A conjectural bound on the second Betti number for hyper-K\"ahler manifolds

TL;DR

This paper proposes a conjectural upper bound on the second Betti number of hyper-K"ahler manifolds based on a natural condition related to their LLV decomposition, extending known cases.

Contribution

It introduces a conjectural bound on the second Betti number assuming a specific LLV decomposition condition holds universally for hyper-K"ahler manifolds.

Findings

Proposes an upper bound on the second Betti number based on the LLV decomposition.

Connects the bound to the dominance of the Verbitsky component in the cohomology.

Provides a conjectural framework extending known cases of hyper-K"ahler manifolds.

Abstract

In previous work, we noted that the known cases of hyper-K\"ahler manifolds satisfy a natural condition on the LLV decomposition of the cohomology; informally, the Verbitsky component is the dominant representation in the LLV decomposition. Assuming this condition holds for all hyper-K\"ahler manifolds, we obtain an upper bound for the second Betti number in terms of the dimension.