 1, b) X2Ir(2)

==

1, and

c) Xlr(4) == L Proof. This follows immediately from the two Lemmas as X is a character. One should notice that Tex = Sym71(L) if

0

e is even.

Corollary 8.7. We have canonical isomorphism a) Q(A)0 8 ~ Jlt'0- 4 as Tex-metT"tc sheaves, b) Q(A)04 ~ Jlt'0-2 as T(2)-metric sheaves, c) and upto sign Q(A)0 2 ~ JIt'®-I as T(4)-metric sheaves.

Proof. The isometry a of Theorem 8.3 is determined upto sign by A. The canonical statement follows and for the rest see Theorem 8.3 and 8.6. 0 Next we will write the classical functional equation of the theta function in our language. There is a homomorphism of sheaves (3 : Q(A) ---+

7r'YA

given by (3(>..)

= >"(60)

.

From the definitions (3 is equivariant for the action T-y on the two sheaves. Thus we have an equivariant mapping 'Y = (32 0 a-I [X-I] : JIt'-1 ~ 7r~YA[X-I]. Let 10 be evalutions at 0, 7r~YA ~ ll'JRe",,(W) which is equivariant for the natural TO/ action by pull-back on i%eas(W)' Thus (100 'Y)dual : i%ea,s(W) ~ JIt'(X) is T-y-equivariant. Let M be the image of the function 1 is TO/-invariant. Then M is a modular form twisted by X for the group Tex but M is essentially (6010)2 dZ I t\ ... t\dz g •

§ 8.3 Generators of the Relevant Discrete Groups Let T be the standard integral symplectic group of the usual form E consisting of integral 2g

X

2g matrices of the form

[~ ~]

where

Let L = 71 2g be the lattice on which T operates naturally. A vector in L is given by (Xl' ... ,X g, YI,'" ,Yg) in coordinates.

§ 8.3 Generators of the Relevant Discrete Groups

73

Let n be a positive integer. Let T(n) be the subgroup of T of matrices of

~;a

the form [I

11n6]

where the

a, p, 6

group of similar elements of the form [I T(4) C

r'

are integral. Let T' be the

'Y and

~~a

I

~4E ]. We have inclusions

C T(2) CT.

[~ ~]

We need another group T(l, 2) which consists of all elements

of

T which preserves the quadratic form Q(x, y) = L:;=l XjYj modulo 2. Clearly T(2) C T(l, 2) C T. We want to find generators of the groups r', T(2), T(l, 2) and T. For I .:::; j .:::; g let Tg be the metric of the transformation of L which sends x i to Yi and Yj to -xi and leaves the other coordinates fixed. What we intend to prove is

Theorem 8.8. a) T' is generated by its elements of the form

[21

~]

and

[~ 2~]

where B is integral and symmetric. b) T(2) is generated by its elements of the form

[~

[~ 2~], [2~ ~]

tAO-I] where B is integral and symmetric and A

= I + 2C

and

where C is

integral. c) T(l, 2) is generated by Tb and

[~ ~]

. ..

,Tg and its elements of the form

[~

tAO-I]

where B is symmetric and has even diagonal.

d) T is generated by Tb ... ,Tg and its elements of the form [ AO and

[~ ~]

where B is symmetric.

Remark. d) may be proven easier directly but we will prove these statements in order. Proof. We begin with a modification of the usual procedure for finding the greatest common divisor of two integers.

Lemma 8.9. Let x be an odd number and y be an even integer. After a finite number of steps using transformations Sl : x ~ x and y ± 2x or S2 : Y ~ Y and x ~ x ± 2y, we may reduce (x, y) to (±d,O) where d = gcd(x, y).

74

Chapter 8. Modular Forms

Proof. Let (x', Y') be equivalent to (x, y) under the group generated by Sl and S2 such that Ix'i + Iy'l is minimal. Clearly as Y' is even and x' is odd, Ix'i < Iy'l if Y' =1= 0 and Iy'l < Ix'i. Thus Y' = O. Also we note that gcd(x, y) is invariant under the group. So the result follows.

0

To prove a) by an easy induction the result will follow if we can prove the following; given two vectors (Xl" ... ,X g, Yll ... ,Yg) = Kl and (UI, ... ,U g, VI, ... ,Vg) = K2 in L such that Xl == 1 == Vb X2 == V2 == Xg == 0((4)) the other entrees are even, and I:XjVj - I:YjUj = 1 there is a transformation in the groups generated by

[2~ ~]

and

[~

2t]

where

B

is integral and

symmetric such that Kl and K2 are transformed to the unit vectors el and eg+l· First if 2 ::; i ::; 9 we use the transformation

until by the lemma Y2 = ... = Yg = O. Then apply W : Xl, Yl f-+ Xl ± 2YI, Yl or Xl, Yl ± 2XI until YI = O. Next we want to make a series of transformations so that Xl divides X2, . .. ,Xg, Yb ... ,Yg. During this process we will preserve the condition XIIYl for 2::; j ::; g. For 2::; j ::; 9 we apply XI,Xj,YbYi ~ XI, xi> YI + 2Xi, Yi + 2XI so that YI = 2Xi. Then we apply process W to make Yl = O. Then by the lemma xl12xj and thus xllxi as Xl is odd. Therefore we have Xl divides all of the Xj and Yj. On the other hand as (Kb K 2 ) = 1, gcd({xj,Yj}) = ±1. Thus Xl = ±1 or, hence, Xl = 1 as Xl == 1(4). Next we repeat the procedure in the beginning so that Yl, ... ,Yg are zero. Then by a W-process we make YI = 2. Then we apply xb Xi, YI, Yi ~ Xl ± 2Yi, Xi ± 2Yb YI, Yi to make Yi zero as Yi == 0(4). Finally we apply W to make YI zero again. Thus KI is now el. SO VI = 1 as E(KbK2) = 1. We want to change K2 to e2 without changing el. IT 9 = 2, make U3 = ... = u g = 0 and U2 = 2 by U : Xl, Xi, YI, Yi ~ XI ± 2Yi, Xi ± 2YI, YI, Yi. Then make V2 = ... = Vg = 0 by X2,Xi,Y2,Yi ~ X2,Xi,Y2 ± 2Xi,Yi ± 2X2. Then make U2 equal zero by U transformation. Then if 9 ~ 1 just make UI = 0 by W s. This accomplishes the task. This proves a). To prove b) we have an injection i : T(2)/T(4) ~ Sym71/471(L 0 71./471.). The image of i is contained in the matrices of the form [1

~~a

1 ~26]

where a, fJ, 'Y, 6 are 9 x 9 matrices with coefficients in 71/271 which preserves E 071/471. Thus - t6 = a and 'Y and fJ are symmetric and the group law for these matrices is addition of the a, fJ, 'Y and 6. We will show that the proposed generators of T(2) generates these subgroups. The

[~

2f] and

[2~ ~]

are

§ 8.3 Generators of the Relevant Discrete Groups

· . t h· D [1+2a III e Image. ror 0 o b VIOUS

75

1 _O t2a] by a ddi· tlOn we may assume t h at

a is upper or lower diagonal or diagonal with only one in the ith-place. The first two cases are given by

[~

t1-

1]

in an obvious way. For the other just

take A to be the diagonal matrix with ones everywhere except for a minus one in the ith-place. The proof of c) is similar. Consider the injection i : T(I,2)/T(2) ~ 01l./21l.(L 0 71/271) where 0 is the orthogonal group of Q 071/271. One shows that the image of the would-be generators generate the orthogonal group. While we are at it we can do the same thing with d). Here we need to show that the generators generate Sp1l./21l.(L 0 71/271). Thus we need to find generators over a field F = 71 /271 in this case. What we need is

Lemma 8.10. a) OF( Q0F) is generated by the image ofTI , ... ,Tg,

[~

t1-

1]

where A is either a permutation matrix or I+F;j, i -1= j, where F;j is the matrix WI·th

. th e ('1,J') p1ace an d on1y ones m

[1

h 0 F;j + 1 Fj;] were I' .../.. r J..

b) Sp(L 0 71/271) is generated by the same plus

i

[~

j

] for all i.

Proof. As char F = 2, E is the bilinear form associated to Q. By the obvious induction on g to prove a) we need to show that if we have two vectors KI = (Xh ... 'X g, YI, ... ,Yg) and K2 = (Uh ... ,U g, VI, ... ,V g) in L071/271 such that E(K I ,K2) = 1 and Q(K I ) = Q(K2) = 0 we can transform them to el and eg+l using our generators. For b) we drop the condition on Q but have more generators at our disposal.

~

Using permutations and T's we may assume that Xl XI,Xj + XI,YI + Yj,Yj we may assume that X2 =

Q(K1 )

=

0 we have YI

=

O. Now TI ... Tg

=

= 1. Using xI> Xj, Yt, Yj ... = Xg = O. Thus as

[~ ~]

and

[~ ~] [~ ~]

1 F;j + 1 Fj;] = [01 1] 0 = [ F;j +1 Fji 0] 1 . Thus we may usc the transforms [o Xh Xj, y., Yj ~ X., Xj, YI + Xj, Yj + Xl i = 1, j = 2, ... ,g, we can assume that Y2 = ... = Yg = O. Thus KI is now el and, hence, VI = 1. As Q(K2) = 0 we have UI = O. We do Xl, Xj, y., Yj ~ Xl +X;, Xj, Yb Yj +YI to make V2 = ... = Vg = O. Then we do Xl, Xj, YI , Yj ~ Xl, Xj + Yb YI + Xj, Yj to make U2 = ... = Ug = O. This proves a).

76

Chapter 8. Modular Forms

. the new transfonn (*) and [01 For b) usmg may make YI = 0 after all x's are zero except UI = 0 if VI = 1. The rest is the same.

1] 0 * [01 Xl

form

[~1 ~]. Tl

[1 Fii

0] . We 1 0

= r(l, 2),

this contains

The group r""A consists of elements of the

[~ ~ ]. The conjugate by [~1 ~]

remaining point is that

=

and similarly we can make

Now we can give the proof of Lemma 8.5. Now rex Til'" ,Tg and hence

01]

has the form

[~ ~ ]. The only

is contained in rex,A. The rest follows directly.

§ 8.4 The Relationship Between

1r~-k4. and H in General

Now when detL(E) is general the sheaf 7r~YA on Reas(W) is locally free. Let A ffi B = L be a decomposition of L such that alA == 1 == alB. The canonical theory of theta function defines an isomorphism

We may multiply '102 by the section -bdZll\ ... I\dz g of H to get an isomorphism M: C[B[E]]02 ®l!'JReas(W) ---+ (7r:YA)0 2 0

I[:

Jf{'.

By the previous reasoning with heights and length of theta function we know that M is an isometry where the inner product on the first sheaf in induced by the natural one on crB[E]] for a definite choice of the constant C. We want to understand the action of rex on the second sheaf in tenns of M. We will prove ~ U where U is the unitary group of crB[E]] 02 such that the action r-y on (7r:Y A)02 0 Jf{' corresponds to the rex-action on crB[E]]0 2 ®I[: i%e",,(W) via ",,-10 (pull-back of functions).

Theorem 8.11. There is a homomorphism"" : r-y

Remark. It can be shown that Image of"" is finite. We will first prove Lemma 8.12. Let U be a finite dimensional Hilbert space. Let II II be the natural extension of its norm to a metNc on U ®I[: l!'Jx where X is a connected complex manifold. The global sections of U ®I[: l!'Jx with constant length are those of the form U 0 1 where u is a vector in U.

§ 8.5 Projective Embedding of Some Moduli Spaces

77

Proof. Let U = en with the usual norm. A section of U ®v (f) x is a vector (h (x ), ... ,Jn (x)) of holomorphic functions of x in X. We need to show that g(x) = I: j Ifj(x )1 2 is constant if and only if the /;,s are constant. We will show that if g(x) has a local maximum at Xo then the /;,s are constants. Choose local coordinates at Xo so that fj(x) is given by a power series I:(n) a(n),jz(n) which converges when IZjl':::; 1 for all j. Then La(o),ja(O),j j

~

L La(m),ja(m),jZn. (m)(m) j

zm

for

IZjl

=

1.

Averaging over U (1) n we get a(O),ja(O),j

L

i

=

L n

L

a(n),ja(n),j . j

Hence a(n),j = 0 if (n) 1= (0). Therefore each fj(x) is locally constant and hence globally because X is connected. 0

Corollary 8.13. Any holomorphic isometry of U ®v (f)x has the form where t/J is an isometry of U.

t/J 0 1

Proof. The isometry preserves the length and holomorphicness of sections. Hence it induces an isometry t/J01 of U01 which determines it by linearity. 0 Theorem follows formally from this corollary.

§ 8.5 Projective Embedding of Some Moduli Spaces Let W be a real symmetric space of dimension 2g with skew-form E. Let LeW be a lattice with elementary divisors ell .. . Ie g • Let A 0 B = L be a decomposition of L. Then as usual we have L~ = A~ ffi B~. Let r be the kernel of the natural mapping Sym71(L) ~ Aut(L~ /L). Here L~ /L = A~ /A ffi B~ / B == A(E) ffi B(E). Let a : L ~ {±1} be the unique solution of the equation a(ll + (2) = a(ll) oa(l2)(-1)E(II,12) and alA == 1 == alB. Then we have the invertible sheaf Y A on B / L. We have the universal theory of theta functions '1 : C[B[E]] ~ 7l'~YA. Then '1 is induced by a linear transform Tj: C[B[E]] ~ r(B/L,YA).

Lemma 8.14. If el

~

2 the image of Tj has no base-points.

Proof. For each J in Reas(W), restriction Tj : e[B] ~ r(X(J),YA,J) is an isomorphism. The assumption means that YA,J is an el -power. Thus this result follows from Lemma 2.7. 0

78

Chapter 8. Modular Forms

Let lP n be the projective space of hyperplane in C[B]. Thus Ob for b in B[E] are a basis of the linear forms on lPn. Also by the lemma we have an analytic mapping t/J : B j L ~ IP n defined by fi if el ~ 2. By the work of the last chapter it follows that t/J is r-equivalent for the trivial action on IPn. If el ~ 3 then L~jL contains the el-torsion in X(J) for all J. Thus by Theorem 9.7 r does not have a fixed point in Reas(W). Hence 7r" : r \ B j L ~ r \ Reas(W) is a family of abelian varieties parameterized by the manifold r \ Reas(W). Now we have an induced analytic mapp~g ~ : r \ B j L ~ lPn. Composing with the zero section we have another ~ : r \ Reas(W) ~ IPn. Theorem 8.15. (Mumford) If el is an even integer ~ 4 then ~ is an embedding. Proof. ~ sends a polarized abelian variety (X(J), H J) with an identification of A(E) ffi B(E) with K(.!l'(l, HJ)) to the point (oblabEB[E]) in IPn. By Theorem 6.12 and 6.13 these null-theta werte determine the equation of X(J) in lP n and the zero of the image. The rest of the marking A(E) ffi B(E) can be found by applying the automorphism Ie of IP n to the image of zero point where k is a point of A(E)ffiB(E). This shows that ~ is injective. Doing the same argument over Spec(C(€]j€2) we see that it is also infinitesimally injective. 0

Let Llb be coordinates in another copy of IPn. The universal family of abelian varieties with marked structure is given by X (a + b, a - b, X) loX (c + d,c - d,X) - X(c + b,a - b,x)loX(c + a,c - a,x) = 0 where X(e,/,x) = I:iEJ XU)Lle+i 0 LlJ+i in the usual situation. As (obla) satisfies these equations, we have the (Mumford-Riemann) X(a+ b,a- b,x)laX(c+d,c- d,xloX(c + d,c - b,X)oX(c + a,e - a,x) = O. These equations plus o~o = Lblo define an algebraic subscheme M.-R. of IP n containing the image of -;j;. One can algebraically prove that image of ~ is an open subscheme of M.-R. and has Zariski closed completement. This is discussed in [4] and [2]. In Igusa's book [1] the boundary of the moduli is studied analytically. As a final result about moduli we will sketch. Theorem 8.16. ~ is an unramified 22g -degree covering of its image. Proof. Let x be a point on a marked abelian variety X. Then by Exercise 1 of Section 6.3 we know that ~(x) = (oblx) defines the image X of X in IP n hut we don't know the zero point of X. The zero point y must satisfy (oble) = (Lble) which means that oble '" obl-e by the inverse formula. By this means that

§ 8.5 Projective Embedding of Some Moduli Spaces

79

e = -e as a point X. Hence the origin of X is only known upto 2-torsion in X. Now if "tF(xd = "tF(X2) then we have an isomorphism of affine space t/J: Xl ~ X 2 which takes 0 to a two torsion point such that t/Jlx.·

Chapter 9. Mappings to Abelian Varieties

§ 9.1

Integration

Let X

= V / L be a complex torus and let S be a smooth connected variety. Let

S. Let f : S ~ V / L be an analytic mapping. Then we flx) ~ fls). have the pull-back mapping of differentials f* : V* = So be a fixed point of

Lemma 9.1. a) f is determined by f(so) b) Conversely given a point x of X fls) these have the form f(so) and (7 in S the element a(A) for A in V* is

res,

J".

rex,

res,

res,

and f* : V* ~ fl s ). and a homomorphism a : V* ~ J* if and only if for a closed path evaluation at an element 1 of L.

Proof. For a) let (S,O) be the Wliversal covering space of (S, so) with covering mapping 7r. Then f 0 7r : S ~ X lifts to an analytic mapping f 07r : S ~ V where f07r = Vo where f(so) = Vo + L. Clearly (f07r(s), A} = foB 7r* J*(A) for all A in V*. Thus j07r is determined by J* and Vo. Hence f is determined by J* and f(so). For b) we define g : S ~ V by (g(s), A} = Jo" 7r*a(A) + k where k + L = x. Thus g is analytic and we want the composition to S ~ V ~ V / L to factor through 7r : S ~ S. This is exactly the given condition. So b) follows easily. 0

A special thing happens when X is abelian. Proposition 9.2. Let f be a meromorphic mapping of a smooth variety into an abelian variety X = V / L. Then f extends to an analytic mapping. Proof. In this case J* : V* ~ meromorphic differentials on S. We need to see that a differential in J*(V*) has no pole along any particular divisor D on S. By Theorem 2.11 we have a projective embedding i : X C IP n where i 0 f(s) = ((7o(s), ... ,(7n(s)) with merom orphic functions. Let (7i(S) have the worst pole of these functions on D. Then i 0 f(s) = ((7o(s)/(7;(s), ... ,(7n(s)/(7;(S)) is regular and non-zero generally along D. Thus f is analytic generally along D and, hence, J*(V*) consists of differentials which are regular along D. 0

82

Chapter 9. Mappings to Abelian Varieties

Exercise 1. Let f : X ~ Y be an analytic mapping between two complex tori. Then f(x) = f(O) + g(x) where g(x) is a homomorphism g: X ~ Y.

§ 9.2 Complete Reducibility of Abelian Varieties Let X be a complex torus and let Y be a subtorus. A complement to Y is a subtorus Z such that X = Y + Z and the intersection Y n Z is finite. In this case we have an isogeny Y ffi Z ~ X and dim Y + dim Z = dim X. Proposition 9.3. If X is an abelian variety then any subabelian variety Y has a complement. Proof. Let X = V / L and Y = W / M where W is a subspace of V and M = W n L. Let H be a polarization of X. Let U = {v E VIH(v,w) = 0 for all w in W}. This is a complex subspace of V and W ffi U = V as H is positive definite. As U = {v E VIE(v, 1m) = 0 for all m in M} where E = ImH. It follows that N = un L is a lattice in U. The subtorus U / N is complementary to Y by construction. 0

A complex torus is called simple if its only proper subtori are zero. With this definition we have Corollary 9.4. An abelian varieties X contains a finite number of simple subabelian varieties Xl, ... ,Xr such that the natural homomorphism Xl ffi ... ffi Xr ~ X is an isogeny. Proof. IT X is simple then there is no problem. Otherwise a minimal non-zero subabelian variety Xl of X is simple. By Proposition 9.3 Xl has a complement Y. As dim Y < dim X we may assume that Y has a finite number of simple sub tori X 2 , ••• ,Xr such that X 2 ffi ... ffi X r . Therefore the second statement of the corollary is true for these Xi'S. 0

Isogeny between two complex tori is an equivalence relation. Clearly it is transitive and contains the identity. The following lemma shows that it is reflexive. Lemma 9.5. Let f : X ~ Y be an isogeny of complex tori. Then there is an isogeny g: Y ~ X such that fog = (degf) ·ly.

§ 9.3 The Characteristic Polynomial of an Endomorphism

83

Proof. We may assume that f : VIL ~ VIM is induced by the identity on V. Then #( MIL) = deg f. Hence (deg f)M C L. Thus we have isogenies deg

I

I

9

VIM - - VI(degf)M ---+ VIL ---+ VIM and our lemma is obvious.

0

For X a complex torus VI L, the ring of endomorphism End(X) is a subring of End 71 (L). Hence End(X) is a finitely generated Zl-module. Let EndO(X) be the CQ-algebra End(X) ®71 CQ. As EndO(X) = {P E Homa::{V, V)IP(L ®L CQ) C L ®71 CQ}, we see that EndO(X) is an isogeny invariant and an isogeny has an inverse in Endo(X). Now let X be an abelian variety. By Corollary 9.4 we may find nonisogenous simple abelian varieties Y1 , ••• ,Yg such that X is isogenous to EBl:5:j:5:g ljn j for some positive integers nl, ... ,ng. Now EndO(X) = ®1:5:j:5:9 M(nj,EndO(y;)) where M(r, A) is the matrix algebra of degree r with coefficients in A. On the other hand EndO(Y,') is a division algebra because Y j is simple. Thus the problem of determining EndO(X) is reduced to the case where Y is simple. I refer the reader to Mumford's book [3] for an introduction to the classification of all such division algebras. The key idea is the Rosatti involution t/J f-+
§ 9.3 The Characteristic Polynomial of an Endomorphism Let f be an endomorphism of a complex torus X = VI L. Then f induces an endomorphism of L. The characteristic polynomial of is called the characteristic polynomial PI(t) of f. As is given by an integral matrix, the coefficients of PI(t) are integers. Furthermore f satisfies the characteristic equation PIU) = 0 as an endomorphism of X. Let g be the dimension of X. Then PI(t) has degree 2g and is given by

i

i

i

i

i

The constant term det is just the scalar by which /\ 2g acts on /\ 2g L H 2gH 1(X,Zl) ~ H2g(X,Zl). It is elementary that deti is the degree of Therefore we have proven

Lemma 9.6. a) PI(t) = deg(t . Ix - f) for all integers t. b) The characteristic roots of are algebraic integers.

i

As an application we will prove.

f.

84

Chapter 9. Mappings to Abelian Varieties

Proposition 9.7. Let X = V / L be an abelian variety with pola16zation H. Let Aut(X, H) be the group of automorphisms o(X which preserve H. Consider the restriction If'n : Aut(X, H) ~ Aut(Xn) for some integer n. a) If n ~ 3, If'n is injective. b) Kernel( 1f'2) is a finite group of idempotents. Proof. We will first prove that Aut(X, H) is a finite group. An automorphism of (X, H) is an automorphism of V which preserve both Hand L. Thus Aut(X, H) = unitary group of H n GL(L). As the intersection of a compact group with a descrete group is finite, Aut( X, H) is finite. Let If' be an element of Kernel( If'n). As If' has finite order, cp is semi-simple and has roots of unity as eigenvalues. Clearly it is enough to show that the order p of an eigenvalue '1 cannot be an odd prime or either two if n ~ 3 or four if n = 2. Now If' - 1 is zero on X n • Thus If' - 1 = na where a is an endomorphism of a. The eigenvalues k of a are algebraic integers as satisfy '1 - 1 = nk. Thus as a is integral, n deg (l('1) divides NormCQ('1- 1). This is impossible. Explicitly if p = 4 and n = 2, NormCQ(±J='I- 1) = 2 is not divisible by 22. If pis an odd prime or n ~ 3, NormCQ('1- 1) = ITl
§ 9.4 The Gauss Mapping Let S be a subvariety of dimension s of a complex torus X = V / L. By translation we identify the tangent space of X at any point with V. Let s be a smooth point of S, then we have the s-dimensional subspace T.S of V where T.S is the tangent space of Sat s. This defines a holomorphic mapping G : S.mooth

----+

Grass. V

into the Grassmann manifold of s-planes in V. The objective of this section is to prove

Proposition 9.10. We have two mutually exclusive possibilities; a) G is generically finite-to-one, and b) S = 7r- 1 S' where S' is a subvariety of an abelian variety X' and X ~ ;X' is a non-trivial surjective homomorphism.

7r :

Proof. First let Y be an irreducible closed analytic subspace of X. The complex torus (Y) spanned is Eo
§ 9.4 The Gauss Mapping

85

of an irreducible compact variety y2i by a morphism. Thus for dimension the above increasing union terminates in an equality (Y) = EO
Corollary 9.11. If D is an irreducible ample divisor on an abelian variety X, the Gauss mapping D.mooth ~ {Hyperplanes in V} is generically finite-to-one. Proof. As D is ample it can not be the pull-back of a divisor by a non-trivial surjection. 0 It would be interesting if this result is also true in characteristic p (even in the principally polarizied case).

Chapter 10. The Linear System

12DI

§ 10.1 When IDI Has No Fixed Components Let D be an ample divisor on an abelian variety X. Recall by Lemma 2.7 the complete linear system 12D I has no base points. Let

1. In other words for any effective divisor F of X, the space T(X, 19x (D - F)) is a proper subRpace of T(X, 19x (D)). We recall from Section 2.4 that a general divisor E in ID I is reduced and has the property that if E + x = E then x = O.

Theorem 10.1. If

IDI

has no fixed components, then

tS

an

embedding. Proof. We will first show that

88

Chapter 10. The Linear System 12DI

component C of (-G) + z, H - k does not contain z. Otherwise k in C would be contained H - Zj i.e. a component of ( -G) + 2z is contained in H which is forbidJen. Thus v is tangent to G at all its points and we are done as in the proof of Theorem 2.11. 0

§ 10.2 Projective Normality of 12DI Let !l' be the invertible sheaf ex (2D). Now !l' is normally generated if the graded ring EBnEZ T(X, !l'0 n ) is generated by T(X, !l'). This is equivalent to

Theorem 10.2. !l' is normally generated if and only if no point of (2)-1 K(JI) is a base-point of ID I. Proof. If n ~ 2 the multiplication T(X,!l') 0 T(X, !l'0 n ) ~ T(X, !l'0 n+l) is always surjective by Theorem 6.8 c). Thus !l' is normally generated if and only if the multiplication M : T(X,!l') 0 T(X,!l') ~ T(X, !l'02) is surjective. The other part of the theorem is equivalent to the statement that 0 is not a base-point of T(X, Jl0 POt) for any two-torsion point a of the dual variety X~. This follows because the Jl0 POt are exactly the Ilheaves T;JI where k runs through (2)-1 K(JI). Thus we want to show that M is surjective if and only iffor all a in (X~)2 there is a section
+ b~,

b~ - b~, X))

= Y(b~, x)lo . Y(b;, X)

for all character X of B(!l'h and b~ and b~ in B(!l'2) which are congruent modulo B(!l'). Recalling from Section 6.1 that the Y's and X's are basis of their respective spaces after we account for the obvious B(!l'h redundances. Fix b~ in B(!l'2) then the image of M intersected with (JY(b~, X) is Y(b~,x)lo . Y(bi,x) where b~ == b~(B(!l')). Thus M is surjective if and only if for all X and bin B(!l'2) Y(b',x)lo =1= 71 for some b' == b(B(!l')). Now by definition these Y(b', X) are the span of the sections of !l'02 = Jl0 4 = 2J( of 2*JI with X 2 -eigenvalue (a,b) 1-+ (e~2(a,b')-I,X(b)). Therefore they are the pull-back by 2 of a basis of Jl0 POt where ~ is the two-torsion point of X~ corresponding to the eigenvalue. As all of X 2 occurs this way we have shown

§ 10.3 The Factorization Theorem

that M is surjective if and only if for each a there is a section
89 J{

0 POi 0

§ 10.3 The Factorization Theorem Let E and F be effective divisors on an abelian variety X such that E + F is ample. We have the obvious inclusion r : T(X, ~Jx(E)) '--+ T(X, 19x (E + F». Theorem 10.3. If r is an isomorphism, then X is isomorphic to product XE X X F and under this isomorphi,~m E and F correspond to divisors of the form E' X X F and XE X F' where E' and F' are ample divisors of XE and XF respectively and F' gives a pnncipal polarization of X F • Proof. By Chap. 2 if we let YE and Y F be the connected components of K(l9 x (E)) and K(l9x(F)) then E = sF/(E') and F = sF 1 (F') where E' is an ample divisor on XE = X/YE and SE : X ~ XE is the projection and similar with F. Thus we have the sum S = (SE,SF) : X ~ X E X X F . We intcm\ to show that oS is an isomorphism and s-I(E' X X F ) = E and S-I(XE X F') = F. Now 19x (E) is trivial on XE and similarly with F. Thus the ample sheaf 19x (E +F) is trivial on the intersection XE nXF = Ker s. Hence the kernel of S is finite. By the Riemann-Roch theorem ~(E+FY = dimT(l9x (E+F)) where x = dimX. Furthermore x~!(EyE = dim T(l9XE (E)) = dimT(l9 x (E)) where XE = dimXE and similarly with F. Therefore J,(E + F)X = ....!..,(E')XE 1 x. XE. and x~,(F')XF ~ 1. By the binomial theorem a = "h(E+F)X = Ec+f=x ~Ec. Ff. As E and F are effective each one of these summand ~ O. We will

=

t.

estimate some. Consider b = ~EXE. -f\Ff where XE + f = x. Now ~EXE = XE. . XE S];;I(_l_,(E')XE) is equivalent to a coset of ker(sE) = S];;I(O) = YEo Thus b =

XE· a(YE · ~;). Now by the projection formula Yt:· Ff = '~F"(YE)' Ff but SF" lYE is a finite morphism onto its image say Z aw_ the degree = #(YE n YF). Thus

~~ = #(YE n YF)f where f = dimZ. As F' is ample (F~\Z) is a positive integer k. Therefore a ~ b ~ a' #(YE n YF) . k. Hence kers = YE n YF = 0 and k = 1 and b is the only non-zero summand of a. Reversing the role of E and F we see that J,Ec...J..,FxF is non-zero where e + XF = x. Thus e = XE C XF' and f = XF. In particular X E X XF has the same dimension as X and S has degree 1. Hence S is an isomorphism. The rest is clear. 0

YE'

Famous special case is Corollary 10.4. Let D be an effective ample division of an abelian variety. If

D gives a principal polarization, (X, D) is a product X(Xi, Di) of principally polarized abelian varieties Xi with irreducible divisors D j •

90

Chapter 10. The Linear System

12DI

Proof. If D = E + F is reducible then r is automatically an isomorphism of lines (!). Hence (X, E + F) factors as in the theorem. Repeat until the divisors 0 are irreducible. Another corollary is useful for studying an ample linear system abelian variety.

IDI

on an

Corollary 10.5. IDI = IMI + F where IMI is a linear system without fixed components on X = X M X X F where F = X M X F' and M = M' X X F where F' gives a pnncipal polarization of XF.

Proof. Let F be the greatest common divisor of IDI. Then let E = M + F for some E in IDI. By construction r : T(X, ex(M)) ~ T(X, ex(E)) is an isomorphism and Theorem 10.3 applies. 0

§ 10.4 The General Case Let D be an effective ample divisor on a abelian variety X. By the last two corollaries we may factor (X, D) as (Y, M) x X(Xi, Pi) where IMI has no base components on Y and Pi is an irreducible principal polarization of Zi. Thus by the Kiinneth formula

By the Theorem 6.4, we know the effect of (-1)* corresponds to the mapping ((-1)* f)(b) = fe-b) on functions on B(Jt0i). When j = 2, B(Jt02) consists entirely of 2-torsion. Hence ( -1)* is the identiy on T( X, J(02). In other words, every section of Jt02 is even. This implies that

~

IP n is an embedding.

§ 10.4 The General Case

91

Proof. We will first show that
O(-i + w) = a_2ze7rH(-i+w,-H)+fH(-2Z,-2z)O(+i + w) = C(w)O(z + w) as 2i E L. In coordinates Q = E ai,i at at; when ai,i is a matrix of constants. Now if O(v) vanishes at + w then

z

Chapter 10. The Linear System 12DI

92

Q(9(v + w)9(v - w))1 z = 22: aij f)~i 9(v + w)\ z f)~i 9(v - w)1 z =

2 2: aii f)Xf) 9( v ) 1_

z+w

i

axf)

i

9( v) I_

z-w

= 22: aii VA, l'l~. 9(v)l_z+w ax f). 9(v)1 _ 1 -z+w = 2C(w) "

L...J

aiif)Xf)

i

9(v)l_ axf) 9(v)l_ . z+w i %+w

Therefore if Q -lOwe would have a nontrivial equation satisfied by the gradient of 9 where 9 = O. This would mean that the image of the Gauss mapping Dsmooth ~ {hyperplane in V} is contained in a quadric hypersurface. This is 0 impossible by Theorem 9.11. Therefore Q = O.

§ 10.5

Projective Normality of

12DI

on X/{±l}

Let JI be an invertible sheaf on an abelian variety X = V / L. We will assume that !l' is symmetric; i.e. JI ~ (-1) * JI. If (a, H) is A ppel-Humbert data for JI this means that a(l) = a( -1)( = a(1)-I) for all I in L or rather a(l) = ±1 for all I in L. We want a geometric interpretation of these signs. Consider the explicit isomorphism i : JI ~ (-1) * JI which corresponds to the substitution f(x) to fe-x) in !l'(a,H). Clearly i 2 = identity. If x in X is a fixed point of the involution -1 of X(x is 2-torsion), then i defines an involution ix of the line 'Jllx. If ix = -1 we say that x is odd and otherwise x is even. If all x in Xn are even (e.g. JI is the square of a symmetric sheaf), then JI is called totally symmetric. If JI is totally symmetric then JI descends to an invertible sheaf M/(±l) on X/(±l). In fact sections of (JI/(±1)0n ) correspond exactly to sections of Jt0n which are invariant under T(X, iOn). We will assume that JI is excellent with respect to a decomposition AffiB = L of L and that T(X, JI) = 1. Then by Theorem 10.6 we have the very ample invertible sheaf (!l'/(±1))2 on X/(±l) where!l' = Jl 02 and we want to know when the graded ring R = Rn = T(X/(±l), (!l' /(±l))n) = T(X, !l'n)+ where + denotes the subspace fixed by T(X, in) is generated by its linear term R 1 • By previous work this is almost true.

EBn>o

EBn

EBn>o

Lemma 10.7. R is generated as a (J-algebra by Rl and R 2.

Proof. In Section 10.4 we have noted that T(X,!l') = T(X,!l')+. Ifm ~ 2 then by Theorem 6.8 c) the multiplication T(X, !l')0T(X, !l'0 m ) ~ !l'0(m+l») is surjective. Hence by taking invariants Rl 0 R m ~ R m+1 is surjective. 0

rex,

§ 10.5 Projective Normality of

12DI on

X/{±l}

93

The main result of this section is

Theorem 10.8. R is generated by Rl as a (J-algebra if and only if a non-zero section of JI does not vanish at any even two-torsion point of X. Proof. By the previous lemma for the first condition we need to see when M : Rl 0 Rl ~ R2 is surjective where R2 = T(X, ,2'1812)+. For the second condition as in the proof of Theorem 10.2 we need to know whether a non-zero section of JI 0 POI does not vanish at zero where JI 0 POI r:::i T; JI where x is an even two-torsion point of x. Let x == ~a + ~b where a and b are in A and B. Then as JI is excellent, a(a + b) = (_l)E(a,b). Thus we want to have E(a, b) even. By Proposition 1.8 the character X' of X 2 given a has the form X'(~Q+

H) =

(_l)E(.lI.,Q)(_l)E(~,.lI.).

Consider a character X of ~B/B, let Y(b,X) = E'IEX2 X(7])OIi+'1 for bin iB/B as usual. Then i*(Y(b,X)) = X(2b)Y(b,X) by the obvious manipulation. Thus R2 corresponds to the span of Y( b, X) when X(2b). Therefore by the same reasoning as in the proof of Theorem 10.2, the first condition is verified if and only if Y(b~,x)lo =1= 0 for all b~ in iB/B such that X(2bD = 1. In this case Y (b~, X) I0 =1= 0 if and only if the non-zero section of Jl0 POI does not vanish at owhere a is given by the character X" ( ~Q+ = (e£!'4( ~Q, b~)-l, X( Now e(~a, bD- 1 = e2"'iE(h,b~)(4)(_1)EC.lI.,4b~) by Lemma 4.2. This corresponds to an even two-torsion point by an easy calculation if and only if X(2bD = 1. 0

H)

Exercise 1. Show that dim R' = ~(2i)g

+ 2; .

H)).

Chapter 11. Abelian Varieties Occurring in Nature

§ 11.1 Hodge Structure Let M be a free abelian group of rank 2g. An elementary structure on M is a complex subspace U of M 0z (J such that M 0z C = U ffi U where - is the linear mapping induced by complex conjugation in (J. Thus dimv U = dimv U = g. Let 7ru and 7r'fJ denote projective of M 0z (J onto U and U. There are two complex tori associated with an elementary structure. Let P(U) be U/7r'fJ(M) and let A(U) be Homv(U,(J)/M.L where M.L is the subgroup of Homv(U, C) consisting of linear functionals A such that A 0 7ru + A(1i[;) == 7r(A) has integral values on M. Lemma 11.1. a) Both P(U) and A(U) are complex tOrt. b) We have a canonical isomorphism

Proof. For P(U) we need to see that 7r'fJ : M ®71 IR ~ U is injective (and hence an isomorphism). Let m be an element of the kernel. Then m = iii and m = (u,O). So m = (u,O) = (O,ii). Thus u = 0 and m = O. For A(U) we need to show that a : M.L ®v IR ~ Homv(U, (J) is an isomorphism. The point is that 7r defines a real isomorphism 7r : Homv(U, C) ~ Homz(M, IR). This follows because 7r(A) satisfies 7r(A)(V) = 7r(A)(V) and hence has the form fL 0 (J for fL in Homz(M, IR). The rest of a) follows easily. For b) by § 1.4 Pic°(P(U)) = HomR(U, IR)/ Homz(7r'fJ(M), Z) where the complex structure on the Hom is given by iK(ii) = -K(iii). Let H(K) be element of Homz(M, IR) defined by H(K) = K 0 7r'fJIM' Consider C : HomlR(U,IR) ~ Homv(U,(J) which equals 7r- 1 H. Clearly C is a real linear isomorphism taking Homz(1rfj{M),Z) to M.L. It remains to check that C(iK) = iC(K). Let A = C(K). By the definition we have 2ReA(u) = K(ii) for all u in U. If ~ = C(iK) we have 2Im~(u) = +2 Re(-i)'(u)) = 2Re(~(-iu)) = (iK)( -iu) = -K(i. iii) = K(ii). Thus Re A = 1m ~ and hence iA = i 0

96

Chapter 11. Abelian Varieties Occurring in Nature

Let V / L be a complex torus. We may canonically construct an elementary structure U on L together with an isomorphism P(U) ~ V/L. Just let U be the kernel (L ®z (J ~ V). Then dim U = dim V. To check that this is an elementary structure note that un U = kernel(L ®z IR ~ V) = {O}. Clearly the above arrow induces an isomorphism U ~ V which identifies 1rfJ(L) with

L. Continuing with these notations let E be a real skew-symmetric form on V. 'lhen we have the natural complex bilinear extension E of E to V ®1R (J =

L®z (J. Lemma 11.2. The following are equivalent a) E(iv, iw) = E(v, w) for all v and w in V, and b) Eluxu == 0, Proof. Any element of U has the form v0i-(iv)01 for some v in V, Now E(u0 i - (iv) 01, w 0i -(iw) 01) = E(v,w)i 2 -E(iv,w)i -E(v,iw)i + E(iv,iw) = (-E(v. w) + E( iv, iw)) + ieEe iv, w) + E( v, iw)), Thus the equivalence is clear,

o If E is integral on L x L, then it is the Chern class of invertible sheaf !l' on V / L if and only if a) or b) is verified. In this case we want to consider when the Hermitian form H on V with imaginary part E is positive definite,

+

Lemma 11.3. H is positive definite if and only if E( u, it) is positive definite. Proof, We just compute E(v 0 i - (iv) 01, -v 0 i - (iv) 0 1) = E(v,v)i

2

+ E(iv,v)i -

= i(E(iv, v) -

E(v,iv)i + E(iv,iv)

E(v,iv)) = i(E(iv, v)

+ E(iv, v» o

= 2iE(iv,v) = 2iH(v,v).

Summing up we have proven Proposition 11.4. There is an equivalence between polarized abelian varieties (V / L, H) and elementary structures U on L together with an integral skewsymmetric form E on L such that Eluxu == 0 and E( u, u) is positive definite on U,

t

The last bunch of data is called a polarized Hodge structure of weight one,

§ 11.2 The Moduli of Polarized Hodge Structure

97

§ 11.2 The Moduli of Polarized Hodge Structure Let E be a fixed nondegenerate abelian integral skew fonn on a rank 2g free abelian group L. Let Hodge( L, E) be the set of all polarized Hodge structure based on (L, E). This consists of all 9 dimensional complex subspaces U of L ®z (J such that 1) U ffi U = L ®z (J, 2) Eluxu == 0, and 3) tE(u,u) is positive definite on U. Actually 2) and 3) = : } 1) because E is zero on un U but tE(u, u) is positive definite. Hence U n U = O. To parameterize these U's we regard them as points in the Grassmannian of g-dimensional subspace of L ®z (J. Then condition 2) means that U is an isotropic subspace. This is a complex analytic condition. In the space of all g-dimensional isotropic subspace is a compact homogeneous space under the complex symplectic group of E. The condition 3) is open and real analytic. Thus the natural moduli space for polarized Hodge structues is an open subset M of the space of all isotropic g-dimensional subspace of L ®z (J. Fortunately we won't need global coordinates to describe M because it lies in a local coordinate neighborhood. The crucial fact is as follows. Let AffiB = L be a decomposition of L.

Lemma 11.5. Let U be a Hodge structure on L. Thus U is the graph of a homomorphism cpu : B ®z C ~ A ®z (J. Proof. We first see that UnA ®z (J. Hence tE(w, iii) is zero because A is isotropic. Hence by 3) w is zero. Thus the projection U ~ B ®z (J is an isomorphism. So U is a graph. 0 The graph cpu gives local coordinates in M. Let us compute all example. Assume that U comes from a polarized abelian variety (V/ L, H) where E = ImH. Let a!, ... ,a g , bI, ... ,bg be a canonical base of L with elementary divisors elle2 •. . Ie g • Let bj = b, 0 elj' Then cpu is given by the matrix (7 such

E

that cpu(bj) = ak 0 (7j where (cpu(bj), b~.) is in U, i.e. bj + Ek ak 0 (7j = O. Thus our matrix (7 is just the usual T attached (V/ L, H) with respect to ai, ... ,a g , b~, . .. ,b~. Thus M is biholomorphic to Reas( L ®z IR) as the coordinates are the same. From the point of view of Hodge structures these coordinates are unnatural because they are taken with respect to coordinates around the isotropic subspace A ®z (J which is not a Hodge structure. The obvious question is, "What happens if we take coordinates around a fixed Hodge structre Uo?".

98

Chapter 11. Abelian Varieties Occurring in Nature

Consider the direct sum Uo ffi U 0 = L

®z (J.

Lemma 11.6. If U is any Hodge structure,

un U o =

{O}.

Proof. -tE(u,u) is positive definite on U but negative definite (why?) on U o. Thus

un U o =

{O}.

0

Thus we may write U as the graph of a homomorphism ¢u : Uo ~ U o. We want to compute the matrix representing ¢u for a basis in this space for a particular Uo . Let Uo be the Hodge structure with (7 = iIg • Thus Ck = bj+aj0i are a basis for Uo and d k = Ck = bj - aj 0 i are a basis of U o• Let ¢U(Ck) = E, W~dk. Also
Lemma 11.7. T=i{I+w)(I-w)-I, andw=(T-lI)(T+iI)-I. The set E of all possible w is the space of symmetric matrices w such that I - will is positive definite. The details may be found in Siegel's book [7] on page 8. The main point of this is that E is a bounded domain. Thus we have biholomorphic identification Reas(L ®z JR) ~ Sg ~ M ~ E.

§ 11.3 The Jacobian of a Riemann Surface Let C be a compact Riemann surface ((J-analytic curve) of genus g. Thus Hl((J,Z) ~ Z as C is oriented. Therefore the cup-product n : HI(C,Z) X HI(C,Z) ~ H2(C,Z) define a skew-symmetric integral-valued form E on HI((J, Z) = L.

If [w] denote the cohomology class of a closed one-form W E([WI]' [W2]) = f()wI /\ W2. Recall that the space of abelian T( C, fle) is a g-dimensional complex subspace and [ ] induces of T(C, fle) into HI(C, C) = HI(C, Z) ®z (J = L 0 (J. Let U be T(C, fle).

on C, then differentials an injection the image of

Proposition 11.8. U is a Hodge structure on L.

Proof. If WI and W2 are holomorphic differentials WI /\W2 is zero. So E([WI]' [W2]) = fewl /\ W2 = O. So P is isotropic. If W is non-zero holomorphic differential,

.

then ~w /\ w is a non-negative measure on (J which is zero at only a finite number of points. Thus tE([w] /\ [w]) = few /\ W > O. Thus we are done. 0

--

§ 11.4 Picard and Albanese Varieties for a Kahler Manifold

99

Let AI>." ,Ag, BI> .. ' ,Bg be a basis of Hl(C, Z) in standard position. Let al>." ,ag, bl>'" ,b g be their dual basis in Hl(C, Z). This is a canonical basis of HI (C, Z). We would like to compute the T-matrix of this Hodge structure. Let WI>'" ,Wg be abelian iategrals normalized such that fBk wi = oj. Then Tt fA Wk. The conditions that T is an element of the Siegel's space Sg are called Riemann's bilinear equations and inequality. Thus P(U) is a principal polarized abelian variety. Hence we have a canonical isomorphism A(U) = PicO(P(U)) and P(U). This abelian variety P(U) is called the Jacobian of C. One must be careful to distinguish between the various equivalent forms of the Jacobian but the autoduality of the Jacobian is an important principle. Next I want to discuss the most direct relationship between the abelian varieties P(U) and A(U) and the curve C. First of all P(U) is the Picard variety Pico(C) of C; i.e. Pico(C) = kerncl(Hl(C, eo) ~ H2(c, Z)) where

e is

2",.-

the boundary in the exact sequence 0 ~ Zc ~ e c c---+ eo ~ O. This follows cohomologically as follows. We have an exact sequence 0 ~ HI (C, Z) ~ Hl(C, ex) ~ Pico(c) ~ 0 by diagram chasing. Now Hl(C, ex) is represented , by the periods of antiholomorphic differentials P and a corresponds to the projection of L on this subspace. For A(P) it is naturally the Albanese variety of C. The complex torus X with a universal analytic mapping S : C ~ X. By Section 9.1 to define S we need to give a C-linear mapping R: Homv(T(C,il),C)* ~ T(C,il) such that for all closed paths 'Y in C, then linear functional x in Hom( T( C, il, C))* defined by f-y R(w) = (x,w) in L.l.; i.e. (X,Wl) + (X,W2) in integral if (WI>W2) is in Hl(C, Z). This is just f-ywi +W2 = b,Wl +W2}. Thus L.l. is the smallest lattice which satisfies the condition. Therefore we have an integral f : C ~ X and it i;; clearly universal.

§ 11.4 Picard and Albanese Varieties for a Kahler Manifold Let Y be a compact Kiihler manifold with Kahler form w. By Hodge Theorem HO(X, ilx) EB HO(X, ilx ) = Hl(X, Z) ®71 C. Thus we have an elementary structure HO(X, ilx ) on Hl(X, Z)/(torsion). Let PicO(X) be P of this Hodge structure and Alb(X) be A of it. Then PicO(X) = {o : Hl(X,e;c) ~ H2(X,Z)} as before as Hl(X,ex) = HO(X,ilx). Similarly with the one dimensional case we have a universal integral f : X ~ Alb(X). One may ask when these tori are abelian. There is a skew-symmetric form E on HI (X, Z) ®71 C given by [0"11\ 0"21\ [w]dim X-l]X. By local calculation we have HO(X,il x ) is isotropic and tE[O",a-] is positive definite of HO(X,il x ). Thus we have all the conditions but that E is integral on HI(X, Z)/(torsion).

100

Chapter 11. Abelian Varieties Occurring in Nature

Now Y is projective variety and w represents first Chern class of an ample line bundle. Then w and hence IE is integral. Therefore we have proven

Proposition 11.9. The Picard and Albanese of a smooth projective vanety are a dual pair of abelian varieties.

Informal Discussions of Immediate Sources

The statement of the Appel-Humbert Theorem 1.5 is due to Mumford [3J. Also I follow his idea for the proof of the existence Theorem 2.2. Some of the material in Chap. 1 is to introduce the reader to the abstract statemcnts used since A. Wei! in the algebraic casco The Scct. 3.4 is modeled on H. Langes lecturcs notes wherc he uses idea::. from Umemura [8J. Thc idea of Chap. 4 comes from Mumford's famous paper [4J. Lemma 4.6 is founded in [5J. The Theorem 5.9 and its application in Chap. 8 to the functional equation of the theta function can be found in Igusa [lJ. The algebraic material is Chap. 6 was initiated by Mumford [4J, added to by Koizumi and Sekiguchi and completed in [2J. Section 8.3 is close to Mumford's discussion in [6J. Proposition 9.7 is due to Serrc. Proposition 9.10 is due to Z. Ran. Scction 10.1 to 10.3 are an adaptation of H. Lange's notes again. He attributes Theorem 10.1 to Obuki. The Theorem 10.6 is due to Sasaki by a different method. Lange and Narasimhan have informed me that they have another proof. The point of vicw of Chap. 11 is from that of Deligne's idea of a Hodge structure.

References

[1] J.-1. Igusa: Theta Functions. Springer, New York 1972 [2] G. Kempf: Projective Coordinate Rings of Abelian Varieties. Algebraic Analysis, Geometry and Number Theory. Edited by J.1. Igusa, Johns Hopkins Press 1989, pp.225-236 [3] D. Mumford: Abelian Varieties. Oxford University Press, Oxford 1970 [4] D. Mumford: On the Equations Defining Abelian Varieties. Invent. math. 1 (1966) 287-354 [5] D. Mumford: Varieties Defined by Quadratic Equations. In: Questions on Algebraic Varieties. Centro. Intern. Mate., Estrivo, Roma, 1970, pp.31-100 [6] D. Mumford: Tata l.€ctures on Theta 1. Prog. in Math., Vol. 28. Birkhauser, Boston 1983 [7] C.L. Siegel: Symplectic Geometry. Academic Press, New York 1964 [8] H. Umemura: Nagoya Math. J. 52 (1983) 97-128

Subject Index

Abelian variety 13 Addition Theorem 46 Albanese variety 99 Appel-Humbert data, 4 Theorem 4 Complex torus 1 ExceIlent sheaf 38 Factors of automorphy 4 Hodge structures 95 Isogeny 2 Jacobian 98

ImDI

where m :2: 3 16 m=2 88 Modular form 69 Mumford's Embedding Theorem 78

PicO(V/ L) 5 Picard variety 99 Poincare sheaf B', 6 cohomology of 27 r.pz

7

Reas(W) 55 Riemann-Roch Theorem 26 Sheaf fi'( a, H), 4 sections of 9 matrix on 9 cohomology of 25 Siegel space Sy 58 Theorem of the cube 8 Theorem of square 14 Theory of theta functions 35 length of 35 analytic theory 39 Theta group H(fi') 29