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

Abstract

We study the cohomology groups H 1 ( X , Θ X ( − m K X ) ) H^1(X,\Theta _X(-mK_X)) , for m ≥ 1 m\geq 1 , where X X is a smooth minimal complex surface of general type, Θ X \Theta _X its holomorphic tangent bundle, and K X K_X its canonical divisor. One of the main results is a precise vanishing criterion for H 1 ( X , Θ X ( − K X ) ) 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 , Θ X ( − K X ) ) H^1(X,\Theta _X (-K_X)) . This interpretation in turn uses higher rank vector bundles on X X . We apply our methods to the long standing conjecture saying that the irregularity of surfaces in P 4 \mathbb {P}^4 is at most 2 2 . We show that if X X has bounded holomorphic Euler characteristic, no irrational pencil, and is embedded in P 4 \mathbb {P}^4 with a sufficiently large degree, then the irregularity of X X is at most 3 3 .