Positivity Cones under Deformations of Complex Structures

TL;DR

This paper explores the properties of positivity cones and the degeneration of the Fr"olicher spectral sequence in complex manifolds, introducing new concepts like the $h$-$ar ext{d}$-lemma and analyzing their behavior under deformations.

Contribution

It introduces a positivity cone in $E_2$-cohomology and an $h$-$ar ext{d}$-lemma analogue, advancing understanding of complex structure deformations.

Findings

Partial degeneration at $E_2$ of the Fr"olicher spectral sequence.

Positivity cone behaves lower semicontinuously under deformations.

The $h$-$ar ext{d}$-lemma property is deformation open.

Abstract

We investigate connections between the sGG property of compact complex manifolds, defined in earlier work by the second author and L. Ugarte by the requirement that every Gauduchon metric be strongly Gauduchon, and a possible degeneration of the Fr\"olicher spectral sequence. In the first approach that we propose, we prove a partial degeneration at $E_2$ and we introduce a positivity cone in the $E_2$-cohomology of bidegree $(n-2,\,n)$ of the manifold that we then prove to behave lower semicontinuously under deformations of the complex structure. In the second approach that we propose, we introduce an analogue of the $\partial\bar\partial$-lemma property of compact complex manifolds for any real non-zero constant $h$ using the partial twisting $d_h$, introduced recently by the second author, of the standard Poincar\'e differential $d$. We then show, among other things, that this $h$-$\partial\bar\partial$-property is deformation open.