Twisted Kodaira-Spencer classes and the geometry of surfaces of general type

We study the cohomology groups $H^1(X,{\mathrm{\Theta }}_X(-m{K}_X))$, for $m\ge 1$, where $X$ is a smooth minimal complex surface of general type, ${\mathrm{\Theta }}_X$ its holomorphic tangent bundle, and $K_X$ its canonical divisor. One of the main results is a precise vanishing criterion for $H^1(X,{\mathrm{\Theta }}_X(-K_X))$. The proof is based on the geometric interpretation of non-zero cohomology classes of $H^1(X,{\mathrm{\Theta }}_X(-K_X))$. This interpretation in turn uses higher rank vector bundles on $X$. We apply our methods to the long standing conjecture saying that the irregularity of surfaces in ${\text{PP}}^4$ is at most 2. We show that if $X$ has prescribed Chern numbers, no irrational pencil, and is embedded in ${\text{PP}}^4$ with a sufficiently large degree, then the irregularity of $X$ is at most 3.