Holomorphic Differential Forms of Higher Degree on Kuga’s Modular Varieties

Šokurov, V V

doi:10.1070/sm1976v030n01abeh001901

Cited by 21 publications

(12 citation statements)

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“…[8] defined mixed cusp forms using the automorphy factor where They proved that the space S2,i(Γ,tι;, χ) of mixed cusp forms of type (2,1) associated to Γ, ω and χ is canonically isomorphic to the space H°(E, Ω 2 ) of holomorphic 2-forms on E. In [14] mixed automorphic forms of type (2, n) for n > 1 were defined using the automorphy factor and it was proved that the space S^nίΓ^x) of mixed cusp forms of type (2,m) associated to Γ, ω and χ is canonically isomorphic to the space H°(E n , Ω n+1 ) of holomorphic (n + l)-forms on the elliptic variety E n , where E n is obtained by resolving the singularities of the compactification of the nfold fiber product of E o over X o . Assuming that Γ C 5L(2, R) with -1 £ Γ and that % is an inclusion Γ °-> SX(2,M), the above result of Hunt and Meyer was proved by Shioda [31] and the higher weight case was proved by Sόkurov [32] (see also [33], [34]). …”

Section: Introductionmentioning

confidence: 94%

Mixed automorphic vector bundles on Shimura varieties

Lee¹

1996

Pacific J. Math.

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Section: Introductionmentioning

confidence: 94%

Mixed automorphic vector bundles on Shimura varieties

Lee¹

1996

Pacific J. Math.

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“…Kuga fiber varieties are closely linked to the theory of automorphic forms (see e.g. [6], [7], [9], [10], [14], [15]). If x is an element of a symmetric space M, there is an isometry S x of M called a symmetry at x such that x is an isolated fixed point of S x and S 2 x = 1.…”

Section: Introductionmentioning

confidence: 99%

Rational equivariant holomorphic maps of symmetric domains

Lee

2002

Archiv der Mathematik

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An equivariant holomorphic map of symmetric domains associated to a homomorphism of semisimple algebraic groups defined over Q is rational if it carries a point belonging to a set determined by an arithmetic subgroup to a point in a similar set. We prove that an equivariant holomorphic map of symmetric domains is rational if the associated Kuga fiber variety does not have a nontrivial deformation. Introduction.In this paper we study the relation between the rationality of equivariant holomorphic maps of symmetric domains and the rigidity of Kuga fiber varieties which are families of polarized abelian varieties parametrized by a locally symmetric arithmetic variety.Let G be a semisimple algebraic group defined over Q that does not contain factors defined over Q and compact over R, and let G = G(R) be the associated real semisimple Lie group. Let K be a maximal compact subgroup of G, and assume that the corresponding symmetric space D = G/K has a G-invariant complex structure. Let V be a vector space over Q of dimension 2m, and let β be a nondegenerate alternating bilinear form on V . We set G = Sp(V, β) and denote by D the symmetric domain associated to G = G (R). Let ρ : G → G be a homomorphism of Lie groups induced by a morphism ρ : G → G of algebraic groups defined over Q. We define the space D ρ to be the set of all holomorphic maps τ : D → D satisfying τ(gz) = ρ(g)τ(z) for all g ∈ G and z ∈ D.Let L be a lattice in V R , and let Γ be a torsion-free subgroup of Sp(L, β) of finite index. We set

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“…Let £" w be the variety obtained from E m by resolving the singularities (see [7] and [8] for details). The map π: E -• X naturally induces the map π m : E m -» X which is a fiber bundle whose generic fiber is the product of m elliptic curves.…”

Section: Elliptic Varietiesmentioning

confidence: 99%

“…In a series of papers ( [7], [8], [9]) Shokurov has constructed elliptic varieties and proved several properties of Kuga's modular varieties which are elliptic varieties of a special kind. If m = 1, E m is simply an elliptic surface and a Kuga's modular variety is an elliptic modular surface of Shioda ([6]).…”

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confidence: 99%