Deformations of Non-Kähler Hyperbolicity Notions and Modifications of Degenerate Balanced Manifolds

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 ℂ X = n, in general degree 2p with 1 ≤ p ≤ 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 p-SKT hyperbolicity, extends SKT hyperbolicity and Gauduchon hyperbolicity to degree 2p. Similarly, the second notion, called 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 p-cyclic hyperbolicity in degree 2p for 2 ≤ p ≤ 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.