The LLV decomposition of hyper-Kaehler cohomology

Abstract

Looijenga--Lunts and Verbitsky showed that the cohomology of a compact hyper-Kähler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so}(4,b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact Kähler manifolds. In this paper, we determine the Looijenga--Lunts--Verbitsky (LLV) decomposition for all known examples of compact hyper-Kähler 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-Kähler manifolds. Specifically, in the $K{3}^{[n]}$ and ${\mathrm{Kum}}_n$ cases, we give generating series for the formal characters of the associated LLV representations, which generalize the well-known Göttsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyper-Kähler 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-Kähler manifolds of O'Grady 10 type. In a different direction, for all known examples of hyper-Kähler manifolds, we establish the so-called Nagai's conjecture on the monodromy of degenerations of hyper-Kähler 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-Kähler manifolds.

48 pages; v4 -- Section 4.1 was rewritten following a referee's suggestion; v3 -- added a discussion on the conjectural behavior of the LLV decomposition in general, added several recent references, in particular, [FFZ19] allows us to recover the cohomology of OG10 (independently of [dCRS19])

References in corpus (4)

• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum
• Derived equivalences of hyperkähler varieties
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Cited by in corpus (6)

• Motivic decompositions for the Hilbert scheme of points of a K3 surface
• On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four
• P=W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds
• Mixed Hodge structures in log symplectic geometry
• A conjectural bound on the second Betti number for hyper-Kähler manifolds
• On the Mumford-Tate conjecture for hyperkähler varieties