Remarks on the Lefschetz standard conjecture for hyperkahler varieties

We study Grothendieck's Lefschetz standard conjecture on a smooth complex projective variety. In degree 2, we reduce it to a local statement concerning local deformations of vector bundles on X. When X is hyperkaehler, we give explicit criteria which imply the conjecture, using Verbitsky's theory of deformations of hyperholomorphic bundles.