On a conjecture of Kashiwara

Abstract

Kashiwara conjectured that the hard Lefshetz theorem and the semisimplicity theorem hold for any semisimple perverse sheaf M on a variety over a field of characteristic 0. He also conjectured that if you apply to such M the nearby cycle functor corresponding to some function then the successive quotients of the monodromy filtration are semisimple. We prove that these conjectures would follow from de Jong's conjecture on representations modulo l of the arithmetic fundamental group of a variety over a finite field.

16 pages, Latex; added references to works by Simpson and Sabbah

Cited by in corpus (12)

• Vanishing Theorems for constructible Sheaves on Abelian Varieties
• The Shafarevich conjecture for hypersurfaces in abelian varieties
• Degenerate Perverse Sheaves on Abelian Varieties
• Parity and symmetry in intersection and ordinary cohomology
• Absolute sets and the Decomposition Theorem
• Density of Arithmetic Representations of Function Fields
• Wild twistor D-modules
• Why certain Tannaka groups attached to abelian varieties are almost connected
• Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian
• Characteristic classes and Hilbert-Poincaré series for perverse sheaves on abelian varieties
• On the conjectures of Braverman-Kazhdan
• A remark on primitive cycles and the Radon transform