Normal Functions over Locally Symmetric Varieties

We classify the irreducible Hermitian real variations of Hodge structure admitting an infinitesimal normal function, and draw conclusions for cycle-class maps on families of abelian varieties with a given Mumford-Tate group.


Introduction
Normal functions are holomorphic horizontal sections of the intermediate Jacobian bundle J(V) associated to a variation of Hodge structure V (of odd weight) over a complex analytic manifold. Given a family of smooth projective varieties X π → S and a cycle Z ∈ CH p (X ) whose restriction Z s := Z · X s to each fiber is homologous to zero, the Abel-Jacobi images AJ p Xs (Z s ) yield such a section ν Z of J(V 2p−1 π ). These geometric normal functions and their singularities are at the heart of the reformulation of the Hodge Conjecture by Green and Griffiths [GG], which has brought about a renewal of interest (cf. [KP] and references therein).
The most famous example arises from the image C + of an algebraic curve C in its Jacobian. Writing ı : J(C) → J(C) for the map sending u → −u, and C − for ı(C + ), the Ceresa cycle Z C := C + − C − ∈ CH 1 (J(C)) is homologous to zero, with AJ g−1 C (Z C ) ∈ J(H 2g−3 (J(C))) of infinite order for very general C of genus at least 3 [Ce]. By studying cohomology of the mapping class group, Hain has proved (up to a factor of 2) that the resulting section essentially generates all normal functions over M g [Ha]. By "essentially" we mean up to sections of Jacobians of level-1 variations, which are families of abelian varieties. In this paper we shall mostly ignore these, which in the geometric case corresponds to considering only the normal functions that factor through the Griffiths group of cycles modulo algebraic equivalence. We shall also work rationally, i.e. consider torsion normal functions to be zero.
A natural question is what happens over A g , or more generally over quotients of the Siegel domain D = III g := Sp 2g (R)/U(g) by a congruence subgroup Γ ≤ Sp 2g (Q). However, Ceresa's normal function does not even survive the (generically 2:1) map from M g to A g , let alone extend to the latter. In fact, a result of Ragunathan [Ra] implies a much more general vanishing phenomenon: if V is a homogeneous VHS (cf. §2.3) over a locally symmetric variety X := Γ\D := Γ\G(R)/K, associated to a nontrivial Q-irrep V of G, and G has Q-rank > 1, then J(V) admits no normal functions over any Zariski open subset of X. Moreover, when the Q-rank is one, the boundary components in the smooth toroidal compactification of X are smooth, and so any admissible normal function would have no singularities to study. This situation improves somewhat when one considers étale neighborhoods of X instead of Zariski ones, motivated by the example of M 3 minus its hyperelliptic locus. To simplify matters, assume that the group G (of any Q-rank), the Hermitian symmetric domain D, and the representation V are as above, but also simple resp. irreducible over R, with V R of highest weight λ. (We also consider some cases where G R = U(1) · simple.) One would like to classify the pairs (D, λ) for which V has odd weight and the resulting J(V) → X admits a normal function after base change to some étale neighborhood. For this to happen, a certain cohomology sheaf ⊕ j≥0 H 1 (j) of infinitesimal normal functions on D (cf. §2.1) must not vanish, and it is the pairs with this property that we shall classify in this paper using a result of Kostant [Ko] (cf. § §2.2, 2.4). We should note that this approach has already been carried out by Nori [No] in the Siegel domain case, so the results below involving D = III g are not new.
Returning to A g , we can ask once more about the loci "supporting" higher-weight normal functions. By this we mean that there is an irreducible homogeneous variation V of odd level > 1 over A g , a subvariety S ı ֒→ A g with étale neighborhood T  → S , and a nontorsion horizontal section of J( * ı * V) over T . Alternatively, one could consider only those V appearing in the cohomology of the universal abelian variety. In either case, the loci ı(S ) are proper subvarieties for g > 3 [No], and are expected to be more like M g than locally symmetric subvarieties. Indeed, M g supports in this sense the Ceresa normal function for any g; another example is the Fano normal function supported by the locus in A 5 of intermediate Jacobians of cubic threefolds [CNP]. The approach in this article is consequently analogous to the search for special subvarieties of M g [MO], which are expected not to exist for large enough g. So the following vanishing result, which follows from our classification, should not be surprising: Theorem. The loci of Weil resp. quaternionic abelian varieties in A g do not support higher weight normal functions for g > 6 resp. 8. In particular, the image of the reduced Abel-Jacobi map is zero for a very general Weil resp. quaternionic abelian variety A of dimension at least 8 resp. 10.
(See Thms. 3.3 and 3.5 in §3.) There are related results for cycles in certain codimensions for the generalized Weil abelian varieties over I p,q (p + q = n + 1, p < q) studied in §4.1, which get weaker as |q − p| grows (cf. Theorem 4.3). However, both here and for the domains IV m parametrizing "spin abelian varieties" (for instance, those arising from the Kuga-Satake construction), one has infinitesimal normal functions for arbitrarily large n and m.
For each of the "infinitesimal normal functions" described in this paper, the next immediate problem is to determine whether it comes from an actual (admissible) normal function, and if so, to geometrically realize it and determine (outside the exceptional cases) its locus of support in A g . We take a stab at this for (I 3,3 , ω 3 ) in §3.5, using the fact that for certain families of Weil abelian 6-folds, the general member is a Prym variety associated to an unramified 3:1 curve covering with base curve of genus g = 4. The normal function extends to the locus of these generalized Pryms in A 6 , and the construction works for g > 4 as well (but the closure of these loci don't contain an I p,p ).
With an admissible normal function in hand, one can also try to determine its zero locus, and its singularities along intersections of toroidal boundary components. But even in the absence of this, one can probably still use Kostant's result, replacing g by subalgebras of the form sl ⊕a 2 , to compute spaces of singularity classes for normal functions over homogeneous variations.
An equally intriguing prospect is to try to extend the computation of infinitesimal normal function spaces (or singularity classes) to the nonclassical case, which is more relevant to the Green-Griffiths program. Here the homogeneous families of Hodge structures only become variations upon restriction to the image of a period map. However, the computation only requires an infinitesimal variation as input, so it may already be interesting to start with tangent spaces to the Schubert VHS of [Ro].
Acknowledgments: The authors were supported by NSF Grant DMS-1361147. This paper was written while MK was a member at the Institute for Advanced Study, and he thanks the IAS for excellent working conditions and the Fund for Mathematics for financial support.

Infinitesimal normal functions and Kostant's theorem
This section contains the abstract underpinnings of the calculations in § §3-4. Apart from §2.4, it is mostly expository and contains the material on infinitesimal invariants, Lie algebra cohomology, and Hermitian VHS that will be used subsequently.

Normal functions and infinitesimal invariants.
Let V → S be a polarized Q-VHS over a complex manifold, pure of weight −1. By abuse of notation, V will denote the underlying holomorphic vector bundle and its sheaf of sections, with Hodge filtration F • ; V is the underlying Q-local system, Q : V × V → Q S the polarization, and ∇ the Gauss-Manin connection.
Define a filtered complex of sheaves the sheaf of (quasi-)horizontal sections of the Jacobian bundle J(V) is Q , this follows from the long-exact sequence of (2.2).
. We recover at once the result of Green and Voisin [Gre], in the form stated by Nori [No]: Proposition 2.3. If all the H 0 (j) and H 1 (j) vanish for j ≥ 0, then NF T ( * V) = {0} for any étale neighborhood  : T → S . Two obvious situations in which the vanishing conditions fail are those of VHS of level one, or level three and "Calabi-Yau type". Recall that if Example 2.5. (level 3 CY) If h = (1, n, n, 1) and 1 < d < 2n, then This makes the broad vanishing results we obtain in this paper somewhat surprising. It also explains why we have to ignore the level-one cases below.
Finally, given a smooth family of varieties π : X → S , let Z ∈ Z r (X s 0 ) Q,hom be a homologically trivial cycle on a very general fiber. Suppose its class in the Griffiths group Griff r ( , and a normal function where R denotes the quotient by the maximal level-one sub-VHS. So we have the Corollary 2.6. If all the H 0 (j) and H 1 (j) vanish for j ≥ 0, then AJ r Xs 0 (Z) = 0 in (2.4).
While there is no converse result, nonvanishing of H 1 (0) in particular seems to be a good predictor of the existence of interesting cycles. Note that its nonvanishing for level-one VHS V has a geometric "origin". Namely, these VHS correspond to the H 1 (or H 2D−1 ) of families of abelian D-folds A π → S . The existence of nontorsion points on the geometric generic fiber over C(S ) yields nontrivial geometric normal functions in NF T ( * V) over some étale neighborhood T . This explains why we only consider the Griffiths group.
2.2. n-cohomology of finite-dimensional representations. Let g be a complex semisimple Lie algebra of rank n, b ⊃ t Borel and Cartan subalgebras, ∆ = ∆(g, t) ⊂ t * the corresponding roots. Denote by ∆ + = ∆(b) the positive roots, Σ = {σ 1 , . . . , σ n } ⊂ ∆ + the simple roots, Ω = {ω 1 , . . . , ω n } ⊂ t * the fundamental weights, and Λ the (weight) lattice they generate. The Killing form B(X, Y ) = Tr (adX • adY ) on g induces a symmetric bilinear form , on Λ, a particular orthonormal basis of which (as in [Kn, App. C]) will be denoted by {e i }. We have in particular ω i , σ j = 1 2 σ j , σ j δ ij . By the Theorem of the highest weight, the irreducible representations {V λ } of g (of finite dimension) are parametrized by their highest weight λ; there is a 1-to-1 correspondence between irreps and weights of the Fix an element E ∈ λ with all 1 2 E(σ i ) non-positive and integral, and let g = ⊕ j∈Z g j,−j be the decomposition into ad(E)-eigenspaces with eigenvalue 2j. For any representation (V, ρ), there is a corresponding decomposition into ρ(E)-eigenspaces, which can have odd or fractional eigenvalues. Write p = ⊕ j≥0 g j,−j , n = ⊕ j<0 g j,−j (so that ∆(n) ⊂ ∆ + ), and g 0 = g 0,0 . We also denote Our main computational tool will be a result of Kostant (cf. [Ko,Thm. 5.14]). It computes the decomposition of the cohomologies of the natural complex under the action of g 0 . To state a version of it (Prop. 2.7 below), let W = W (g, t) and W 0 = W (g 0 , t) be the Weyl groups, and consider the set . Writing ∆ = ∆ c ∐ ∆ n for the decomposition into compact and noncompact roots, we assume 1 2 E(∆ c ) ⊂ 2Z and Let w 0 ∈ W denote the unique element with w 0 (∆ + ) = ∆ − , and write −τ for the induced involution on Λ; for g simple of Hermitian type other than A n ,D 2m+1 , or E 6 , τ = id Λ . In the non-real case, we have V λ ∼ = V τ (λ) , and we say V λ is complex resp. quaternionic when λ = τ (λ) resp. = τ (λ). To distinguish the real and quaternionic cases, we use the following Proposition 2.8. [GGK] For It is to these representations that we shall apply Prop. 2.7. Note that sinceṼ λ has an underlying R-vector spaceṼ λ R (which is irreducible as a representation of g R ), it can be viewed (up to Tate twist) as an R-Hodge structure via the action of E, whose eigenvalues are regarded as "p − q" (on Hodge type (p, q)).
Since differences of weights of V λ lie in the root lattice (which E sends into 2Z) and −τ Moreover, when E(λ) is odd there exists (up to scale) a unique ginvariant alternating bilinear form Q onṼ λ R , which polarizes this Hodge structure. Note that in the non-real cases, Q pairs V λ and V τ (λ) .
Remark 2.9. The level of the complex summands (as complex Hodge structures) is ) . This is the minimal level that can be achieved using half-twists, which are discussed in §4.

Homogeneous VHS over locally symmetric varieties. Let
G be a semisimple Q-algebraic group of Hermitian type, such that G R has a compact maximal torus T R ; and write g R ⊃ t R for the Lie algebras (with complexifications g ⊇ t). Choose a cocharacter χ 0 : Denoting the composition G m χ 0 → T C ֒→ G C by ϕ 0 , the corresponding Hermitian symmetric domain is the orbit under conjugation by the group of real points G(R): with G 0 (R) a maximal compact subgroup. Taking Γ ≤ G(Q) a torsionfree congruence subgroup, the locally symmetric variety X := Γ\D is in fact a projective variety by the Baily-Borel theorem. The gives a (−B)polarized Q-VHS of weight zero and level two over D, with Hodge decompositions g = ⊕ j=−1,0,1 g j,−j Ad(g)E at gϕ 0 g −1 ∈ D; this descends to X. More generally (with A = Q or R), given an A-representation ρ : G → Aut(V A , Q) (Q a (−1) k -symmetric bilinear form) such that ρ • ϕ is an A-PHS, we get a homogeneous A-PVHS V A (with weight of parity k) over X, called a Hermitian VHS. In every case the Hodge decomposition is induced by dρ (Ad(g)E); that is, a weight subspace For groups other than E 6 and E 7 , most 2 Hermitian Q-PVHS arise from the relative cohomology of various canonical families of abelian varieties; in some cases (cf. [FL, ?]) one has also families of Calabi-Yau varieties. While these structures may matter for questions of geometric origin (i.e. algebraic cycles), the behavior of the {H k (j)} depends only on V R . Moreover, any Hermitian PVHS of weight −1 on X decomposes over R into a direct sum of the irreducible homogeneous real variations V λ R arising (as above) from the PHS on (Ṽ λ ,ρ λ ) described at the end of §2.2. Indeed, since Γ ≤ G(R) is Zariski-dense, we have an equivalence between real Hermitian VHS V R over X and representations of G(R). Before turning to our main calculation, we remind the reader what forms D and E can take when G C is simple. First, (2.11) forces ∆ n ∩ Σ to be a singleton {σ I }, which must additionally be a special simple root. That is, if we write g = V λ ad , λ ad = n i=1 M ad i σ i , then σ I must be one of the simple roots σ i for which M ad i = 1. Further, the choice of σ I determines: the decomposition ∆ = ∆ c ∐ ∆ n , and thus the real form g R ; and the Hodge structure at the base point ϕ 0 , by (2.12) E(σ I ) = −2 , E(σ j ) = 0 (∀j = I).
This leads to the classification (up to isomorphism) of the irreducible Hermitian symmetric domains of noncompact type: Here R is the root system, G(R) is the simply connected group (which has allṼ λ as representations), and d = dim C D = dim C X. (See [LZ] for more details.) Remark 2.10. We have omitted some cases to avoid redundancy due to exceptional isomorphisms and conjugate-isomorphisms. The latter are induced by the action of τ on σ I . For A n , τ exchanges σ i and σ n+1−i (∀i); for D n , n odd, τ exchanges σ n−1 and σ n ; for E 6 , τ exchanges σ 6 ↔ σ 1 and σ 5 ↔ σ 3 ; and in all other cases the action is trivial. The exceptional isomorphisms are III 1 ∼ = I 1,1 ∼ = IV 1 ∼ = II 2 , III 2 ∼ = IV 3 , I 2,2 ∼ = IV 4 , II 3 ∼ = I 1,3 , II 4 ∼ = IV 6 (triality for D 4 ), and IV 2 ∼ = III 1 × III 1 . For this reason we consider only A n≥2 , B n≥3 , C n≥1 , and D n≥4 .
In each case, the real variations arising from H 1 of the canonical families of abelian varieties over X are (half-twists of 3 ) theṼ λ with − 1 2 (E(λ) + E(τ (λ))) = 1 (cf. (2.10)). All such λ take the form ω i , with the possibilities corresponding to so-called "symplectic nodes" (cf. [LZ]). This will be recalled case by case where relevant in § §3-4. The variations of Calabi-Yau type are even simpler to describe: they are (again, up to half-twist for A n , D n odd , E 6 ) precisely theṼ kωI for k ≥ 1 [FL].
2.4. The main calculation. We now apply Proposition 2.7 to compute the {H k (j)} for the Hermitian variationsṼ λ (over X or D) with E(λ) odd. Of course, we can work with (its summands) the complex variations V λ C , and do the computation at one point ϕ 0 ∈ D. Write (2.13) 3 this is only relevant for A n , cf. §4 Proof. First, we note that (2.14) follows from the identification of the Lie algebra cohomology complex ∧ • n ∨ ⊗ V λ , d with the associated graded ⊕ j Gr j F C • ,∇ under the isomorphism n ∨ ∼ = Ω 1 D,ϕ 0 . To see this identification, extend v ∈ (V λ ) j,−j−1 to a section of (V λ ) j,−j−1 byṽ =ρ λ (g).v, and write (for Next, we compare Hodge gradings on the two sides of (2.14). For X * ∈ n ∨ and v ∈ Gr j−1 consists of all the weight spaces with weights ξ for which E(ξ) = 2j +1. Since g 0 commutes with E, these are just the V w·λ 0 for which E(w · λ) = 2j + 1. The result follows.

Analysis of (mostly) tube domain cases
In this section we study the real variationsṼ λ R arising from dominant integral λ with E(λ) an odd integer. Though we (more or less) carry this out for all the domains in the table 2.3, the results are of interest mainly when D is of tube type (I p.p , II 2m , III n , IV m , or EVIII). As above, we have E(σ I ) − 2, E(σ j =I ) = 0, and write λ = n i=1 m i ω i ; note that s sends σ I → −σ I and fixes all ω j =I . To streamline the discussion of examples, we make the In all the cases below where we obtain such an assertion, one knows that all level-one real sub-VHS are in fact defined over Q.
We can now turn to the richer remaining classical cases.
Definition 3.2. By a universal quaternionic abelian variety, we shall mean any family A IIn → Γ\II n of abelian 2n-folds whose H 1 recovers V ω 1 R . Such families admit an embedding of a definite rational quaternion algebra Q into End(A ) Q . (The Mumford-Tate group G is a Q-form of G R and so, by considering its fixed 2-tensors, Q is a Q-form of H. See [vGV] for more details on quaternionic abelian varieties.) There are natural embeddings II n ֒→ III 2n which yield countably many "quaternionic subfamilies" of A III 2n . Now from s(ω n ) = ω n−2 − ω n we find Imposing −E(λ) odd > 1, n ≥ 4, and µ(λ) ≥ 0 yields λ = ω 1 + aω 4 , ω 3 + aω 4 (n = 4, a > 0) which is quaternionic, and λ = ω 6 (n = 6) which is real.
Theorem 3.3. No étale pullback of a universal quaternionic abelian variety of (relative) dimension 2n admits reduced normal functions outside the case n = 4.
Proof. It remains to deal with n = 6. The point is that only V ω 5 +ω 6 and V 2ω 5 ,V 2ω 6 occur in H * rel (A II 6 ) ∼ = * (V ω 1 ⊕ V ω 1 ); the half-spin variations 4 V ω 5 , V ω 6 do not. (See [FH] for the equivalent fact on representations of SO(12).) What is special about the quaternionic 8-folds that might yield AJnontrivial elements of the Griffiths group? According to [vGV], there exist families A II 4 whose general member arises as a quaternionic Prym variety, associated to a certain 8 : 1 unramified cover of a general genus 3 curve (plausible as dim M 3 = dim II 4 = 6). Since 3 V ω 1 = V ω 3 +ω 4 , the (non-Calabi-Yau) variationṼ ω 3 +ω 4 R occurs in H 3 of A II 4 , and it seems reasonable to expect that one can construct AJ -nontrivial 1cycles from the Abel-Prym image of the genus 17 cover curves. For n = 6, it seems to be an open problem to give a simple motivic construction of V ω 6 R , so we cannot speculate about cycles in this case.
So restricting to p = n+1 2 (n odd), i.e. the case I p,p (p ≥ 2), we have We remark that all representations are either real (m j = m n+1−j ∀j) or complex, and that the sole level 1 variation isṼ ω 1 = V ω 1 ⊕ V ωn .
Definition 3.4. A universal Weil abelian variety is a family A Ip,p → Γ\I p,p of abelian 2p-folds (of dimension p 2 ) whose H 1 recoversṼ ω 1 R . Such families admit an embedding of an imaginary quadratic field into End(A ) Q , and produce countably many subfamilies of A III 2p .
Theorem 3.5. No étale pullback of a universal Weil abelian variety of (relative) dimension 2p admits reduced normal functions outside the cases p = 2, 3.
Definition 3.6. A universal spin abelian variety is an abelian family A → Γ\IV m with H 1 (over R) a number of copies ofṼ ω n−1 R andṼ ωn R (m = 2n − 2 even) resp.Ṽ ωn R (m = 2n − 1 odd). The minimum possible (relative) dimension of A is clearly 2 n (m and n−1 2 odd), 2 n−1 (m and n+1 2 odd; m even and 4 ∤ n + 2), or 2 n−2 (m even and 4|n+ 2). However, the main natural source of spin abelian varieties is the Kuga-Satake construction (cf. [vG2]), which produces varieties of much higher dimension: for m ≤ 19, one has families of K3 surfaces X with H 2 tr (X ) ∼ = V ω 1 , and Clifford algebras produce an embedding H 2 So the situation is in marked contrast to those encountered above: 5 Proposition 3.7. The relative cohomology of any Kuga-Satake family of spin abelian varieties over IV m (any m ≥ 7) admits infinitesimal (reduced) normal functions.
Proof. We only need to show that (say) V ω 1 +ωn occurs in H 3 (A ) = 3 H 1 (A ). This is done by considering the decomposition of (V ωn ) ⊗2 or V ωn ⊗ V ω n−1 ; e.g. for m odd, (There are enough copies of V ωn that we can consider tensor rather than wedge powers.) Finally, we remark that the triality isomorphism for D 4 exhibits the universal quaternionic abelian 8-folds as "minimal" spin 8-folds, by equatingṼ ω 4 R → IV 6 andṼ ω 1 R → II 4 . Specialization under IV 5 ֒→ IV 6 shrinks the Mumford-Tate group to Spin(2, 5) and gives geometric realizations ofṼ ω 3 R → IV 5 (as noticed by [vGV]). All these cases admit infinitesimal normal functions.

Half-twists and non-tube cases
In this section we consider a slight generalization of the homogeneous variations described in §2.3, by enlarging our simple G C toG C = G m · G C (or G R to U(1) · G(R)) and takingẼ to have a component in the abelian factor ofg. This allows us to shift the Hodge grading of a complex summand (e.g. to make it integral), which is necessary in order to study the cohomology of abelian varieties of generalized Weil type and to obtain all Calabi-Yau variations. We restrict our investigation to these examples and proceed with a minimum of formality.
Given an irrep V λ of g and E ∈ t as before, we takeẼ = (E, 1) ∈ g ⊕ C =g, and define representations ofg by We write whether or not V λ or V λ a 2 is complex; this is a variant of van Geemen's half-twist [vG1] that preserves the weight. As a real Hodge structure, (4.1) has level When (4.2) is odd, define over X = Γ\D has a nonzero H 1 (j) (j ≥ 0). (As above, we shall say it has an infinitesimal normal function.) If both invariants are negative, any underlying Q-VHS admits no nontrivial reduced normal function on an étale neighborhood of X.
Since p + d − j − n ≥ 0 (with d ≤ n 2 , p ≤ n+1 2 ) boils down to the level one cases we identified, the last two entries can be ignored. We have 1 − d ≥ 0 and p = j or n − 2d + j in the cases (d, p) = (1, 1), (1, 2), (1, 3); and j − d − p ≥ 0 for some j =⇒ 2d + 1 ≥ 2p − 1. Hence we conclude that the variations of level > 1 in H odd of A admitting infinitesimal normal functions therefore lie in Note that p = n+1 2 (k = 0) is the Weil abelian setting dispensed with in §3.5. Omitting this case, we can restate our conclusion as follows: Theorem 4.3. For k ≤ n − 7, no étale pullback of a universal k-Weil abelian (n+1)-fold admits reduced normal functions arising from cycles of dimension less than n−k−1 2 or codimension less than n−k+1 2 . As to why (or whether) greater disparity in the signature of the Hermitian form should lead to more AJ -nontrivial cycles in the Griffiths group, we can say nothing yet.

Comments on the Hermitian Calabi-Yau VHS.
For the tube domain cases, the CY variations are nothing but the V kωI R (described for k = 1 by Gross [Gro]). In the remaining cases (after [SZ, FL]), where V kωI is complex, we choose the shift inṼ kωI a 2 to minimize the (odd) level while having the CY property.

Proposition 4.4.
Amongst the minimal-level Calabi-Yau variations over irreducible Hermitian symmetric domains other than I 1,n or I 2,n−1 , only those of level three admit infinitesimal normal functions.