Deformations of Non-K\"ahler Hyperbolicity Notions and Modifications of Degenerate Balanced Manifolds

TL;DR

This paper introduces new hyperbolicity notions for non-K"ahler manifolds, explores their relationships with classical hyperbolicity, and studies their stability under deformations, advancing understanding of complex geometric structures.

Contribution

It proposes two novel hyperbolicity concepts for non-K"ahler manifolds, analyzes their connections to existing notions, and proves their stability under smooth deformations.

Findings

Introduction of $p$-SKT hyperbolicity and $p$-HS hyperbolicity.

Relationships established between analytic and geometric hyperbolicity notions.

Proved openness of $p$-HS and $p$-K"ahler hyperbolicity under deformations.

Abstract

We study deformation properties of balanced hyperbolicity, with a particular emphasis on degenerate balanced manifolds and their behavior under smooth modifications. From a different perspective, we introduce two new notions of hyperbolicity for compact complex non-K\"ahler manifolds $X$ of complex dimension $\dim_{\mathbb{C}}X=n$, in general degree $2p$ with $1 \leq p \leq n-1$. These notions are motivated by the work of D.~Popovici and H.~Kasuya on partial hyperbolicity in arbitrary degree and by the work of F.~Haggui and S.~Marouani on $p$-K\"ahler hyperbolicity. The first notion, called \emph{$p$-SKT hyperbolicity}, extends SKT hyperbolicity and Gauduchon hyperbolicity to degree $2p$. Similarly, the second notion, called \emph{$p$-HS hyperbolicity}, generalizes the notion of strongly Gauduchon hyperbolicity introduced by Y.~Ma. We then analyze the relationships between these analytic notions and geometric notions of hyperbolicity, namely Brody/Kobayashi hyperbolicity and \emph{$p$-cyclic hyperbolicity} in degree $2p$ for $2 \leq p \leq n-1$. In addition, we study the behavior of $p$-HS hyperbolicity and $p$-K\"ahler hyperbolicity under holomorphic deformations, establishing openness results for these properties.