Remarks on the Lefschetz standard conjecture and hyperkähler varieties

Abstract

We study the Lefschetz standard conjecture on a smooth complex projective variety $X$. In degree $2$, we reduce it to a local statement concerning local deformations of vector bundles on $X$. When $X$ is hyperkähler, we show that the existence of nontrivial deformations of stable hyperholomorphic bundles implies the Lefschetz standard conjecture in codimension $2$.