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Register a new userA characterization of varieties whose universal cover is a bounded symmetric domain without ball factors
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Authors
Fabrizio Catanese
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We give two characterizations of varieties whose universal cover is a bounded symmetric domain without ball factors in terms of the existence of a holomorphic endomorphism s of the tensor product Totimes T of the tangent bundle T with the cotangent bundle T. To such a curvature type tensor s one associates the first Mok characteristic cone S, obtained by projecting on T the intersection of ker (s) with the space of rank 1 tensors. The simpler characterization requires that the projective scheme associated to S be a finite union of projective varieties of given dimensions and codimensions in their linear spans which must be skew and generate.
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Consider a family f:A --> U of g-dimensional abelian varieties over a quasiprojective manifold U. Suppose that the induced map from U to the moduli scheme of polarized abelian varieties is generically finite and that there is a projective manifold Y, containing U as the complement of a normal crossing divisor S, such that the sheaf of logarithmic one forms is nef and that its determinant is ample with respect to U. We characterize whether $U$ is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map from U to the moduli scheme or by the existence of CM points. More precisely, we show that U is a Shimura variety, if and only if two conditions hold. First, each irreducible local subsystem V of the complex weight one variation of Hodge structures is either unitary or satisfies the Arakelov equality. Secondly, for each factor M in the universal cover of U whose tangent bundle behaves like the one of a complex ball, an iterated Kodaira-Spencer map associated with V has minimal possible length in the direction of M.
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