The LLV decomposition of hyper-Kaehler cohomology

Mark Green, Yoon-Joo Kim, Radu Laza, Colleen Robles

Published 2019-06-08Version 1

Looijenga-Lunts and Verbitsky showed that the cohomology of a compact hyper-Kaehler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact Kaehler manifolds. In this paper, we determine the Looijenga-Lunts-Verbitsky (LLV) decomposition for all known examples of compact hyper-Kaehler manifolds. As an application, we compute the Hodge numbers of the exceptional OG10 example starting only from the knowledge of the Euler number $e(X)$, and the vanishing of the odd cohomology of $X$. In a different direction, we establish the so-called Nagai's conjecture for all known examples of hyper-Kaehler manifolds. More importantly, we prove that, in general, Nagai's conjecture is equivalent to a representation theoretic condition on the LLV decomposition of the cohomology $H^*(X)$. We then notice that all known examples of hyper-Kaehler manifolds satisfy a stronger, more natural condition on the LLV decomposition of $H^*(X)$: the Verbitsky component is the dominant representation in the LLV decomposition of $H^*(X)$.