Properties of Holomorphic $p$-Contact Manifolds

TL;DR

This paper advances the understanding of compact holomorphic p-contact manifolds by exploring non-Kähler hyperbolicity, introducing a new differential calculus, and establishing deformation unobstructedness, contributing to non-Kähler mirror symmetry research.

Contribution

It introduces the notion of p-contact deformations and proves an unobstructedness theorem, expanding the theory of holomorphic p-contact manifolds and their deformations.

Findings

Expanded the study to include non-Kähler hyperbolicity.

Proposed a new differential calculus based on the Lie derivative.

Proved a Bogomolov-Tian-Todorov-type unobstructedness theorem.

Abstract

We continue the study of compact holomorphic $p$-contact manifolds $X$ that we introduced recently by expanding the discussion to include non-K\"ahler hyperbolicity issues and a differential calculus based on what we call the Lie derivative with respect to a $(0,\,q)$-form with values in the holomorphic tangent bundle of $X$. We also propose the notion of $p$-contact deformations for which we prove a Bogomolov-Tian-Todorov-type unobstructedness theorem to order two. This kind of small deformations of the complex structure is related to the essential horizontal deformations that we introduced in our previous work and forms part of a wider on-going project aimed at developing a non-K\"ahler mirror symmetry theory that was first tested on the Iwasawa manifold and subsequently on Calabi-Yau page-$1$-$\partial\bar\partial$-manifolds.