Deformations of noncompact Calabi--Yau manifolds, families and diamonds

TL;DR

This paper introduces a novel deformation theory for noncompact Calabi--Yau manifolds, providing new tools and examples for understanding their complex structures and Hodge invariants.

Contribution

It develops a new deformation framework tailored for noncompact manifolds and illustrates it with diverse examples including toric Calabi--Yau threefolds and cotangent bundles.

Findings

New deformation notion suited for noncompact manifolds

Explicit examples with computed Hodge and KKP diamonds

Enhanced understanding of complex structure variations in noncompact settings

Abstract

We introduce a new notion of deformation of complex structure, which we use as an adaptation of Kodaira's theory of deformations, but that is better suited to the study of noncompact manifolds. We present several families of deformations illustrating this new approach. Our examples include toric Calabi--Yau threefolds, cotangent bundles of flag manifolds, and semisimple adjoint orbits, and we describe their Hodge theoretical invariants, depicting Hodge diamonds and KKP diamonds.