Extension formulas and deformation invariance of Hodge numbers

TL;DR

This paper introduces a canonical isomorphism for complex differential forms on compact complex manifolds, generalizes extension formulas, and proves deformation invariance of Hodge numbers for specific classes of manifolds.

Contribution

It provides a new canonical isomorphism and extends existing formulas, leading to deformation invariance results without relying on traditional inequalities or Betti number invariance.

Findings

Established a canonical isomorphism between differential forms on manifolds and their deformations.

Generalized an extension formula for complex differential forms.

Proved deformation invariance of Hodge numbers for certain classes of complex manifolds.

Abstract

We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the second author. As a direct corollary of the extension formulas, we prove several deformation invariance theorems for Hodge numbers on some certain classes of complex manifolds, without use of Fr\"{o}licher inequality or the topological invariance of Betti numbers.