Relative Calabi-Yau structures and ice quivers with potential

TL;DR

This paper generalizes Van den Bergh's theorem, showing that morphisms with relative Calabi-Yau structures are equivalent to Ginzburg morphisms, connecting algebraic and geometric frameworks in higher-dimensional Calabi-Yau contexts.

Contribution

It extends Van den Bergh's classification to relative Calabi-Yau morphisms, establishing their equivalence to Ginzburg morphisms under broad conditions.

Findings

Any morphism with a relative Calabi-Yau structure is equivalent to a Ginzburg morphism.

In dimension 3 over an algebraically closed field of characteristic 0, such morphisms are given by ice quivers with potential.

The result links algebraic structures to non-commutative symplectic geometry.

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 non-commutative analogue of Joyce-Safronov's Lagrangian neighbourhood theorem in derived symplectic geometry.