A Sharp Bound for the Second Betti Number of Hyperk\"ahler Sixfolds

Abstract

Let $X$ be a smooth compact irreducible hyperk\"ahler manifold of dimension six. We prove that the second Betti number satisfies $b_2(X) \le 23$. Furthermore, we show that the third cohomology vanishes whenever $b_2(X) \ge 19$, and that all odd cohomology vanishes when $b_2(X) = 23$. If the second Chern class lies in the subalgebra generated by $H^2(X,\mathbb{Q})$, then $b_2(X) \in \{5,8,23\}$. The proof uses the Looijenga--Lunts--Verbitsky decomposition, curvature inequalities for Segre classes, and the Hodge--Riemann bilinear relations on the projective cotangent bundle.