Geography and the Number of Moduli of Surfaces of General Type

The paper considers a relationship between the Chern numbers $K^2_X,\,c_2(X)$ of a smooth minimal surface $X$ of general type and the dimension of the space of infinitesimal deformations of $X$, i.e. $\HI$, where $\TET$ is the holomorphic tangent bundle of $X$. We prove that if the ratio of the Chern numbers $\AL(X) = \frac{c_2(X)}{K^2_X} \leq \TE$ and $K_X$ is ample then $$\HI \leq 9(\BY).$$ On the geometric side it is shown that a smooth surface of general type $X$ with $\AL(X)\leq \TE$ and $\HI \geq 3$ has two distinguished effective divisors $ F$ and $E$ such that $\HH$ admits a direct sum decomposition $\HH = V_1 \oplus V_0$, where $V_1$ is identified with a subspace of $H^0(\OO_X (F))$ while $V_0$ is identified with a subspace of $H^0(\TET \otimes \OO_E (E))$. This gives a geometric interpretation of the cohomology classes in $\HH$ and allows to bound the dimension of $V_0$ (resp. $V_1$) in terms of geometry of the divisor $E$ (resp. $F$).

The main idea of the paper is to use the natural identification $$\HH = Ext^1(\OM, \OO_X)$$ where $\OM$ is the holomorphic cotangent bundle of $X$. Then the "universal" extension gives rise to a certain vector bundle whose study constitutes the essential part of the paper.

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Published 1 January 2005