The paper under review is a sequel of J. Chen et al. [Algebra Number Theory 13, No. 9, 2005–2055 (2019; Zbl 1490.16015)] by the same authors. In the present paper it is recalled the rather technical definition of the Frobenius-Perron dimension of an endofunctor \(\sigma:{\mathcal T}\to{\mathcal T}\), where \({\mathcal T}\) is a hom-finite \({\mathbb K}\)-category (\({\mathbb K}\) is a base field). If \({\mathcal T}\) is triangulated, e.g., it is the bounded derived category of coherent sheaves over a projective scheme \(\mathrm{D}^b(\mathrm{coh}\mathbb{X})\), then the endofunctor \(\sigma\) is actually the shift functor of \(\mathcal T\), and we denote by \(\mathrm{fp}(\mathcal{T})\) the Frobenius-Perron dimension.
The first main result says that is \(\mathbb X\) is a weighted projective line that is either domestic or tubular, then \(\mathrm{fp}\ \mathrm{D}^b(\mathrm{coh}{\mathbb X})=1\). Next there are introduced Frobenius-Perron versions for the Calabi-Yau dimension \(\mathrm{fpcy}({\mathcal T})\), respectively for the Kodaira dimension \(\mathrm{fp}\kappa({\mathcal T})\) for a suitable triangulated category \({\mathcal T}\). It is shown that they coincide with the classical invariants for \({\mathcal T}=\mathrm{D}^b(\mathrm{coh}{\mathbb X})\), where \(\mathbb X\) is a smooth projective scheme. The paper contains a number of examples where are computed the above defined invariants, e.g., \(\mathrm{fp}({\mathcal T})\), for \(\mathcal T\) being the bounded derived category of finite dimensional representations of some quivers, or \(\mathrm{fpcy}\ \mathrm{D}^b(\mathrm{coh}{\mathbb X})\), for \(\mathbb X\) being the noncommutative projective scheme associated to a noetherian connected graded Artin-Schelter Gorenstein algebra of injective dimension \(\geq2\).