The LLV decomposition of hyper-Kaehler cohomology

Looijenga--Lunts and Verbitsky showed that the cohomology of a compact hyper-K\"ahler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact K\"ahler manifolds. In this paper, we determine the Looijenga--Lunts--Verbitsky (LLV) decomposition for all known examples of compact hyper-K\"ahler manifolds, and propose a general conjecture on the weights occurring in the LLV decomposition, which in particular determines strong bounds on the second Betti number $b_2(X)$ of hyper-K\"ahler manifolds. Specifically, in the $K3^{[n]}$ and $\mathrm{Kum}_n$ cases, we give generating series for the formal characters of the associated LLV representations, which generalize the well-known G\"ottsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyper-K\"ahler manifolds. For the two exceptional cases of O'Grady we refine the known results on their cohomology. In particular, we note that the LLV decomposition leads to a simple proof for the Hodge numbers of hyper-K\"ahler manifolds of O'Grady 10 type. In a different direction, for all known examples of hyper-K\"ahler manifolds, we establish the so-called Nagai's conjecture on the monodromy of degenerations of hyper-K\"ahler manifolds. More consequentially, we note that Nagai's conjecture is a first step towards a more general and more natural conjecture, that we state here. Finally, we prove that this new conjecture is satisfied by the known types of hyper-K\"ahler manifolds.