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

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 $(c_2(X),K_X^2)$ subject to $${\tau }_X:=2c{h}_2({\Theta }_X)=K_X^2-2c_2(X)>0$$ 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.

182 pages, 3 figures