Boundness of $b_2$ for hyperk\"ahler manifolds with vanishing odd-Betti numbers

TL;DR

This paper proves that the second Betti number $b_2$ is bounded for certain hyperk"ahler manifolds with vanishing odd-Betti numbers, providing explicit bounds in specific dimensions and conjecturing a general upper limit.

Contribution

It establishes the boundedness of $b_2$ for hyperk"ahler manifolds with vanishing odd-Betti numbers and verifies bounds in dimensions eight and ten.

Findings

$b_2$ is bounded in dimension eight and ten

Explicit bounds are 24 and 25 respectively

Conjecture for a general upper boundary

Abstract

We prove that $b_2$ is bounded for hyperk\"ahler manifolds with vanishing odd-Betti numbers. The explicit upper boundary is conjectured. Following the method described by Sawon we prove that $b_2$ is bounded in dimension eight and ten in the case of vanishing odd-Betti numbers by 24 and 25 respectively.