Obstructions of deforming complex structures and cohomology contractions

TL;DR

This paper provides an analytic approach to the Kodaira principle, demonstrating that under certain conditions, all obstructions to deforming complex structures vanish, leading to new criteria for unobstructedness in complex geometry.

Contribution

It develops a refined power series method for deformations and extends the Kodaira principle using the Frölicher spectral sequence, offering new unobstructedness criteria.

Findings

Obstructions lie in the kernel of contraction maps under partial vanishing conditions.

Refined Kodaira principle extends previous results.

New unobstructedness criteria for manifolds with trivial canonical bundle.

Abstract

The Kodaira principle asserts that suitable cohomological contraction maps annihilate obstructions to deforming complex structures. In this paper, we revisit these phenomena from a purely analytic point of view, developing a refined power series method for the deformation of $(p,q)$-forms and complex structures. Working with the Fr\"olicher spectral sequence, we show that under natural partial vanishing conditions on its differentials, all obstruction classes lie in the kernel of the corresponding contraction maps. This yields a refined Kodaira principle that recovers and strictly extends the known results. As a main application, we obtain new unobstructedness criteria for compact complex manifolds with trivial canonical bundle.