Some Properties of Balanced Hyperbolic Compact Complex Manifolds

TL;DR

This paper establishes vanishing theorems, a Hard Lefschetz-type theorem, and non-existence results for certain currents on balanced hyperbolic compact complex manifolds, advancing understanding of their geometric and cohomological properties.

Contribution

It introduces new vanishing theorems and a Hard Lefschetz-type theorem for balanced hyperbolic manifolds, expanding the theoretical framework of complex geometry.

Findings

Vanishing theorems for cohomology of balanced hyperbolic manifolds

A Hard Lefschetz-type theorem for certain compact complex balanced manifolds

Non-existence of specific $L^1$ currents on universal covers

Abstract

We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the $L^2$ harmonic spaces on the universal cover of these manifolds. Other results include a Hard Lefschetz-type theorem for certain compact complex balanced manifolds and the non-existence of certain $L^1$ currents on the universal covering space of a balanced hyperbolic manifold.