Deformed Calabi-Yau completions

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.