An integral condition involving $\overline\partial$-harmonic $(0,1)$-forms

TL;DR

This paper investigates integral conditions involving $ar{ ext{d}}$-harmonic $(0,1)$-forms on compact almost complex manifolds, establishing automatic satisfaction under strongly Gauduchon metrics and exploring implications for integrable and non-integrable structures.

Contribution

It proves the integral condition is automatically satisfied for strongly Gauduchon metrics and shows equivalence to being strongly Gauduchon in the integrable case, with applications to complex surfaces.

Findings

Integral condition holds automatically for strongly Gauduchon metrics.

In integrable case, the condition is equivalent to being strongly Gauduchon.

Existence of non-integrable examples satisfying the condition without compatible almost-K"ahler metrics.

Abstract

We study compact almost complex manifolds admitting a Hermitian metric satisfying an integral condition involving $\overline \partial$-harmonic $(0,1)$-forms. We prove that this integral condition is automatically satisfied, if the Hermitian metric on the compact almost complex manifold is strongly Gauduchon. Under the further assumption that the almost complex structure is integrable, we show that the integral condition for a Gauduchon metric is equivalent to be strongly Gauduchon. In particular, a compact complex surface with a Gauduchon metric satisfying the integral condition is automatically K\"ahler. If we drop the integrability assumption on the complex structure, we show that there exists a compact almost complex $4$-dimensional manifold with a Hermitian metric satisfying the integral condition, but which does not admit any compatible almost-K\"ahler metric.