Orbits in stability manifolds and categorical entropy: applications to varieties with finite Albanese morphisms

Source abstract

This paper proves the Gromov--Yomdin property for smooth projective varieties with finite Albanese morphisms. Our approach is to study the orbits of stability conditions under the action of autoequivalences on Bridgeland stability manifolds. For a finite generating set of the derived category, we consider the locus on which all its objects are semistable. Using this locus, we formulate an orbit escape principle, which provides a sufficient condition for an autoequivalence to satisfy the Gromov--Yomdin equality. Together with the stability of simple semihomogeneous bundles with respect to arbitrary numerical stability conditions, this principle readily yields the result for abelian varieties. We then reduce the case of varieties with finite Albanese morphisms to the abelian case by passing to suitable finite étale covers. Finally, we construct autoequivalences with positive categorical entropy on every abelian variety and show that a smooth projective variety with a finite Albanese morphism admits such an autoequivalence if and only if it is not of general type.