Relative Calabi-Yau structures and ice quivers with potential

Abstract

In 2015, Van den Bergh showed that complete $3$-Calabi-Yau algebras over an algebraically closed field of characteristic $0$ are equivalent to Ginzburg dg algebras associated with quivers with potential. He also proved the natural generalisation to higher dimensions and non-algebraically closed ground fields. The relative version of the notion of Ginzburg dg algebra is that of Ginzburg morphism. For example, every ice quiver with potential gives rise to a Ginzburg morphism. We generalise Van den Bergh's theorem by showing that, under suitable assumptions, any morphism with a relative Calabi-Yau structure is equivalent to a Ginzburg(-Lazaroiu) morphism. In particular, in dimension $3$ and over an algebraically closed ground field of characteristic $0$, it is given by an ice quiver with potential. Thanks to the work of Bozec-Calaque-Scherotzke, this result can also be viewed as a noncommutative analogue of Joyce-Safronov's Lagrangian neighbourhood theorem in derived symplectic geometry.

43 pages; v2: more accurate historical account in introduction, reference to Joyce-Safronov's work added, many minor changes; v3: references added, readability improved; v4: published in Advances in Mathematics