Versal dg deformation of Calabi--Yau manifolds

TL;DR

This paper establishes a deep connection between the deformation theory of higher dimensional Calabi--Yau manifolds and their dg categories of perfect complexes, showing a natural isomorphism of deformation functors and exploring the uniqueness and properties of versal Morita deformations.

Contribution

It proves the equivalence of deformation theories for Calabi--Yau manifolds and their dg categories, and introduces the concept of the generic fiber of versal Morita deformations.

Findings

Deformation functors for manifolds and dg categories are naturally isomorphic.

Versal Morita deformations are unique up to certain equivalences.

The generic fiber of the versal deformation is quasi-equivalent to the dg category of perfect complexes.

Abstract

We prove the equivalence of the deformation theory for a higher dimensional Calabi--Yau manifold and that for its dg category of perfect complexes by giving a natural isomorphism of the deformation functors. As a consequence, the dg category of perfect complexes on a versal deformation of the original manifold provides a versal Morita deformation of its dg category of perfect complexes. Besides the classical uniqueness up to \'etale neiborhood of the base, we prove another sort of uniqueness of versal Morita deformations. Namely, given a pair of derived-equivalent higher dimensional Calabi--Yau manifolds, the dg categories of perfect complexes of their algebraic deformations over a common base, which always exist, become quasi-equivalent close to effectivizations. Then the base change along the corresponding first order approximation yields quasi-equivalent versal Morita deformations. We introduce the generic fiber of the versal Morita deformation as a Drinfeld quotient, which is quasi-equivalent to the dg category of perfect complexes on the generic fiber of the versal deformation.