  For a projective $2n$-dimensional irreducible holomorphic symplectic manifold
$Y$ of generalized Kummer deformation type and $j>1$ the smallest divisor of
$n+1$, we prove the Lefschetz standard conjectures in degree $<2(n+1)(j-1)/j$.
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.

Algebraic Geometry