Higher-Degree Holomorphic Contact Structures

TL;DR

This paper introduces new classes of holomorphic contact and symplectic manifolds, explores their properties, and establishes structure and unobstructedness theorems, extending classical deformation results to these generalized structures.

Contribution

It defines holomorphic p-contact and s-symplectic manifolds, provides examples, and proves structure and unobstructedness theorems generalizing classical deformation results.

Findings

Introduction of holomorphic p-contact and s-symplectic manifolds

Establishment of structure theorems for these manifolds

Proof of unobstructedness theorems extending classical deformation results

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.