Remarks on the Lefschetz standard conjecture and hyperk\"ahler varieties

TL;DR

This paper investigates the Lefschetz standard conjecture for complex projective varieties, especially hyperk"ahler varieties, linking it to deformations of vector bundles and hyperholomorphic bundles.

Contribution

It reduces the conjecture to a local deformation problem and establishes a connection between hyperholomorphic bundle deformations and the conjecture for hyperk"ahler varieties.

Findings

Reduction of the conjecture to local deformation statements in degree 2

Existence of nontrivial hyperholomorphic bundle deformations implies the conjecture for hyperk"ahler varieties

Provides new insights into the relationship between vector bundle deformations and algebraic cycles

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 deformations of vector bundles on X. When X is hyperk\"ahler, we show that the existence of nontrivial deformations of stable hyperholomorphic bundles implies the Lefschetz standard conjecture in codimension 2.