Co-polarised Deformations of Gauduchon Calabi-Yau $\partial\bar{\partial}$-Manifolds and Deformation of $p$-SKT $h$-$\partial\bar{\partial}$-Manifolds

TL;DR

This paper investigates the deformation theory of Calabi-Yau $ ext{ extbackslash}partial ext{ extbackslash}bar{ extbackslash}partial$-manifolds with Gauduchon metrics and introduces the $hp$-Hermitian symplectic form to study the stability of $p$-SKT $h$-$ ext{ extbackslash}partial ext{ extbackslash}bar{ extbackslash}partial$-properties.

Contribution

It establishes the deformation openness of the $p$-SKT $h$-$ ext{ extbackslash}partial ext{ extbackslash}bar{ extbackslash}partial$-property and introduces the concept of $hp$-Hermitian symplectic form.

Findings

Deformation of co-polarised Calabi-Yau $ ext{ extbackslash}partial ext{ extbackslash}bar{ extbackslash}partial$-manifolds is studied.

The $p$-SKT $h$-$ ext{ extbackslash}partial ext{ extbackslash}bar{ extbackslash}partial$-property is shown to be deformation open.

Introduction of $hp$-Hermitian symplectic form as a tool for deformation analysis.

Abstract

The main result of this paper is to study the local deformations of Calabi-Yau $\partial\bar{\partial}$-manifold that are co-polarised by the Gauduchon metric by considering the subfamily of co-polarised fibres by the class of Aeppli/De Rham-Gauduchon cohomology of Gauduchon metric given at the beginning on the central fibre. In the latter part, we prove that the $p$-SKT $h$-$\partial\bar{\partial}$-property is deformation open by constructing and studying a new notion called $hp$-Hermitian symplectic ($hp$-HS) form.