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**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)].

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)].

Reviewer: Rita Pardini (Pisa)