Deformations of Non-Kähler Hyperbolicity Notions and Modifications of Degenerate Balanced Manifolds
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arXiv
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| Format: | Preprint |
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2025
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| author | Khelifati, Abdelouahab |
| author_facet | Khelifati, Abdelouahab |
| contents | 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-Kähler 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 $p$-Kähler 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 $p$-Kähler hyperbolicity under holomorphic deformations, establishing openness results for these properties. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_19284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deformations of Non-Kähler Hyperbolicity Notions and Modifications of Degenerate Balanced Manifolds Khelifati, Abdelouahab Complex Variables Algebraic Geometry Differential Geometry 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-Kähler 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 $p$-Kähler 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 $p$-Kähler hyperbolicity under holomorphic deformations, establishing openness results for these properties. |
| title | Deformations of Non-Kähler Hyperbolicity Notions and Modifications of Degenerate Balanced Manifolds |
| topic | Complex Variables Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2512.19284 |