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

TL;DR

This paper investigates the cohomology of twisted Kodaira-Spencer classes on surfaces of general type, providing vanishing criteria and applying these results to bound the irregularity of certain embedded surfaces.

Contribution

It introduces a new geometric interpretation of cohomology classes using higher rank vector bundles and applies this to longstanding conjectures about surface irregularity.

Findings

Established a vanishing criterion for H^1(X, Θ_X (-K_X)).

Linked cohomology classes to geometric structures via vector bundles.

Bound the irregularity of certain surfaces in P^4 to at most 3.

Abstract

We study the cohomology groups $H^1(X,\Theta_X(-mK_X))$, for $m\geq1$, 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 $H^1(X,\Theta_X (-K_X))$. 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 $\PP^4$ is at most 2. We show that if $X$ has prescribed Chern numbers, no irrational pencil, and is embedded in $\PP^4$ with a sufficiently large degree, then the irregularity of $X$ is at most 3.