Shimura Varieties in the Torelli Locus via Galois Coverings

Abstract

Given a family of Galois coverings of the projective line, we give a simple sufficient condition ensuring that the closure of the image of the family via the period mapping is a special (or Shimura) subvariety of Ag. By a computer program we get the list of all families in genus g ≤ 9 satisfying our condition. There are no families with g = 8, 9, all of them are in genus g ≤ 7. These examples are related to a conjecture of Oort. Among them we get the cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot, Rohde, Moonen and others) and the abelian non-cyclic examples found by Moonen-Oort. We get 7 new non-abelian examples.

Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

“…Here we find 6 families of Galois coverings, all with g ′ = 1 and g = 2, 3, 4 and we show that these are the only families with g ′ = 1 satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of Ag, while the other examples arise from certain Shimura subvarieties of Ag already obtained as families of Galois coverings of P 1 in [13]. Finally we prove that if a family satisfies this sufficient condition with g ′ ≥ 1, then g ≤ 6g ′ + 1.…”

“…The purpose of this paper is to continue the investigation started in [13] of those special subvarieties of A g contained in the Torelli locus arising from families of Jacobians of Galois coverings f : C → C ′ where C ′ is a smooth complex projective curve of genus g ′ ≥ 1, g = g(C). In [13] the authors systematically studied families of Galois covering of P 1 following the previous work done by Moonen [28] in the cyclic case and initiated in [38,30,10,36] (see also the survey [29, §5]).…”

“…In [13] the authors systematically studied families of Galois covering of P 1 following the previous work done by Moonen [28] in the cyclic case and initiated in [38,30,10,36] (see also the survey [29, §5]). …”

Vector bundles on curves coming from variation of Hodge structures

“…The main point is that (see [11] and [18], especially for more details concerning the relation with Shimura curves) the dual of H 1 (C, Θ C ) G equals H 0 (2K C ) G , while the tangent space to the symmetry preserving deformations of the Abelian varieties is given by…”

“…This situation leads to a finite number of cases, which were classified by Moonen in [39] (see [18] for groups more general than cyclic groups).…”