Table Of ContentShimura Varieties of W@~ght Two Hodae Structures
by
James A. Carlson and Carlos Simpson
.I Introduction
Consider a variation of Hodge structure of weight two,
v : Y ~ F\D.
Griffiths transversality [6,7] asserts that vectors tangent to the image
of v lie in the so-called horizontal tangent bundle Th(F\D), which has
fiber dimension h2,°h ,I,I so that
(i.i) dim v(Y) <_ ,2h 1 hl, 0
When h2, 0 > 1 the horizontal vectors define a non-integrable
distribution in the holomorphic tangent bundle. But v(D) is an integral
submanifold for Th, so that additional restrictions hold, namely
i l,lh0,2h
(1.2) dimv(X) ~
In the category of abstract variations of Hodge structure this bound is
best possible [2]. The purpose of this note is to show that the bound
is also best possible in the category of geometric variations of Hodge
structure.
To make the notion "geometric" precise, we follow [8]: Let AF be
the category whose objects are smooth families of algebraic varieties,
X/Y = ~[ : X ) Y], and whose morphisms are given by commutative
diagrams of morphisms of varieties. Let GVHS 0 be the category whose
objects are the variations of Hodge structure Rk~,C defined by objects
of AF and whose morphisms are those induced from AF. Let GVHS be the
smallest abelian category which contains GVHS 0 and which is closed under
formation of tensor products and duals.
The weight two variations to be constructed satisfy a symmetry
condition: their Hodge structures admit an automorphism J such that )i(
j2 = -i, (ii) S(Jx, Jy) = S(x,y), where S is the polarizing form, and
(iii) JIH 2,0 = +i. We shall call such an object a J-Hodae structure.
J-Hodge structures are classified by a space with a transitive U(p,q)
action with isotropy subgroup U(p)×U(q), so that there is a natural
identification with the generalized unit ball,
m t
(1.4) B = { complex p×q matrices I ZZ < I }
Pq
(See [9], p. 527, domains of type A III). Moreover, the natural family
of J-Hodge structures over Bpq actually defines a variation of Hodge
structures with h 2,0 = p and h I,I = 2q. Since dimBpq = pq =
(i/2)h2,0hl, ,l the bound of (1.2) is satisfied. We shall also refer to
variations of J-Hodge structures as ~li~.
To show that unitary structures arise from geometry, we consider
unitary weight one structures: structures admitting an automorphism J as
above, but where the dimension of the +i-eigenspace of JIL 1,0 (call it
p) is arbitrary. Let E be the Hodge structure on 1 H of the elliptic
curve with period lattice generated by 1 and i. Then E carries a
natural J-structure, induced by multiplication by i on the underlying
curve. Let L®E be the tensor product of these two unitary structures,
and observe that J~J is a morphism with eigenvalues ±I. Define the
Prvm structure associated to L to be the -i eigenspace of J®J:
(1.5) P(L) = kernel [J®J + I: L®E ) L~E ].
This structure is unitary with respect to J®l. The main result is
then:
Theorem (1.6) If (H,J) is a J-Hodge structure of weight two, then there
exists a unitary weight one structure (L,J) such that
(P(L), J®l) _= (L,J) .
Corollary (1.7) Every J-Hodge structure of weight two arises from
geometry.
2. Unitarv variations of Hod~e structure
To describe the variations which realize the bounds we fix the
following terminology. A ~ is a pair (A,S) consisting of a
torsion-free abelian group A and a non-degenerate integer-valued
bilinear form S on A. A complex structure on a lattice is an
endomorphism J of A satisfying (i) S(Jx, Jy) = S(x,y) for all x and y,
and (ii) j2 = -i. We shall also call a lattice with a complex structure
a J-lattice. A morphism of J-lattices #: A 1 ---9 2 A is a group
homomorphism satisfying )i( J2~ = ~Jl, (ii) S 2(~x,~y) = S l(x,y) if the
two forms are either both symmetric or both antisymmetric, (iii)
S2(~x,~y) = Sl(JX, y) otherwise. Note that in general the bilinear
form
(2.1) JS(x,y) = S(Jx, y)
is symmetric if J is antisymmetric (and viceversa) . A J-polarized Hodae
structure consists of )i( a polarized Hodge structure and (ii) a J-
structure on the underlying lattice, where J is a morphism of Hodge
structures.
J-lattices exist. For a trivial example, take a lattice (L,S),
consider the doubled lattice (L,S+) = (L@gL, S~S), then set J(x,y) =
(_~-l(y), ~(x)), where ~ : L ---9 L is an isometry. The 8 E lattice
4
gives a nontrivial (indecomposable) example. To see this, view 8 E in Q8
with root system A given in the standard basis by
i ± ± e ej (I ~ j < i ~ 8) ;
(1/2) (~ Eiei) , where i e = ±i, ~i = -I °
Let J be the linear transformation of Q8 defined by
J(Xl, ... ,x ,4 5 , x ... , x )8 = (-x 5 , ... , -x 8 , x ,I ... ,x )4
and observe that J(~) = ~ , so that J restricts to an automorphism of
the 8 E lattice of the required type. One may construct pairs (S,J) with
associated hermitian form of prescribed signature (p,q), as in the
matrix example below:
p 2 1 [ J 0
, J = , where Jp =
S+ = 0 -I 2 0 Jq
p
Let D = D(A,S,n,h) be the classifying space for S-polarized Hodge
structures of weight n with Hodge numbers hP, q = hP,q. (We suppose here
that hP, q = 0 for p < 0.) According to [7], D is a homogeneous space
for a real Lie group G(D) = SO(AR, ) ~ S SO(a,b,R), where in general K A =
A®K. The discrete subgroup F = SO(Az,S) acts on D to give an analytic
quotient space F\D which is quasiprojective in the Hermitian symmetric
case [i].
Fix a complex structure J on (A,S), fix integers ~P,q, and let
D(J,~) be the set J-polarized Hodge structures in D for which the
dimension of the +i-eigenspace of JIHP,q is ~P,q. Thus D(J,~) is a
homogeneous space for the real Lie group G(D,J) = SO(AR, S, J) consisting
of elements of SO(AR, )S which commute with J. To identify this group
consider the decomposition C = A A+ ~A_ into the ±i-eigenspaces of ,J an
orthogonal decomposition relative to the hermitian form h(x,y) =
ihS(x,y) associated to S. Then
(2.2) the restriction map g ) glA+ defines an isomorphism of
G(D,J) with the unitary group of h on A+ --- an indefinite
unitary group of type U(c, d).
Once again, analytic quotients F(J)\D(J,~) are defined, where F(J)
denotes the (arithmetic) subgroup of G(D,J) which preserves the lattice. ~
When the isotropy subgroup of a reference structure F*, written
G(D,J,F*), is maximal compact, i.e., of the form U(c)×U(d), then D(J) is
hermitian symmetric and the discrete quotient is quasiprojective.
We now study the construction D(J) in the weight one case. Let z L
denote the underlying lattice, and let L = (Lz,S,F*) denote an S-
polarized Hodge structure of weight one and genus g: dim 1,0 = g. L Let p
denote the dimension of the +i-eigenspace of J on L, and write Hg for D,
Hg(J,p) for D(J,~), where Hg is the Siegel space of genus g.
Theorem (2.3) Let Hg(J,p) be a weight one unitary space. Then
)a( the isotropy group of a reference structure is isomorphic to
U(p)×U(q), where p+q = g,
)b( Hg(J,p) is hermitian symmetric,
)c( Hg(J,p) is a complex submanifold of Hg.
Proof : Because J is a morphism of Hodge structure, one has a
decomposition L+ = L+ 1,0 9~ 1 L+0, Let h(x, )y = iS (x,y) be the Hermitian
form associated to S. The Riemann bilinear relations imply that h is
positive on L+ 1,0 and negative on L+ 0,I, hence is of signature (p,q) .
Since an element g E G(Hg, J) preserves the type decomposition on L+, it
maps to an element of U(p)XU(q); the map G(Hg, J,L) ~. > U(p)XU(q) is
then easily verified to be an isomorphism.
To see that Hg(J,p) is Hermitian symmetric, observe that the
associated special unitary group, SU(p,q) also acts transitively, with
isotropy group S(U(p)xU(q)). This latter group is maximal compact with
one-dimensional center, as required.
To show that Hg(J,p) is a complex submanifold of Hg one may use the
fact that the imbedding is defined by an imbedding of Lie groups in
which the (one-dimensional) centers of the isotropy groups correspond:
(2.4 )a. Hg --_- Sp(g)/U(g)
(2.4 )b. Hg (J,p) = SU ,p( q)/S (U )p( xU (q))
where g = p+q. One may also give a direct argument. The complex
structure on Hg is that given as an open subset of a subvariety of a
Grassmanian:
(2.5) Hg C Hg = {F C C L I dim F = g and S IF = 0 } ,
where F = 1,0. L The locus Hg(J,p) is defined by the Schubert conditions
(2.6) dim(F ~r L+) = p and dim(F N L_) = q ,
and so is a complex submanifold.
The discrete quotient Y = F(J)\Hg(J,p) is quasiprojective, and is
an instance of a Shimura variety [10,11,12,13]. Replacing F by a
subgroup of finite index which acts without fixed points, one may
construct a family of Abelian varieties A/Y which admits a globally
defined endomorphism J of square -I. Thus Y classifies "Abelian
varieties with additional structure" (i.e. J).
We study next the weight two case:
Theorem (2.7) Let D(J) be a weight two unitary space with 2,0 = h p,
h I,I = 2q, and JIH 2,0 = +i. Then
(a) the isotropy group of a reference structure is isomorphic to
U(p)×U(q),
)b( D(J) is Hermitian symmetric,
)c( D(J) is a complex submanifold of D,
)d( D(J) is tangent to the horizontal distribution of D,
)e( D(J) defines a variation of Hodge structure of maximal
dimension.
Proof: Denote the underlying lattice by z H and fix a reference Hodge
filtration F~ E D(J) . Since J preserves H I,I, there is an eigenspace
decomposition I,I = H H+ I,I • H_ I,I Let c H = H+ ~ H_ be the eigenspace
decomposition on the complexification of the lattice, and let h(x,y) =
-S(x,y) be the Hermitian form associated to S. By the Hodge-Riemann
bilinear relations, hIH+ has signature (p,q) ° An element g E G(J,F*)
preserves the refined type decomposition
(2.8) C H = [H2, 0 ~ H+I, ]I ~ 1 ~ [H.I, H0,2],
where the first term in brackets is H+ and the second term is H_.
Therefore gIH+ lies in U(p)xU(q). This proves (a), and )b( follows,
since the isotropy group of the associated special unitary group is
maximal compact with one-dimensional center.
To prove holomorphicity, note that by (2.8) D(J) is (once again),
defined by a Schubert condition: 2 C F H+.
To prove horizontality, we use the dual Schubert condition H+ =
[H 2,0 ~ H+ I,I] c I. F Consider therefore a holomorphic curve of
filtrations F*(t), and let ~(t) E F2(t) define a holomorphic section of
the Hodge bundle 2. F Then J~(t) = +i~(t) . Differentiation yields
Jd~/dt = +id~/dt, so that Jd~/dt E H+ C Fl(t) . Thus dF2/dt c F ,I as
required.
The last assertion follows from (1.2), since the dimension of
U(p,q)/U(p)×(U(q) is pq = (i/2)h2,0hl, .I
3. The Prvm construction
In this section we prove theorem (1.6). The first step is to
define canonical operators which change the weight of a J-Hodge
structure. Suppose given a J-Hodge structure of weight two, H =
(Hz, S, F*, J). Define a new filtration JF* by
(3.1) JF 1 = 0 H2, )~ 1 H_I, JF0 = HC 0
Then (Hz,JS,JF*,J) is a J-polarized Hodge structure of weight one. One
thinks of this as follows: The +i eigenspace of J has formal type
(I,0), while the -i eigenspace has formal type (0, I). Subtract formal
type from actual type to get the type of the "J-twisted" Hodge
structure. According to this rule, JF 1 has type (I,0).
Given a J-poldrized Hodge structure of weight one, H = (Hz, S,F*,J),
define a new filtration by
(3.2) JF 2 = 0 H+I, JF 1 = H+ 1,0 • H_ 1,0 ~ H+ 0'I JF 0 = c. H
The correct definitions are arrived at by the same rule, except that one
adds actual and formal types to get the new type. The new object
(Hz, JS,JF*,J) is a J-polarized Hodge structure of weight two. Note that
J(JH) = H, so that the operation H ---~ JH defines a natural
isomorphism between the categories of J-Hodge structures of weights one
and two, respectively.
The next (and final) step is the following:
Lemma (3.3) For every J-polarized Hodge structure of weight 2 there is
a canonical isomorphism
: H ) P(JH)
Proof: Let P+ and P_ be the projections of C H onto the +i and -i
eigenspaces of J. Let E = (Ez, SE, F*, J) be the J-Hodge structure of
the elliptic curve with period ratio i. Let (e, e') be a symplectic
basis for EZ, and let ~ = e + ie' generate El, 0. Define a linear
transformation
(3.4) ~ C : H ) H C®E C
by the formula
(3.5) ¢(v) = P+ )v( ~(~ + P_ )v( ®~.
Because J®J(~(v)) = -~(v), the image of # lies in the complex vector
space of P(JH). Since the terms of ~(v) lie in distinct eigenspaces of
10
J~l, ~(v) vanishes if and only if b~th P+(v) and P_(v) vanish, i.e., if
and only if v = 0. Consequently
(3.6) ~ : H ) P(JH) is an isomorphism of vector spaces.
To see that ~ preserves the integral structure, substitute the
relations P±(v) = (I/2) (J ± )i into the definition of ~(v) to obtain
(3.7) ~ )v( = Jv~e - v®e',
and observe that J preserves the integral structures.
We verify that ~ preserves polarizations, where L®E carries, up to
scale factor, the canonical polarization of a tensor product,
(3.8) S(x®y, x'®y') = (-i/2)JS(x,x')SE(y,y )'
To this end substitute the definition of ~(v) and ~(v') in the
definition of S and use the fact that the +i and -i eigenspaces of J are
S~-orthogonal to obtain
S(~(v),~(v')) = (-I/2) [Js(P÷v, P_v')SE(0),~ ) + JS(P_v,P+v')SE((0,~) ]
= [ S(P+v,P v') + S(P_v,P÷v') ]
= S(v,v') .
Since ~ preserves the integral structures, it is defined over the
real numbers. Thus, to show that ~ preserves Hodge filtration, it
suffices to show that #(H2, )0 C (H~E) 2,0. If v is in H2, 0 then P+(v) = v
and P_(v) = 0, so that ~(v) = v®G) E (H®E)2, ,0 as required.
