Higher-Degree Holomorphic Contact Structures

Abstract

We introduce the classes of holomorphic $p$-contact manifolds and holomorphic $s$-symplectic manifolds that generalise the classical holomorphic contact and holomorphic symplectic structures. After observing their basic properties and exhibiting a wide range of examples, we give two types of general conceptual results involving the former class of manifolds: structure theorems and unobstructedness theorems. The latter type generalises to our context the classical Bogomolov-Tian-Todorov theorem for a type of small deformations of complex structures that generalise the small essential deformations previously introduced for the Iwasawa manifold and for Calabi-Yau page-$1$-$\partial \bar{\partial }$-manifolds.

33 pages. This is essentially the first part of our original submission that is being updated. The second part has been posted as the new submission arXiv:2511.10818v1 [math.DG]