## Geography and the number of moduli of surfaces of general type.(English)Zbl 1103.14022

Let $$X$$ be a minimal complex surface of general type. The “number of moduli” $$m(X)$$ of $$X$$ is the dimension of the base $$B$$ of the semiuniversal family of deformations of $$X$$, which coincides with the dimension of the moduli space of surfaces of general type at the point corresponding to $$X$$. By construction, $$B$$ is a germ of analytic space with tangent space canonically isomorphic to $$H^1(T_X)$$, hence $$m(X)$$ is bounded above by $$h^1(T_X)$$. Let $$K^2_X$$ and $$c_2(X)$$ be the numerical invariants of $$X$$. It is well known that they satisfy the inequalities: $c_2(X)/5-36\leq K^2_X \leq 3c_2(X).$ Since for a surface of general type one has $$h^0(T_X)=0$$, a naïve lower bound for $$h^1(T_X)$$ is given by $$\chi(T_X)=(5c_2(X)-7K^2_X)/6)$$, the so-called “expected number of moduli”. So one expects that $$m(X)$$ decrease as the ratio $$K^2/c_2$$ increases. Indeed, by a famous result of S.-T. Yau [Proc. Natl. Acad. Sci. USA 74, 1798–1799 (1977; Zbl 0355.32028)], surfaces on the upper line $$K^2=3c_2$$ are uniformized by the unit ball in $${\mathbb C}^2$$, hence their number of moduli is equal to 0. On the other hand, E. Horikawa [Ann. Math. (2) 104, 357–387 (1976; Zbl 0339.14024)] has classified the surfaces on the lower line $$K^2=c_2/5-36$$ and he has also computed $$h^1(T_X)>0$$ and shown the equality $$m(X)=h^1(T_X)$$ for these surfaces.
The main result of this paper points in the same direction. The author considers surfaces with $$K_X$$ ample and satisfying $$K^2>8c_2/3$$, (thus “close” to the line $$K^2=3c_2$$) and under these assumptions he proves the following inequality: $h^1(T_X)\leq 9(c_2(X)-K^2_X).$ The strategy of the proof is to consider the extension of vector bundles on $$X$$: $0\to H^1(T_X)\otimes {\mathcal O}_X\to {\mathcal T} \to \Omega_X\to 0,$ canonically associated to the space $$H^1(T_X)$$ and study the stability of $${\mathcal T}$$. In the case of irregular surfaces, bounds for $$m(X)$$ have been obtained by F. Catanese [J. Differ. Geom. 19, 483–515; Appendix: 513–514 (1984; Zbl 0549.14012)] and by G. P. Pirola, F. Zucconi [J. Algebr. Geom. 12, No. 3, 535–572 (2003; Zbl 1083.14515)].

### MSC:

 14J29 Surfaces of general type 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli

### Citations:

Zbl 0355.32028; Zbl 0339.14024; Zbl 0549.14012; Zbl 1083.14515
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