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 non-commutative analogue of Joyce-Safronov's
Lagrangian neighbourhood theorem in derived symplectic geometry.Comment: 39 pages; v2: more accurate historical account in introduction,
reference to Joyce-Safronov's work added, many minor change