The LLV decomposition of hyper-Kähler cohomology (the known cases and the general conjectural behavior)

Looijenga–Lunts and Verbitsky showed that the cohomology of a compact hyper-Kahler manifold X admits a natural action by the Lie algebra $$\mathfrak {so} (4, b_2(X)-2)$$ , generalizing the Hard Lefschetz decomposition for compact Kahler manifolds. In this paper, we determine the Looijenga–Lunts–Verbitsky (LLV) decomposition for all known examples of compact hyper-Kahler 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-Kahler manifolds (see Kim and Laza in Bull Soc Math Fr 148(3):467–480, 2020). 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 Gottsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyper-Kahler manifolds. For the two exceptional cases of O’Grady (OG6 and OG10) 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-Kahler manifolds of $$\mathrm {OG}10$$ type. In a different direction, for all known examples of hyper-Kahler manifolds, we establish the so-called Nagai’s conjecture on the monodromy of degenerations of hyper-Kahler 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-Kahler manifolds.