Degeneration at $E_2$ of Certain Spectral Sequences

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.