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-Kähler manifold X admits a natural action by the Lie algebra so(4 , b2(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 b2(X) of hyper-Kähler manifolds (see Kim and Laza in Bull Soc Math Fr 148(3):467–480, 2020). Specifically, in the K3 [n] and 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 (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-Kähler manifolds of OG 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.
Green, M; Kim, YJ; Laza, R; Robles, C
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