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. An immediate 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 cohomological decomposition theorem.
