Degeneration at E2 of certain spectral sequences

Abstract

We propose a Hodge theory for the spaces [Formula: see text] featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold [Formula: see text] or in the spectral sequence associated with a pair [Formula: see text] of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on [Formula: see text] whose kernel in every bidegree [Formula: see text] is isomorphic to [Formula: see text] in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for [Formula: see text] to give sufficient conditions for the degeneration at [Formula: see text] of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on [Formula: see text]. For example, in the Frölicher case, we prove degeneration at [Formula: see text] if there exists an SKT metric [Formula: see text] (i.e. such that [Formula: see text]) whose torsion is small compared to the spectral gap of the elliptic operator [Formula: see text] defined by [Formula: see text]. In the foliated case, we obtain degeneration at [Formula: see text] under a hypothesis involving the Laplacians [Formula: see text] and [Formula: see text] associated with the splitting [Formula: see text] induced by the foliated structure.