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

TL;DR

This paper characterizes spherical objects on Kodaira cycles using topological models, computes auto-equivalence groups, and links these to vector bundle classifications, extending and answering questions in derived category theory.

Contribution

It provides a topological parametrization of spherical objects, computes auto-equivalence groups, and relates these to vector bundle classifications on Kodaira cycles.

Findings

Spherical objects correspond to closed curves on the punctured torus.

Auto-equivalence groups act transitively on spherical objects.

Simple vector bundles are uniquely determined by multi-degree, rank, and determinant.

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 and extends earlier results by Burban-Kreussler and Lekili-Polishchuk. The description of auto-equivalences is further used to establish faithfulness of a mapping class group action defined by Sibilla. 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 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.