Markman, EyalFoster, Josiah2024-12-062024-12-062024-0910.7275/55146https://hdl.handle.net/20.500.14394/55146For 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.Attribution 4.0 InternationalAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/standard conjectures, algebraic cycles, derived equivalences, hyperholomorphic sheavesDissertation (Open Access)https://orcid.org/0009-0005-0589-4052