Polarisation of SKT Calabi-Yau $\partial\bar\partial$-manifolds by Aeppli classes

TL;DR

This paper studies how small deformations of certain complex manifolds with special metrics are classified and related through Aeppli and Bott-Chern cohomology classes, revealing new geometric correspondences.

Contribution

It introduces the concept of polarisation by Aeppli classes for $ ext{SKT}$ manifolds and establishes a correspondence with primitive Bott-Chern classes in their deformation space.

Findings

Establishes a correspondence between polarised manifolds and primitive Bott-Chern classes.

Analyzes the existence of primitive elements in Bott-Chern classes.

Compares metrics on the base space of polarised subfamilies within the Kuranishi family.

Abstract

Given a $\partial\bar\partial$-manifold $X$ with trivial canonical bundle and carrying a metric $\omega$ such that $\partial\bar\partial\omega=0$, we introduce the concept of small deformations of $X$ polarised by the Aeppli cohomology class $[\omega]_A$ of an SKT metric $\omega$. There is a correspondence between the manifolds polarised by $[\omega]_A$ in the Kuranishi family of $X$ and the Bott-Chern classes that are primitive in a sense that we define. We also investigate the existence of a primitive element in an arbitrary Bott-Chern primitive class and compare the metrics on the base space of the subfamily of manifolds polarised by $[\omega]_A$ within the Kuranishi family.