Looijenga—Lunts and Verbitsky (LLV) have shown that the cohomology of
a compact hyper-Kaehler manifold admits the action of a big Lie
algebra g, generalizing the usual sl(2) Hard Lefschetz action. We
compute the LLV decomposition of the cohomology for the known classes
of hyper-Kaehler manifolds (i.e. K3^n, Kim_n, OG6, and OG10). As an
application, we easily recover the Hodge numbers of the exceptional
example OG10. In a different direction, we establish the so-called
Nagai’s conjecture (on the nilpotency index for higher degree
monodromy operators) for the known cases. More interestingly, based
on the known examples, we conjecture a new restriction on the
cohomology of compact hyper-Kaehler manifolds, which in particular
implies the vanishing of the odd cohomology as soon as the second
Betti number is large enough relative to the dimension.
This is joint work with M. Green, Y. Kim, and C. Robles.