Footnotes to papers of O'Grady and Markman

TL;DR

This paper generalizes results about hyper-K"ahler manifolds of generalized Kummer type and proves the algebraicity of their Kuga-Satake correspondence using recent results, broadening understanding in complex geometry.

Contribution

It extends O'Grady's results to all hyper-K"ahler manifolds with nonzero third Betti number and establishes the algebraicity of the Kuga-Satake correspondence for these manifolds.

Findings

Generalization of O'Grady's results to broader hyper-K"ahler classes

Proof of algebraicity of the Kuga-Satake correspondence for generalized Kummer type manifolds

Application of Markman's results to establish algebraicity

Abstract

In this paper, we first generalize to any hyper-K\"ahler manifold with nonzero third Betti number results proved by O'Grady for hyper-K\"ahler manifolds of generalized Kummer type. In the second part, we restrict to hyper-K\"ahler manifolds of generalized Kummer type and prove, using results of Markman, that their Kuga-Satake correspondence is algebraic.