Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

Authors: Daniel Naie and Igor Reider
Journal: J. Algebraic Geom. 23 (2014), 165-200
Published electronically: August 6, 2013
MathSciNet review: 3121851
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Abstract: 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$.

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Additional Information

Daniel Naie
Affiliation: Université d’Angers, Département de Mathématiques, 49045 Angers, France

Igor Reider
Affiliation: Université d’Angers, Departement de Mathematique, 2, Bd. Lavoisier, 49045 Angers, France

Received by editor(s): February 8, 2011
Received by editor(s) in revised form: July 21, 2011
Published electronically: August 6, 2013
Dedicated: Dedicated to Fedor Bogomolov on his 65th birthday
Article copyright: © Copyright 2013 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.