Limit Mixed Hodge Structures of Hyperkähler Manifolds

Abstract

The purpose of this note is to give an account of a well-known folklore result: the Hodge structure on the second cohomology of a compact hyperkähler manifold uniquely determines Hodge structures on all higher cohomology groups. We discuss the precise statement and its proof, which are somewhat difficult to locate in the literature.

P = W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds

“…Theorem 3 also offers conceptual explanations to the main results in [9]. As is remarked in [4,Introduction], a recent result of Soldatenkov [12,Theorem 3.8] shows that limiting mixed Hodge structure for type III degenerations is of Hodge-Tate type. In particular, we have…”

“…, is precisely [12,Theorem 4.6]. Whereas the original statement requires 2 ( ) ≥ 5 to ensure the existence of an element ∈ 2 ( , Q) with ( ) = 0, in our situation is readily given by the Lagrangian fibration :…”

“…As in [12,Lemma 4.1], we obtain a nilpotent operator , = ∧ ∈ ( 2 ( , Q), ) whose action on 2 ( , Q) satisfies Im( , ) = , , Im…”

Teichmüller spaces and Torelli theorems for hyperkähler manifolds

“…We shall refer to this as a spin orientation on H . We make both the orientation on M and spin orientation on H as part of our initial data and so we shall only consider hyperkählerian structures that induce the given orientation (but as Soldatenkov [29] has noted, this is in fact automatically the case) and spin orientation (for which the same property might hold-by a theorem of Donaldson this is the case for K 3 surfaces). This spin orientation determines for every positive oriented 2-plane in H R , a positive cone: ⊥ has signature (1, n) and so the set of positive vectors in ⊥ make up an antipodal pair of open cones and the spin orientation singles out one of them.…”

“…We here treat a twistor deformation as if its base (a projective line) were a Shimura variety (which it certainly is not), as this not only is helpful in deriving the Torelli theorem, but also yields a simple way to formulate-and leads to a short way to obtain-a recent result of Soldatenkov [29] (qualified by him as 'folklore') and Green-Kim-Laza-Robles [9] on the period map for the full cohomology of a hyperkählerian manifold. Strictly speaking this last application is independent of the Torelli theorem, but we included it, because this merely comes as a bonus after the ground work done here.…”