  As a generalization of a Calabi-Yau category, we will say a k-linear
Hom-finite triangulated category is fractionally Calabi-Yau if it admits a
Serre functor S and there is an n > 0 with S^n = [m]. An abelian category will
be called fractionally Calabi-Yau is its bounded derived category is. We
provide a classification up to derived equivalence of abelian hereditary
fractionally Calabi-Yau categories (for algebraically closed k). They are: the
category of finite dimensional representations of a Dynkin quiver, the category
of finite dimensional nilpotent representations of a cycle, and the category of
coherent sheaves on an elliptic curve or a weighted projective line of tubular
type. To obtain this classification, we introduce generalized 1-spherical
objects and use them to obtain results about tubes in hereditary categories
(which are not necessarily fractionally Calabi-Yau).
