  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.
