Spherical objects, transitivity and auto-equivalences of Kodaira cycles via gentle algebras

Abstract

This paper studies the class of spherical objects over any Kodaira $n$-cycle of projective lines and provides a parametrization of their isomorphism classes in terms of closed curves on the $n$-punctured torus without self-intersections. Employing recent results on gentle algebras, we derive a topological model for the bounded derived category of any Kodaira cycle. The groups of triangle auto-equivalences of these categories are computed and are shown to act transitively on isomorphism classes of spherical objects. This answers a question by Polishchuk (2002) and extends earlier results by Burban–Kreußler (2012) and Lekili–Polishchuk (2019). The description of auto-equivalences is further used to establish faithfulness of a mapping class group action defined by Sibilla (2014). The final part describes the closed curves which correspond to vector bundles and simple vector bundles. This leads to an alternative proof of a result by Bodnarchuk–Drozd–Greuel (2012) which states that simple vector bundles on cycles of projective lines are uniquely determined by their multi-degree, rank and determinant. As a by-product, we obtain a closed formula for the cyclic sequence of any simple vector bundle on $C_n$ as introduced by Burban–Drozd–Greuel (2001).