Bridgeland stability conditions and the tangent bundle of surfaces of general type

TL;DR

This paper explores how Bridgeland stability conditions can be used to analyze the deformation space and moduli of complex surfaces of general type, revealing instability of the tangent bundle in certain conditions.

Contribution

It introduces a novel approach linking Bridgeland stability to the geometry of surfaces of general type, especially through the instability of the tangent bundle in specific cases.

Findings

$ heta_X[1]$ is Bridgeland unstable under certain conditions

Harder-Narasimhan filtrations offer new geometric insights

Evidence supports the use of stability conditions in surface classification

Abstract

Let $X$ be a smooth compact complex surface with the canonical divisor $K_X$ ample and let $\Theta_X$ be its holomorphic tangent bundle. Bridgeland stability conditions are used to study the space $H^1 (\Theta_X)$ of infinitesimal deformations of complex structures of $X$ and its relation to the geometry/topology of $X$. The main observation is that for $X$ with $H^1 (\Theta_X)$ nonzero and the Chern numbers $(c_2 (X), K^2_X)$ subject to $$ \tau_X :=2ch_2 (\Theta_X)=K^2_X -2c_2(X) >0 $$ the object $\Theta_X [1]$ of the derived category of bounded complexes of coherent sheaves on $X$ is Bridgeland unstable in a certain part of the space of Bridgeland stability conditions. The Harder-Narasimhan filtrations of $\Theta_X [1]$ for those stability conditions are expected to provide new insights into geometry of surfaces of general type and the study of their moduli. The paper provides a certain body of evidence that this is indeed the case.