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

We study the cohomology groups $H^1(X,\Theta _X(-mK_X))$, for $m\geq 1$, where $X$ is a smooth minimal complex surface of general type, $\Theta _X$ its holomorphic tangent bundle, and $K_X$ its canonical divisor. One of the main results is a precise vanishing criterion for $H^1(X,\Theta _X (-K_X))$ (Theorem 1.1).

The proof is based on the geometric interpretation of non-zero cohomology classes of $H^1(X,\Theta _X (-K_X))$. 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 $\mathbb {P}^4$ is at most $2$. We show that if $X$ has bounded holomorphic Euler characteristic, no irrational pencil, and is embedded in $\mathbb {P}^4$ with a sufficiently large degree, then the irregularity of $X$ is at most $3$.