##
**On the moduli spaces of surfaces of general type. Appendix: Letter of E. Bombieri written to the author.**
*(English)*
Zbl 0549.14012

J. Differ. Geom. 19, 483-515 (1984); appendix 513-514 (1984).

The present paper is devoted to the study of general properties of the moduli spaces of surfaces of general type (i.e., given a minimal model S of a surface of general type, one looks at the moduli space M(S) parametrizing all the isomorphism classes of complex structures on the 4- dimensional oriented compact topological manifold underlying S: M(S) is a quasi projective variety by a theorem of D. Gieseker [Invent. Math. 43, 233-282 (1977; Zbl 0389.14006)].

Theorem A proves that for each natural number n there are positive integers \(d_ 1<d_ 2<...<d_ n\) and a surface of general type S such that M(S) has irreducible components \(Y_ 1,...,Y_ n\) of respective dimensions \(d_ 1,...,d_ n\) (this is in sharp contrast with the case of a curve C of genus \(g\geq 2\) where M(C) is irreducible of dimension 3g- 3). In view of this result, in the paper is given an upper bound for the dimension of an irreducible component of M(S), to wit, if d(S) is the dimension of M(S) at the point corresponding to S, theorem B proves that d(S) is at most \(10\chi +3c^ 2_ 1+108,\) and (theorem C) at most \(10\chi +q+1\) if S contains a smooth canonical curve (where \(\chi =1- q+p_ g,\) one should notice that \(\chi\), q, \(c^ 2_ 1\) are topological invariants, in fact, and that a lower bound for d(S) is given by the Kodaira-Spencer-Kuranishi theory of deformations, giving: d(S) is at least \(10\chi -2c^ 2_ 1).\)

Finally, the last chapter studies irregular surfaces without irrational pencils, since in the literature there are scattered correct and incorrect results: using some of these results, G. Castelnuovo claimed [Atti. Accad. Naz. Lincei, VIII. Ser., Rend., CE. Sci. Fiz. Mat. Nat. 7, 3-11 (1949; Zbl 0035.372)] that for those surfaces d(S) is at most \(p_ g+2q\). In this paper a counterexample is given (showing that at least, asymptotically on \(p_ g\), the coefficient 4 is needed in front of \(p_ g)\), but theorem D proves that if q is at least 3, and there exist 2 holomorphic 1-forms whose wedge product defines a reduced irreducible curve, then d(S) is at most \(p_ g+3q-3\), and moreover \(c^ 2_ 1\) is at least 6\(\chi\).

The first chapter deals with smooth Galois covers with abelian group, studying their fundamental groups [the method used is to prove that the fundamental group of the complement of the branch locus is abelian: theorem 1.6 is, for instance, a generalization of a previous result of R. Mandelbaum and B. Moishezon, Trans. Am. Math. Soc. 260, 195-222 (1980; Zbl 0465.57014), but has, in the meantime, been further generalized by Nori].

The main ingredients to prove theorem A are a careful study of the deformations of Galois covers in the special case where the group is \(({\mathbb{Z}}/2)^ 2\), plus an application of Freedman’s result on diffeomorphisms of 4-manifolds [M. H. Freedman, J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)], and a number-theoretic lemma proved by E. Bombieri in the appendix to this paper: in fact, the surfaces considered for theorem A, are just \(({\mathbb{Z}}/2)^ 2\) Galois covers of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\).

Theorem A proves that for each natural number n there are positive integers \(d_ 1<d_ 2<...<d_ n\) and a surface of general type S such that M(S) has irreducible components \(Y_ 1,...,Y_ n\) of respective dimensions \(d_ 1,...,d_ n\) (this is in sharp contrast with the case of a curve C of genus \(g\geq 2\) where M(C) is irreducible of dimension 3g- 3). In view of this result, in the paper is given an upper bound for the dimension of an irreducible component of M(S), to wit, if d(S) is the dimension of M(S) at the point corresponding to S, theorem B proves that d(S) is at most \(10\chi +3c^ 2_ 1+108,\) and (theorem C) at most \(10\chi +q+1\) if S contains a smooth canonical curve (where \(\chi =1- q+p_ g,\) one should notice that \(\chi\), q, \(c^ 2_ 1\) are topological invariants, in fact, and that a lower bound for d(S) is given by the Kodaira-Spencer-Kuranishi theory of deformations, giving: d(S) is at least \(10\chi -2c^ 2_ 1).\)

Finally, the last chapter studies irregular surfaces without irrational pencils, since in the literature there are scattered correct and incorrect results: using some of these results, G. Castelnuovo claimed [Atti. Accad. Naz. Lincei, VIII. Ser., Rend., CE. Sci. Fiz. Mat. Nat. 7, 3-11 (1949; Zbl 0035.372)] that for those surfaces d(S) is at most \(p_ g+2q\). In this paper a counterexample is given (showing that at least, asymptotically on \(p_ g\), the coefficient 4 is needed in front of \(p_ g)\), but theorem D proves that if q is at least 3, and there exist 2 holomorphic 1-forms whose wedge product defines a reduced irreducible curve, then d(S) is at most \(p_ g+3q-3\), and moreover \(c^ 2_ 1\) is at least 6\(\chi\).

The first chapter deals with smooth Galois covers with abelian group, studying their fundamental groups [the method used is to prove that the fundamental group of the complement of the branch locus is abelian: theorem 1.6 is, for instance, a generalization of a previous result of R. Mandelbaum and B. Moishezon, Trans. Am. Math. Soc. 260, 195-222 (1980; Zbl 0465.57014), but has, in the meantime, been further generalized by Nori].

The main ingredients to prove theorem A are a careful study of the deformations of Galois covers in the special case where the group is \(({\mathbb{Z}}/2)^ 2\), plus an application of Freedman’s result on diffeomorphisms of 4-manifolds [M. H. Freedman, J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)], and a number-theoretic lemma proved by E. Bombieri in the appendix to this paper: in fact, the surfaces considered for theorem A, are just \(({\mathbb{Z}}/2)^ 2\) Galois covers of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\).

### MSC:

14J10 | Families, moduli, classification: algebraic theory |

14D22 | Fine and coarse moduli spaces |

14E20 | Coverings in algebraic geometry |

14J15 | Moduli, classification: analytic theory; relations with modular forms |

14J25 | Special surfaces |

32G13 | Complex-analytic moduli problems |

14D15 | Formal methods and deformations in algebraic geometry |