Geography and the Number of Moduli of Surfaces of General Type

Abstract

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. $h^1(\Theta_X)$, where $\Theta_X$ is the holomorphic tangent bundle of $X$. We prove that if the ratio of the Chern numbers $\alpha(X) = \frac{c_2(X)}{K^2_X} \leq \frac38$ and $K_X$ is ample then $$h^1(\Theta_X) \leq 9(3c_2-K^2).$$ On the geometric side it is shown that a smooth surface of general type $X$ with $\alpha(X)\leq \frac38$ and $h^1(\Theta_X) \geq 3$ has two distinguished effective divisors $ F$ and $E$ such that $H^1(\Theta_X)$ admits a direct sum decomposition $H^1(\Theta_X) = V_1 \oplus V_0$, where $V_1$ is identified with a subspace of $H^0(\mathcal{O}_X (F))$ while $V_0$ is identified with a subspace of $H^0(\Theta_X \otimes \mathcal{O}_E (E))$. This gives a geometric interpretation of the cohomology classes in $H^1(\Theta_X)$ 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 $$H^1(\Theta_X) = Ext^1(\Omega_X, \mathcal{O}_X)$$ where $\Omega_X$ 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.