Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Holomorphic Deformations of Hyperbolicity Notions on Compact Complex Manifolds

Abdelouahab Khelifati

TL;DR

The paper investigates how balanced hyperbolicity notions behave under deformations on compact complex manifolds, introducing $p$-SKT hyperbolicity and $p$-HS hyperbolicity as degree-$p$ analogues of SKT/Gauduchon and sG hyperbolicity. It relates these analytic notions to geometric hyperbolicity concepts such as Kobayashi and $p$-cyclic hyperbolicity and establishes deformation openness results, including stability under smooth modifications and under $(n-1,n)$-th weak $arar{ ext{d}}$-Lemmas. Key contributions include generalizing deformation stability to higher degrees, proving implications among the $p$-hyperbolic notions, and exploring how these properties interact with Bott-Chern cohomology and cohomological conditions. The results deepen our understanding of non-Kähler hyperbolicity and provide a framework for further exploration of deformation theory in complex geometry, with conjectural links between $p$-SKT, $p$-HS, and $p$-Kähler hyperbolicity under weaker hypotheses.

Abstract

We investigate deformation properties of balanced hyperbolicity, with particular emphasis on degenerate balanced manifolds and their behavior under modifications. In this context, we introduce two new notions of hyperbolicity for compact non-Kähler manifolds $X$ of complex dimension $\dim_{\mathbb{C}}X=n$ in degree $1 \leq p \leq n-1$, inspired by the work of F. Haggui and S. Marouani on $p$-Kähler hyperbolicity. The first notion, called \emph{p-SKT hyperbolicity}, generalizes the notions of SKT hyperbolicity and Gauduchon hyperbolicity introduced by S. Marouani. The second notion, called \emph{p-HS hyperbolicity}, extends the notion of sG hyperbolicity defined by Y. Ma. We investigate the relationship between these notions of analytic nature and their geometric counterparts, namely Kobayashi hyperbolicity and \emph{p-cyclic hyperbolicity} for $2 \leq p \leq n-1$, and we examine the openness under holomorphic deformations of both $p$-HS hyperbolicity and $p$-Kähler hyperbolicity.

Holomorphic Deformations of Hyperbolicity Notions on Compact Complex Manifolds

TL;DR

The paper investigates how balanced hyperbolicity notions behave under deformations on compact complex manifolds, introducing -SKT hyperbolicity and -HS hyperbolicity as degree- analogues of SKT/Gauduchon and sG hyperbolicity. It relates these analytic notions to geometric hyperbolicity concepts such as Kobayashi and -cyclic hyperbolicity and establishes deformation openness results, including stability under smooth modifications and under -th weak -Lemmas. Key contributions include generalizing deformation stability to higher degrees, proving implications among the -hyperbolic notions, and exploring how these properties interact with Bott-Chern cohomology and cohomological conditions. The results deepen our understanding of non-Kähler hyperbolicity and provide a framework for further exploration of deformation theory in complex geometry, with conjectural links between -SKT, -HS, and -Kähler hyperbolicity under weaker hypotheses.

Abstract

We investigate deformation properties of balanced hyperbolicity, with particular emphasis on degenerate balanced manifolds and their behavior under modifications. In this context, we introduce two new notions of hyperbolicity for compact non-Kähler manifolds of complex dimension in degree , inspired by the work of F. Haggui and S. Marouani on -Kähler hyperbolicity. The first notion, called \emph{p-SKT hyperbolicity}, generalizes the notions of SKT hyperbolicity and Gauduchon hyperbolicity introduced by S. Marouani. The second notion, called \emph{p-HS hyperbolicity}, extends the notion of sG hyperbolicity defined by Y. Ma. We investigate the relationship between these notions of analytic nature and their geometric counterparts, namely Kobayashi hyperbolicity and \emph{p-cyclic hyperbolicity} for , and we examine the openness under holomorphic deformations of both -HS hyperbolicity and -Kähler hyperbolicity.
Paper Structure (3 sections, 23 theorems, 37 equations)

This paper contains 3 sections, 23 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Deformations of Balanced hyperbolic manifolds
  3. p-SKT and p-HS hyperbolicity

Key Result

Proposition 2.2

A compact complex manifold $X$ is degenerate balanced if and only if it admits no non-zero $d$-closed $(1,1)$-current $T\geq 0$.

Theorems & Definitions (64)

  • Definition 2.1: Balanced hyperbolicity, marouani2023balanced
  • Proposition 2.2: popovici2015aeppli, Proposition 5.4
  • Proposition 2.3
  • proof
  • proof
  • Remark 2.5
  • Definition 2.6: popovici2013deformation, Definition 4.1 & Proposition 4.2
  • Definition 2.7: ma2024strongly, Definition 2.2
  • Definition 2.8: ma2024strongly, Definition 2.8
  • Proposition 2.9
  • ...and 54 more