2023/03/25 by Foster, Josiah · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2303.14327
For a projective 2n-dimensional irreducible holomorphic symplectic manifold Y of generalized Kummer deformation type and j the smallest prime number dividing n+1, we prove the Lefschetz standard conjectures in degrees <2(n+1)(j-1)/j. We show that the restriction homomorphism from the cohomology of a projective deformation of a moduli space of Gieseker-stable sheaves on an Abelian surface to the cohomology of Y is surjective in these degrees. A corollary is that the Lefschetz standard conjectures hold for Y when n+1 is prime. The proofs rely on Markman's description of the monodromy of generalized Kummer varieties and construction of a universal family of moduli spaces of sheaves, Verbitsky's theory of hyperholomorphic sheaves, and the decomposition theorem.