Mathematics – Algebraic Geometry
Scientific paper
2011-09-06
Mathematics
Algebraic Geometry
32 pages References added and some minor changes in the last section
Scientific paper
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.
Naie Daniel
Reider Igor
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