Chern degree functions

TL;DR

The paper introduces Chern degree functions for complexes of coherent sheaves on polarized surfaces, analyzing their properties and relation to stability conditions, with specific results for abelian surfaces.

Contribution

It defines and studies the properties of Chern degree functions, extending stability condition analysis to boundary cases and relating them to cohomological rank functions on abelian surfaces.

Findings

Chern degree functions extend to continuous real-valued functions.

They are differentiable in relation to stability conditions.

On abelian surfaces, they coincide with Jiang-Pareschi's cohomological rank functions.

Abstract

We introduce Chern degree functions for complexes of coherent sheaves on a polarized surface, which encode information given by stability conditions on the boundary of the $(\alpha, \beta)$-plane. We prove that these functions extend to continuous real valued functions and we study their differentiability in terms of stability. For abelian surfaces, Chern degree functions coincide with the cohomological rank functions defined by Jiang-Pareschi. We illustrate in some geometrical situations a general strategy to compute these functions.