A Sharp Bound for the Second Betti Number of Hyperkähler Sixfolds

Let $X$ be a smooth compact irreducible hyperkähler 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.