On irreducible symplectic 4-folds numerically equivalent to $(K3)^{[2]}$

TL;DR

This paper investigates the classification of certain hyper-K"ahler fourfolds with specific topological properties, providing partial proofs related to O'Grady's conjecture on their structure.

Contribution

It proves special cases of O'Grady's conjecture concerning hyper-K"ahler fourfolds that are numerically equivalent to the Hilbert scheme of two points on a K3 surface.

Findings

Partial validation of O'Grady's conjecture for specific cases.

Identification of conditions under which hyper-K"ahler fourfolds are classified.

Advancement in understanding the structure of hyper-K"ahler fourfolds with $b_2=23$.

Abstract

We address the problem of classification of hyper-K\"ahler fourfolds with $b_2=23$. In particular we prove some special cases of the Conjecture of O'Grady about hyper-K\"ahler $4$-folds numerically equivalent to the Hilbert scheme of two points on a $K3$ surface.