Degeneration at $E_2$ of Certain Spectral Sequences

TL;DR

This paper develops a Hodge theory for the second page of certain spectral sequences on compact complex manifolds, providing conditions for their degeneration at E2 based on metric properties.

Contribution

It introduces a Laplace-type operator linked to Hermitian metrics that characterizes E2 spaces and establishes new criteria for spectral sequence degeneration.

Findings

Hodge isomorphism for E2 spaces established

Degeneration at E2 occurs under specific metric conditions

Conditions involve SKT metrics and spectral gaps of elliptic operators

Abstract

We propose a Hodge theory for the spaces $E_2^{p,\,q}$ featuring at the second step either in the Fr\"olicher spectral sequence of an arbitrary compact complex manifold $X$ or in the spectral sequence associated with a pair $(N,\,F)$ 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 $X$ whose kernel in every bidegree $(p,\,q)$ is isomorphic to $E_2^{p,\,q}$ 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 $E_2^{p,\,q}$ to give sufficient conditions for the degeneration at $E_2$ of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on $X$. For example, in the Fr\"olicher case we prove degeneration at $E_2$ if there exists an SKT metric $\omega$ (i.e. such that $\partial\bar\partial\omega=0$) whose torsion is small compared to the spectral gap of the elliptic operator $\Delta' + \Delta"$ defined by $\omega$. In the foliated case, we obtain degeneration at $E_2$ under a hypothesis involving the Laplacians $\Delta'_N$ and $\Delta'_F$ associated with the splitting $\partial = \partial_N + \partial_F$ induced by the foliated structure.