Density of members with extra Hodge cycles in a family of Hodge structures

Ching-Li Chai
1998 Asian Journal of Mathematics  
Let MK(G,X) be a Shimura variety over C; let L be a Q-subgroup of G whose G((Q)-conjugacy class corresponds to a prescribed type of Hodge cycles. For every subvariety V of MK(G,X), denote by S(V,L) the subset of points of V whose Mumford-Tate group is contained in a (^((QJ-conjugate of L. We define an invariant c(G,X,L) depending only on the G(R)-conjugacy of L. The main result says that for every subvariety V of codimension at most c(G,X,L), the subset S(V, L) is dense in V in the metric
more » ... in the metric topology. The value of c(G r ,X, L) is tabulated in many examples. 0. Introduction. The question we address in this paper is part of a more general question: Given a variation of Hodge Q-structures over a complex analytic variety V, do the points of V corresponding to members in this family having extra Hodge cycles of a given type form a dense subset of V? A special case of this question was raised in [1] . Let V C Ag be a subvariety of the moduli space Ag of principally polarized abelian varieties of dimension g over C, and for any integer A; between 1 and g -1 denote by Sk{V) the subset of V consisting of all points x G V such that the corresponding abelian variety A x has an abelian subvariety of dimension k. In [1], Colombo and Pirola gave a sufficient condition for Skiy) to be dense in V with respect to the metric topology. They also proved that if V is a subvariety of the Jacobian locus TQ/lg) of codimension at most g -1, then Si(V) is dense in V. Here T(Mp) denotes the locus of jacobians of complete smooth curves of genus g in Ag. Using the same criterion, Izadi showed in [5] that if V is either a subvariety of Ag of dimension at most #, or V is a subvariety of T(M^) of dimension at most p, then 5i(V) is dense in V. This paper was started by the attempt to understand the meaning of the results of [1, 5] in a more general context. We replace the parameter space A g by a Shimura variety MK(G,X), and allow Hodge cycles more general than idempotent endomorphisms of a polarized abelian variety to arrive at the more general question stated at the beginning; see §1 for the precise formulation. In our setting, the subset Sk(V) of V C A g is replaced by a subset S(V,L) of V C MK(G,X), where L is a reductive subgroup of G defined over Q. We define an invariant c(G, X, L) G N which has the property that S(V,L) is dense in V if V has codimension at most c(G,X,L) in MK(G,X). This generalizes [1, 5] . The invariant c(G,X,L) depends only on the G(]R)-conjugacy class of L in G] its value in the case considered in [1, 5] is g. Our method differs from those used in [1, 5] only at the end, where we further linearized the problem to compute c(G, X, L). This is the content of §2; it allows us to replace the geometric arguments used in [1, 5] by a simpler linear algebra argument, which makes the invariant c(G, X, L) easy to compute. We illustrate this in §3 and tabulate the value of c(G, X, L) in many cases. For instance in the situation considered in [1, 5] , our result says that if V C Ag is a subvariety of codimension at most g, then Sk{V) is dense in V for any k between 1 and g -1. Another example of our result is that for every subvariety V in ^9 of codimension at most 10, the subset of points of *
doi:10.4310/ajm.1998.v2.n3.a1 fatcat:jbs5nb4pgffstm2vjvju7zcsjm