The Standard Conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces

TL;DR

This paper proves the standard conjectures for a class of complex projective varieties that are deformation equivalent to Hilbert schemes of K3 surfaces, using hyperholomorphic sheaves and cohomology algebra analysis.

Contribution

It establishes the standard conjectures for these varieties, extending known results to a broader class of hyperkähler manifolds.

Findings

Standard conjectures hold for deformations of Hilbert schemes of K3 surfaces.

Use of Verbitsky's hyperholomorphic sheaves theory in the proof.

Analysis of cohomology algebra of Hilbert schemes of K3 surfaces.

Abstract

We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky's theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.