Deformed Aeppli cohomology: canonical deformations and jumping formulas

TL;DR

This paper investigates how Aeppli cohomology varies in complex manifolds under canonical deformations, establishing jumping formulas and conditions for unobstructedness related to the $ar{ ext{d}}$-lemma.

Contribution

It introduces canonical Aeppli deformations, proves jumping formulas for their cohomology, and links unobstructedness to weak $ar{ ext{d}}ar{ ext{d}}$-lemma conditions.

Findings

Proved jumping formulas for deformed Aeppli cohomology.

Established conditions for constant cohomology dimension under deformations.

Showed unobstructedness of Bott-Chern/Aeppli deformations under weak $ar{ ext{d}}ar{ ext{d}}$-lemma.

Abstract

Given a complex analytic family of complex manifolds, we consider canonical Aeppli deformations of $(p,q)$-forms and study its relations to the varying of dimension of the deformed Aeppli cohomology $\dim H^{\bullet,\bullet}_{A\phi(t)}(X)$. In particular, we prove the jumping formula for the deformed Aeppli cohomology $H^{\bullet,\bullet}_{A\phi(t)}(X)$. As a direct consequence, $\dim H^{p,q}_{A\phi(t)}(X)$ remains constant iff the Bott-Chern deformations of $(n-p,n-q)$-forms and the Aeppli deformations of $(n-p-1,n-q-1)$-forms are canonically unobstructed. Furthermore, the Bott-Chern/Aeppli deformations are shown to be unobstructed if some weak forms of $\partial\bar{\partial}$-lemma is satisfied.