Math @ Duke

Publications [#358294] of Colleen M Robles
Papers Published
 Green, M; Kim, YJ; Laza, R; Robles, C, The LLV decomposition of hyperKähler cohomology (the known cases and the general conjectural behavior),
Mathematische Annalen
(January, 2021) [doi]
(last updated on 2021/12/03)
Abstract: Looijenga–Lunts and Verbitsky showed that the cohomology of a compact hyperKähler 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 hyperKähler 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 hyperKähler 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 wellknown Göttsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyperKähler 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 hyperKähler manifolds of OG 10 type. In a different direction, for all known examples of hyperKähler manifolds, we establish the socalled Nagai’s conjecture on the monodromy of degenerations of hyperKähler 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 hyperKähler manifolds.


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