Deformed Calabi-Yau Completions

TL;DR

This paper introduces deformed Calabi-Yau completions of dg categories, demonstrating their Calabi-Yau property, compatibility with derived equivalences, and applications to Ginzburg dg algebras, cluster algebras, and quiver mutations.

Contribution

It defines deformed Calabi-Yau completions, proves their Calabi-Yau property, and connects them to Ginzburg dg algebras and cluster theory, expanding the understanding of Calabi-Yau structures in algebraic geometry.

Findings

Deformed Calabi-Yau completions are Calabi-Yau.

Ginzburg dg algebras are examples of deformed Calabi-Yau completions.

Deformed 3-Calabi-Yau completions relate to Ginzburg dg algebras and cluster-tilted algebras.

Abstract

We define and investigate deformed n-Calabi-Yau completions of homologically smooth differential graded (=dg) categories. Important examples are: deformed preprojective algebras of connected non Dynkin quivers, Ginzburg dg algebras associated to quivers with potentials and dg categories associated to the category of coherent sheaves on the canonical bundle of a smooth variety. We show that deformed Calabi-Yau completions do have the Calabi-Yau property and that their construction is compatible with derived equivalences and with localizations. In particular, Ginzburg dg algebras have the Calabi-Yau property. We show that deformed 3-Calabi-Yau completions of algebras of global dimension at most 2 are quasi-isomorphic to Ginzburg dg algebras and apply this to the study of cluster-tilted algebras and to the construction of derived equivalences associated to mutations of quivers with potentials. In the appendix, Michel Van den Bergh uses non commutative differential geometry to give an alternative proof of the fact that Ginzburg dg algebras have the Calabi-Yau property.