Vanishing Theorems for constructible Sheaves on Abelian Varieties

Abstract

We show that the hypercohomology of most character twists of perverse sheaves on a complex abelian variety vanishes in all non-zero degrees. As a consequence we obtain a vanishing theorem for constructible sheaves and a relative vanishing theorem for a homomorphism between abelian varieties. Our proof relies on a Tannakian description for convolution products of perverse sheaves, and with future applications in mind we discuss the basic properties of the arising Tannaka groups.

References in corpus (11)

• Revêtements étales et groupe fondamental (SGA 1)
• Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules
• Semisimple algebraic tensor categories
• Degenerate Perverse Sheaves on Abelian Varieties
• Brill-Noether Sheaves
• On Subvarieties of Abelian Varieties with degenerate Gauss mapping
• A Remark on Rigidity of BN-sheaves
• Why certain Tannaka groups attached to abelian varieties are almost connected
• The symmetric Square of the Theta Divisor in Genus 4
• Tannakian Categories attached to abelian Varieties
• Holonomic complexes on abelian varieties, Part I

Cited by in corpus (23)

• Perverse sheaves on semiabelian varieties
• The Shafarevich conjecture for hypersurfaces in abelian varieties
• Degenerate Perverse Sheaves on Abelian Varieties
• Semisimple Super Tannakian Categories with a small Tensor Generator
• Cubic threefolds, Fano surfaces and the monodromy of the Gauss map
• On Subvarieties of Abelian Varieties with degenerate Gauss mapping
• Perverse sheaves on semi-abelian varieties
• Why certain Tannaka groups attached to abelian varieties are almost connected
• Characteristic cycles and the microlocal geometry of the Gauss map, II
• The symmetric Square of the Theta Divisor in Genus 4
• On the Rigidity of BN-sheaves
• On a family of surfaces of general type attached to abelian fourfolds and the Weyl group W(E_6)
• Remarks on the nonvanishing of cohomology groups for perverse sheaves on abelian varieties
• Vanishing theorems for perverse sheaves on abelian varieties, revisited
• Summands of theta divisors on Jacobians
• Characteristic classes and Hilbert-Poincaré series for perverse sheaves on abelian varieties
• Mellin transformation, propagation, and abelian duality spaces
• Holonomic complexes on abelian varieties, Part I
• Vanishing theorems for abelian varieties over finite fields
• Generic vanishing theory via mixed Hodge modules
• Generic Vanishing, 1-forms, and Topology of Albanese Maps
• Holonomic D-modules on abelian varieties
• Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution