Calabi–Yau and fractional Calabi–Yau categories

We discuss Calabi–Yau and fractional Calabi–Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi–Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of applications of this construction and discuss some general properties of Calabi–Yau categories.

Lefschetz Decompositions for Quotient Varieties

In an earlier paper, the authors constructed an explicit Chow-Künneth decomposition for the quotient of an abelian variety by the action of a finite group. In the present paper, the authors extend the techniques used there to obtain an explicit Lefschetz decomposition for such quotient varieties.