  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 application of this construction and discuss some general
properties of Calabi-Yau categories.

Algebraic Geometry