FIBRED SURFACES, VARIETIES ISOGENOUS TO A PRODUCT AND RELATED MODULI SPACES

by

Fabrizio CATANESE

This article is dedicated to the memory of Fernando Serrano

Contents of the paper:

1. Fundamental groups cannot determine.

2. Basic results on fibred surfaces, and differentiable invariance of the fibre.

3. Surfaces isogenous to a product.

4. The fundamental group of surfaces isogenous to a product, and the irreducibility of their moduli space.

5. Topological characterizations of surfaces carrying a constant moduli fibration.

6. The fundamental group and moduli spaces.

7. Results in the higher dimensional case.

0. Introduction

The study of fibrations of a smooth algebraic surface S over a smooth algebraic curve B lies at the heart of the classification theory and of the geometry of algebraic surfaces. Our main results in this paper ( Theorems D,E) concern a topological characterization of surfaces admitting a constant moduli fibration . These results are intimately related to a series of variations on the same theme ( also in higher dimension ) , and to a circle of allied techniques which we present in the course of the paper . The first new result we have in this direction, and which is presented in section 2 after we recall several known results, (some classical, some quite recent) is the following theorem on the differentiable invariance of the genus of fibres (cf. theorem 2.9 for a more precise statement).

Theorem A. Let f: S → B be a fibration of the surface S over a curve B of genus b ≥ 2, and with fibre F of genus g > 1. Assume that S′ is another smooth compact Kähler surface which is orientedly diffeomorphic to S . Then the subspace of H^{1}(S′, IC) corresponding to f^{∗}(H^{1}(B, IC)) determines a fibration f′: S′ → B′ with the same invariants as f ( namely, the base curve B’ has genus b, and the fibre genus is equal to g ).

The result above, whose proof uses the results of Seiberg Witten theory, was conjectured in [Cat4], of which the present paper is in some sense a continuation.

In fact, as in that paper one of the main themes was the topological characterization of the existence of fibrations (in dimension 2 and greater), here, as already remarked, the leit motiv is the topological characterization of constant moduli fibrations. Recall that f: S → B is said to be a constant moduli fibration if all the smooth fibres are isomorphic, and that in this case S is bimeromorphic to a quotient C_{1} × C_{2}/G of a product of curves by the action of a finite group G. The first case which deserves attention is the one where the group G acts freely, and we say then that S is strongly isogenous to a product. More generally, we shall say that a complex manifold X is isogenous to a higher product if it admits a finite unramified covering which is isomorphic to a product of curves of respective genera at least 2. Indeed, as we see in 3.11, this weaker property is equivalent to the stronger property of having a Galois such cover (thus the notions of a variety isogenous, resp. strongly isogenous to a product are equivalent). Our second result in the case of surfaces is (see theorems 3.4 , 4.13, 4.14.)

Theorem B. A surface S is isogenous to a higher product if and only if π_{1}(S) admits a finite index subgroup Γ isomorphic to a direct product Π_{g_{1}} × Π_{g_{2}} of fundamental groups of curves of genera g_{1}, g_{2} > 1, and moreover, d denoting the index of Γ, either

e) e(S) = \frac{4(g_{1}−1)(g_{2}−1)}{d} or

k) S is minimal and K_{S}^{2} = \frac{8(g_{1}−1)(g_{2}−1)}{d} or

c) S is minimal and χ(O_{S}) = \frac{(g_{1}−1)(g_{2}−1)}{d}. Setting Π = Π_{g_{1}} × Π_{g_{2}} , one can indeed relax the hypothesis Γ ≅ Π to the weaker assumption that Γ/Γ_{3} ≅ Π/Π_{3} , where, for a group Γ , Γ_{3} is the third group in the nilpotent series ( i.e., Γ_{2} is the commutator subgroup [Γ, Γ] and Γ_{3} = [Γ, Γ_{2}]). Moreover, let Π be the fundamental group of a surface S isogenous to a product as above. Then the moduli space 𝔐(Π, e(S)) of the surfaces with fundamental group isomorphic to Π , and e.g. Euler number = e(S) = 4(g_{1} − 1)(g_{2} − 1)/d is irreducible.

In fact the result can be made stronger, as to classify the fundamental groups Π which occur. For instance we have (see theorem 4.13):

Theorem C. Let Π be a group such that i) there is an exact sequence

1 → Π_{g_{1}} × Π_{g_{2}} → Π → G → 1,

where G is a finite group of order d,

b) the individual factors Π_{g_{1}}, Π_{g_{2}} are normal in Π,

c) the induced quotients

1 → Π_{g_{i}} → Π(i) → G → 1

are such that c1) there is no element of Π mapping in each Π(i) to an element of finite order, c’) by the above exact sequences G embeds in Out(Π_{g_{i}}).

Then the moduli space of the surfaces S with π_{1}(S) ≅ Π, and with Euler number e(S) = \frac{4(g_{1}−1)(g_{2}−1)}{d} is non empty and irreducible.

Theorems B), C) are a sharpening in two directions of two earlier theorems of Jost and Yau ([J-Y 1,2]) asserting that if S′ is homotopically equivalent to a surface S whose universal cover is the product H^{2} of two discs, then also S′ is a quotient H^{2}/Π (Π = π_{1}(S)). Secondly, Jost and Yau show that if Π acts irreducibly (i.e., S is not isogenous to a product, in our terminology) , then S′ is either biholomorphic or antibiholomorphic to S ( strong rigidity). The assumption of Jost and Yau on S′ is clearly that π_{1}(S′) = π_{1}(S), and moreover that π_{j}(S′) = 0 for j ≥ 2. The advantage of theorem B) is thus to involve the Euler number, which is easy to compute, unlike the higher homotopy groups. Also, we complement the strong rigidity result by showing weak rigidity (irreducibility of the moduli space) and an existence result in the reducible case: here we rely on the properties of Teichmüller spaces, and more precisely on a variant of the solution of the Nielsen realization problem (cf. section 4) given in [Tro]. Moreover, we show indeed (cf. theorem 7.7, and remark 7.8) that we can get the first result of Jost and Yau also in the irreducible case with the weaker assumptions π_{1}(S′) = π_{1}(S), and e(S) = e(S′) (or, S′ is minimal and K_{S′}^{2} = K_{S}^{2}). Of course, one could ask whether the result of theorem B could be further improved, and we show in section 1 that this is not the case. Namely, we show that only an assumption on the fundamental group cannot be sufficient. Because for any surface X (corollary 1.3 considers for simplicity the case where X is minimal and non ruled) there exists a finite (ramified) covering S → X, such that S is of general type with K_{S} very ample, π_{1}(S) ≅ π_{1}(X), but S is not birationally equivalent to X. We can obtain such a surface S with negative index, but it is not clear whether it is possible to achieve positive index.

The above remark ( based on a simple construction used by Bogomolov ) has some implications on the moduli spaces of surfaces of general type. For those, the main numerical invariants are the holomorphic Euler Poincaré characteristic χ(O_{S}) and the self intersection K^{2} = K_{S}^{2} of the canonical divisor K_{S} of the minimal model S (Noether’s formula gives, for the Euler number, e(S) = 12χ(O_{S}) − K_{S}^{2}). Given a pair of invariants x, y ≥ 1 the surfaces of general type S with those invariants (i.e. χ(O_{S}) = x, K_{S}^{2} = y) are parametrized by a moduli space 𝔐(x, y) which is a quasi projective variety. The following (cf. [Miy], [Yau1,2]).

\underline{Noether’s inequality:} K_{S}^{2} ≥ 2p_{g} − 4, whence K_{S}^{2} ≥ 2χ − 6 and

\underline{Bogomolov-Miyaoka-Yau inequality:} K_{S}^{2} ≤ 9χ

impose essentially the only restrictions on x, y. It is known (cf. [Cat5], [Ch]) that the number i(x, y) of irreducible components of 𝔐(x, y) has a limsup which grows exponentially (y^{\frac{1}{4} y} ≤ lim sup i(x, y) ≤ C y^{cy^{2}}), but most of the examples are based on the case of surfaces which are simply connected. There are of course two parts in which the world of surfaces of general type is divided: based (instead that upon political faith) on the property of the fundamental group of being either finite or infinite. Proposition 6.11 shows that if we take two families X , Y of surfaces, parametrized by smooth respective bases B, B′, and with surjective Kodaira Spencer map, then by the Bogomolov construction we get also a family S parametrized by a smooth base T, and with surjective Kodaira Spencer map.

Whence, from each irreducible component of the moduli space of 1-connected surfaces Y, we produce an irreducible component of the moduli space of surfaces S with π_{1}(S) ≅ π_{1}(X) (here, we can take e.g. X isogenous to a product). Therefore, the “geographical” meaning of theorem B) is that π_{1}(S) determines the geometry only if we restrict ourselves to move on certain lines, x = constant, y = constant,or y − 12x = constant. It would be interesting to show that being on the watershed line (y = 8x) and having fundamental group of a surface isogenous to a product does not imply that S is isogenous to a product (in a similar vein Moishezon and Teicher in [M-T] have shown the existence of simply connected surfaces on the watershed line, contrary to a previous expectation). We find that the Bogomolov construction deserves to be better analyzed. For instance, assume that we “marry” two 1- connected (orientedly) homeomorphic surfaces Y, Y′ with such an X (having infinite fundamental group): can we have an example where Y, Y′ are not diffeomorphic but the resulting surfaces S, S′ are diffeomorphic? Moreover (cf. [Cat1,2], [Man1,2]) assume that we consider (1-connected) surfaces Y, Y′ for which the moduli space has several connected components, and let us take X isogenous to a product. Then, as observed, we produce new irreducible components of the moduli space for S, S′: are these also connected components? Leaving aside speculations and going to the technique, our method is founded on the classical Hodge Theory rather than on Jost and Yau’s method of harmonic maps. In this way the algebraic geometric methods come into play (offering easier proofs and more precise results), and the advantage of doing so is particularly evident in the non compact case, where we can make full use of the assumption of quasi projectivity. One of the basic tools in the characterization of the more general case of surfaces carrying a constant moduli fibration are two theorems of I. Bauer ([Ba]) and one of D. Arapura ( [Ara]). The first gives a bijection between the maps of a quasi projective manifold U = X − D to a quasiprojective curve C (with first Betti number b_{1} ≥ 3) and the real maximal isotropic subspaces V of H^{1}(U, IR), the second one ( obtained independently by Arapura but under the weaker assumption b_{1} ≥ 2) generalizes results of several authors in the compact case and asserts the existence of such maps under the assumption on π_{1}(U) of admitting a surjection onto a non abelian free group.

With this in our hand, we get our two main results on the ”topological” characterization of constant moduli fibrations (cf. theorems 5.4, 5.7 for more complete statements). We write ”topological” with quotation marks because, as we show in remark 2.5 , the existence of these fibrations is not a topological property of a surface S ( actually the property is not even invariant by deformation ). But indeed this property is a combined topological property , based on the interplay of the Hausdorff topology and the analytic Zariski topology .

Theorem D. Let U′ = S′ − D be a quasi projective surface, and assume that U′ is proper homotopically equivalent to the good locus Y^{0} of a constant moduli fibration f : S → B having fibre genus g ≥ 2 and base genus b ≥ 2 . Then S′ carries a constant moduli fibration with the same invariants as S. Moreover, the birational classes of all such surfaces S′ form an irreducible subvariety of the moduli space.

The previous result is clearly in the spirit of the one by Jost-Yau , whereas the next theorem E is a generalization of theorem B. Except that( in the proof only, as pointed out by L. Kaup) we have to replace the Euler number by the Euler number in Borel Moore homology (this is the Euler number of the Alexandroff compactification, diminished by 1), and moreover we have also to consider the fundamental group at infinity (this is a disjoint union of groups, one for each end of U).

Theorem E. Let U′ = S′ − D be a quasi projective surface. Assume that π_{1}(U′) ≅ π_{1}(Y^{0}), where Y^{0} is the good locus of a constant moduli fibration f : S → B having fibre genus g ≥ 2 and base genus b ≥ 2, and that π_{1}^{∞}(U′) ≅ π_{1}^{∞}(Y^{0}), compatibly with the natural homomorphism π_{1}^{∞} → π_{1}. Assume further that e_{BM}(U′) = e_{BM}(Y^{0}) ( or, equivalently,e(U′) = e(Y^{0}) ) . Then S′ also carries a constant moduli fibration which , in case S′ is minimal, is a deformation of the one of S.

In section 3 we proceed to a detailed classification of the surfaces isogenous to product, which allows a precise description of their fundamental group and of their moduli spaces. Surfaces isogenous to a product are divided into 3 major types:

M) Surfaces of mixed type, quotient of a product C × C by a group of automor-

phisms exchanging the two factors D) Surfaces of double type for which the image of the Albanese map is either a

product of curves, or a curve, or a point (in this last class, which we label by D^{0}), are contained some surfaces constructed by Beauville, and which turn out, by theorem C, to be strongly rigid: cf. 3.22).

GH) Generalized Hyperelliptic surfaces, quotients C_{1} × C_{2}/G such that C_{2}/G ≅

IP^{1}, while G acts freely on C_{1}. This last case, generalizing the one of the classical hyperelliptic surfaces ( where C_{1}, C_{2} have genus 1) , which play such a central role in the Enriques classification of algebraic surfaces, admits another very easy characterization (cf. 3.18).

Theorem F. Let S be a surface with index 0, e(S) > 0, irregularity q ≥ 2 and Albanese map a pencil. S is a Generalized Hyperelliptic surface if and only if the genus g of the Albanese fibres satisfies g = 1 + \frac{χ(S)}{(q−1)}.

Now, an entirely similar classification for varieties isogenous to a product can be given also in higher dimension, but we preferred to stick to the 2-dimensional case just to limit ourselves to giving the flavour of the arguments used and to avoid a more complicated algebraic apparatus in the mixed case.

On the other hand, one can extend many of the above results from the surface case to higher dimension (here, to avoid touching upon the theory of minimal models, we restrict ourselves to imposing the natural condition that K should be ample).

This was first done by Mok ([Mok1,2]) who extended the results of Jost and Yau to quotients of polydiscs, again under an assumption of homotopy equivalence. In section 7 we show that one can relax these assumptions also in higher dimension.

Theorem G. Let X be a compact complex manifold of dimension n with ample canonical bundle and assume that

i) Γ = π_{1}(X) admits a finite index subgroup Γ′ isomorphic to Π_{g_{1}} × Π_{g_{2}} × … × Π_{g_{n}}, with g_{1}, …, g_{n} > 1,

ii) the image of H^{2n}(Γ′, IZ) in H^{2n}(X′, IZ) is non zero.

k) K_{X}^{n} = \frac{n!2^{n}(g_{1}−1)(g_{2}−1)...(g_{n}−1)}{d}. Then X is isogenous to a higher product, and the moduli space of such varieties is irreducible (weak rigidity).

Theorem H. Let X be a compact manifold of dimension n with ample canonical bundle, and assume that

i’) π_{1}(X) ≅ Γ, where Γ is a cocompact torsion free subgroup of Aut (H^{n}) ( i.e.,

Γ ≅ π_{1}(W′), where H^{n}/Γ = W′ is a compact manifold). ii’) H^{2n}(Γ, IZ) → H^{2n}(X, IZ) is an isomorphism, or ii”) H^{2n}(Γ, IZ) → H^{2n}(X, IZ) is non zero and K_{X}^{n} = K_{W′}^{n}.

Then also X is biholomorphic to a quotient W = H^{n}/Γ. It would be interesting to consider the higher dimensional topological characterization of varieties bimeromorphic to a quotient of a product of curves (cf. however remark 5.5). For the time being we felt that treating here all the possible generalizations would have been too demanding for the reader (and for the writer).

Acknowledgement: The present research was begun when the author was visiting the M.S.R.I. in Berkeley during the special year 1995-96 on Several Complex Variables. The results of the paper were obtained while the author was Professore distaccato at the Accademia dei Lincei, and in the framework of the AGE project HCM, contract ERBCHRXCT 940557. The final version was written at the University of Göttingen. But, in retrospective, I started to think carefully about constant moduli fibrations already years ago, after some conversations I had with F. Serrano, and which marked a fruitful interaction between our points of view. I would like to thank also F. Zucconi for providing an initial stimulus which brought me to theorem 3.18, then S. Luo for pointing out the previous work of Jost and Yau, and especially Sidney Frankel for mentioning the convexity properties of Teichmüller space. Finally, thanks to T. Lehmkuhl for pointing out an embarassing error in the Göttingen preprint of october 1997, and to E. Viehweg for asking whether one could relax an assumption in the statement of Theorem B.

Some Notation

Π_{g} = the fundamental group of a compact complex curve C of genus g. e(S) = the topological Euler number. e_{BM}(Y) = the Euler number in Borel Moore homology. H = the unit disc in IC, H = {z | |z| < 1}. In a complex vector space V with a real structure, Ū denotes the complex conjugate of the subspace U.

1. Fundamental groups cannot determine.

We start this section by reminding the reader of a consequence of the Lefschetz theorem on hyperplane sections.

(1.1) Lemma. Let X, Y be smooth complete algebraic surfaces: then there exists a smooth complete surface S ⊂ X × Y, mapping finitely to both X and Y with degree > 0, and such that π_{1}(S) ≅ π_{1}(X) × π_{1}(Y).

Proof. Let A be (the pull back of) a very ample divisor on X, and B a very ample divisor on Y: then A + B is very ample on X × Y, and any surface S which is a general complete intersection of two divisors H, H′ in |A + B| is smooth and has π_{1}(S) ≅ π_{1}(X) × π_{1}(Y), by the theorems of Bertini and Lefschetz.

The degree of S → X equals the intersection product B^{2} , thus we simply have to exclude the case where Y is IP^{2}, B is the hyperplane divisor, and similarly for X. To show e.g. the finiteness of S → X we argue as follows:

\underline{Claim 1.} For H general, H → X has all fibres of dimension 1.

\underline{Proof.} H = \{(x, y) | ∑_{j=0}^{n} α_{j}(x)β_{j}(y) = 0\} , where α_{0}, α_{1}, …, α_{n} are sections of

H^{0}(O_{X}(A)), and β_{0}, β_{1}, …, β_{n} form a basis of H^{0}(O_{Y} (B)). It suffices to show that for each p in X, {H | H ⊃ {p} × Y } has codimension ≥ 3. But H ⊃ {p} × Y iff α_{j}(p) = 0 for each j, thus we get a linear subspace of codimension n + 1 ≥ 3.

\underline{Claim 2.} For H′ general, H′ does not contain any component D of a fibre H ∩(\{p\}× Y) of H → X.

\underline{Proof.} If H′ = \{(x, y) | ∑_{j=0}^{n} α′_{j}(x)β_{j}(y) = 0\}, then H′ ⊃ D iff ∑_{j=o}^{n} α′_{j}(p)β_{j}(y) ≡

0 on D. Thus {H′ ⊃ D} is a linear subspace of codimension 3 unless D is embedded as a line. Therefore we can find a H′ as desired, unless (H being irreducible) all the fibres of H → X are lines, which was excluded by setting B^{2} > 1. □

(1.2) Remark. Let X, Y, S be as in the construction of lemma 1.1. Assume X, Y are minimal: then also S is minimal, and indeed S has a nef canonical divisor K_{S}, which is indeed very ample if X and Y are not ruled. Moreover, if we set χ = χ(O_{X}), χ′ = χ(O_{Y}), χ(S) = χ(O_{S}), we have

χ(S) = χB^{2} + χ′A^{2} + c(A, B),

K_{S}^{2} = K_{X}^{2}B^{2} + K_{Y}^{2} A^{2} + 8c(A, B) − 4A^{2}B^{2},

where

c(A, B) = \frac{7}{2}A^{2}B^{2} + \frac{3}{2}\{(AK_{X})B^{2} + A^{2}(BK_{Y})\} + \frac{1}{2}(AK_{X})(BK_{Y}).

Proof. Let Z = X × Y, use the Koszul sequence

(∗)  0 → O_{Z}(−2A − 2B) → O_{Z}(−A − B)^{2} → O_{Z} → O_{S} → 0

to calculate χ(O_{S}), and then keep in mind that

K_{S} = (K_{X} + 2A + K_{Y} + 2B)|_{S} .

E.g. by a more general result of Fujita ([Fu], thm. 11.2, page 93) K_{X} + 2A is nef on X, since the pair (X, A) is not the pair (IP^{2}, O_{IP}^{2}(1)), and similarly for (Y, B). Clearly if X, Y are not ruled then K_{X} and K_{Y} are nef, thus K_{S} is then very ample.

□

(1.3) Corollary. Let X be a non ruled minimal surface. Then there exists a finite (ramified) covering S → X of degree > 1, such that S is of general type with K_{S} very ample, π_{1}(S) ≅ π_{1}(X), but S is not birationally equivalent to X. We can moreover obtain that S has negative index, i.e., K_{S}^{2} − 8χ(S) < 0.

Proof. Take X, Y as in remark 1.2 but with Y simply connected, and A sufficiently very ample. Then, in particular, K_{S}^{2} − 8χ(S) = B^{2}{K_{X}^{2} − 8χ(X)} + A^{2}{K_{Y}^{2} − 8χ(Y)} − 4A^{2}B^{2}. Clearly, if the index of Y is non positive (there is no problem to find such a surface), for A >> 0, K_{S}^{2} − 8χ(S) is negative. □

(1.4) Remarks.

i) Indeed (cf. [Miy2], prop. 3) the above construction can yield a surface of general type S with negative index and Ω_{S}^{1} ample.

ii) If X has positive (resp. zero index) we can ask whether we can moreover obtain also the case that S has positive index (resp. zero index): it would suffice to find a 1-connected Y with a very ample divisor B such that {K_{Y}^{2} − 8χ(Y)} ≥ 4B^{2} (can one find this ?, cf. [Chen1,2]).

iii) If X is a Yau surface, i.e. K_{X}^{2} = 9χ(X) (by [Ya1,2] and [Miy2] then X is a quotient X = B/Γ of the unit ball in IC^{2}) then certainly S does not satisfy K_{S}^{2} = 9χ(S). In fact, then S would also be homeomorphic to X (indeed isomorphic, by Mostow’s rigidity [Mos]) whereas K_{S}^{2} > K_{X}^{2}.

2. Basic results on fibred surfaces, and differentiable invariance of the fibre

For the benefit of reader, we collect together a series of results which will be used in the sequel: throughout, by a fibration, we shall mean a surjective morphism with connected fibres between complete smooth varieties.

Later on , when our varieties will no longer be complete, but only quasi-projective, a fibration will denote a surjective morphism with irreducible general fibre. In case where we have f: S → B, with S is a surface and B a curve of genus b, we shall say that f is minimal if there is no (−1) curve contained in a fibre, and we will denote by g the arithmetic genus of a(ny) fibre F. Notice that, for g ≥ 1, f is minimal iff K_{S} is f-nef, i.e., K_{S} has intersection number ≥ 0 with each component of a fibre. In the case g ≥ 1 follows then easily that the minimal model of f is unique. F will also be called a genus b pencil of curves of genus g, and a higher genus pencil if b > 1 (cf. [Ca4]). The oldest result is the

(2.1) Theorem of Zeuthen-Segre. Let f: S → B be a genus b pencil of curves of genus g: then we have the following inequality for the topological Euler-Poincaré characteristic of S: e(S) ≥ 4(g − 1)(b − 1). If g ≥ 2, then equality holds if and only if f is a topological fibre bundle.

(2.2) Theorem of Arakelov. Let f: S → B be a minimal genus b pencil of curves of genus g: then we have the following inequality for the self intersection of the canonical divisor of S: K_{S}^{2} ≥ 8(g − 1)(b − 1). If g ≥ 2, then equality holds only if f has constant moduli, meaning that all the smooth fibres are isomorphic.

The two above theorems were combined nicely by Beauville ([Bea2]) in order to yield a simple description, first conjectured by Castelnuovo, of the surfaces for which holds the inequality p_{g} ≤ 2(q − 4) (see [Ros], [Com], [Jon]).

(2.3) Theorem of Beauville. Let f: S → B be a minimal genus b pencil of curves of genus g ≥ 2: then χ(S) ≥ (b − 1)(g − 1), equality holding if and only if f is a holomorphic fibre bundle.

(2.4) Remarks.

i) Let f: S → B be a constant moduli fibration: then f is isotrivial, in the sense that there is a Galois base change C_{2} → B = C_{2}/G such that the pull back of f can be relatively blown down to a product projection C_{1} × C_{2} → C_{2}. We recall that the pull back is here the minimal resolution of singularities of the normalization of the fibre product S ×_{B} C_{2}. In this situation the smooth fibres are isomorphic to C_{1} and if g ≥ 1 we have a biregular action of G on C_{1} × C_{2}. We denote by Y the quotient C_{1} × C_{2}/G.

ii) The case of a holomorphic bundle (also called etale bundle) is the one where

G acts freely on the curve C_{2}. iii) The case of a quasi bundle (according to the definition of Serrano this means

that every singular fibre is a multiple of a smooth curve), gives rise to the notion of a surface isogenous to a product (cf. section 3): this case is the one where G acts freely on the product.

iv) Let f: S → B be more generally a constant moduli minimal fibration with g ≥ 1: then the birational map ε: S → Y = (C_{1} × C_{2})/G does not need to be a morphism (cf. e.g. the Comessatti’s example analysed in [C-C], pages 96-99). We can however say something to characterize the case where ε is not a morphism. Let π: Z → Y be a minimal resolution of singularities: then there is a birational map η: S → Z commuting with the projection onto B. Since f is minimal, it follows that Z → B is not minimal. Assume E is a (−1) curve contained in a fibre: then by assumption E is not π-exceptional, whence E is the central curve, the unique curve mapping onto the curve E′ in Y, quotient of a fibre C_{1} × {p} by the stabilizer G_{p} of p. Let a E+ ∑_{j=1}^{n} m_{j}D_{j} be the fibre containing p, where a is the order of G_{p}. Then

E^{2} = −1 if and only if E ∑_{j} m_{j}D_{j} = a. Let us analyze the singularities of Y

in the points of E′: these correspond to a fixed point (p′, p) for the action of G_{p}, thus we get a cyclic subgroup H of G_{p} of order n. H has a canonical generator given by the (\frac{a}{n})th power of the fixed generator of G_{p} which acts on the tangent space at p by the standard primitive a-th root of 1. In terms of the standard primitive n-th root ζ of 1, the local action on the product surface of our generator of H is a diagonal action with eigenvalues ζ^{b}, ζ, and according to [B-P-V], pages 80-85, the resolution of the cyclic quotient singularity yields a Hirzebruch-Jung string with exactly one curve D meeting E. Moreover, D meets E transversally at only one point, and its multiplicity in the fibre equals (\frac{a}{n})q, where 0 < q < n, and q satisfies the congruence qb ≡ 1(modn). Therefore, we have that E is a (−1) curve if and only if

i) the quotient of the action of G_{p} on C_{1} yields IP^{1}, ii) the sum over the fixed points of the action of G_{p} on C_{1} gives

∑_{i=1}^{r} q_{i}(\frac{a}{n_{i}}) = a.

Recall that the Hurwitz formula has to be satisfied:

2 g − 2 = −2a + ∑_{i=1}^{r}(n_{i} − 1)(\frac{a}{n_{i}}) .

We can conversely see that for each choice of such numbers q_{i}, n_{i} satisfying the above two equations we get a corresponding covering of IP^{1}. In fact, setting q_{i}(\frac{a}{n_{i}}) = m_{i}, it suffices to take the covering

{(w, z) | w^{a} = ∑_{i=1}^{r}(z − z_{i})m_{i}} .

For instance, when a is prime, then a = n_{i}, thus ∑_{i=1}^{r} q_{i} = a, and for q_{i} = 1 we

get the genus g of a plane curve of degree a (Fermat curve).

(2.5) Remark. Let f: S → B be a constant moduli minimal fibration: then a converse to Arakelov’s theorem does not hold, namely the equality K_{S}^{2} = 8(g − 1)(b − 1) does not need to hold. In fact what happens is that a small deformation of a constant moduli fibration does not need to be again a constant moduli fibration. The simplest example of this phenomenon is given by an isotrivial surface S which is a resolution of Y = C_{1} × C_{2}/G, with G = IZ/2. We can view Y as a singular double cover of C′_{1} × C′_{2} (where C′_{i} = C_{i}/G), with a branch curve B which is a union of vertical and horizontal curves. Deforming B to a smooth divisor in the linear system |B| (this is possible by Bertini’s theorem), we get a deformation of S having still a map to C′_{2} = C_{2}/G, but which is no longer a constant moduli fibration. As a matter of fact, in this example, if the genus b of C′_{2} = C_{2}/G is at least 2, then every deformation of S carries a fibration onto a curve of genus b, as implied by the following (cf. [Cat4], thm. 1.10, page 268).

(2.6) Isotropic subspace Theorem. Let X be a compact Kähler manifold. Then the correspondence which associates, to a fibration f: X → B, where B is a curve of genus b ≥ 2, the subspace f^{∗}(H^{1}(B, IC)) induces a bijection between

i) isomorphism classes of fibrations f: X → B where B is a curve of genus b ≥ 2 (f ≅ f′ iff there is an isomorphism ψ such that ψf = f′)

ii) 2b-dimensional subspaces V which can be written as U ⊕ Ū where U is a maximal isotropic subspace for the bilinear map H^{1}(X, IC) × H^{1}(X, IC) → H^{2}(X, IC) given by cup product.

(2.7) Remarks.

a) Of course one also obtains another bijection by associating to f the subspace f^{∗}(H^{1}(B, IR)) of H^{1}(X, IR) (respectively f^{∗}(H^{1}(B, IZ)) of H^{1}(X, IZ)).

b) Moreover, any maximal isotropic subspace U of dimension b ≥ 2 determines

a fibration f, whence it is such that U ∩ Ū = 0. It was shown in [Cat6] that the isotropic subspace theorem easily implies the following theorem ([Gro]).

(2.8) Gromov’s Few Relations Theorem. Let X be a compact Kähler manifold and assume there is a surjection π_{1}(X) → Γ, where Γ has a presentation with n generators and m relations, where n ≥ m + 2. Then there is a curve B of genus b, where b ≥ 2 and 2b ≥ n − m, and a fibration f: X → B such that f^{∗}(H^{1}(B, IC)) ⊃ H^{1}(Γ, IC) ⊂ H^{1}(X, IC).

Theorem 2.6 implies in particular that if two surfaces S, S′ are homeomorphic under φ: S′ → S, and f: S → B is a genus b pencil with b ≥ 2, then V′ = φ^{∗}(f^{∗}(H^{1}(B, IC))) determines a genus b pencil f′: S′ → B′. In this situation, let g be the genus of the fibres of f, resp. g′ the genus of the fibres of f′. Is it then true that g = g′? Using Seiberg-Witten theory we can answer this question under a stronger assumption

(2.9) Theorem on the differentiable invariance of the fibre genus in higher genus pencils. Let S, S′ be compact Kähler minimal surfaces, φ: S′ → S an orientation preserving diffeomorphism, and f: S → B a genus b pencil with b ≥ 2. Then the genus b pencil f′: S′ → B′ determined by V′ = φ^{∗}(f^{∗}(H^{1}(B, IC))) is such that the genus g′ of the fibres equals the genus g of the fibres of f.

Proof. We can w.l.o.g. identify S, S′ as differentiable manifolds. Looking at the Jacobian variety J(B) as a differentiable manifold J, we have two maps a, resp. a′ of S into J. a is defined as the composition of f with the Albanese embedding of B into J(B) and one moment’s reflection shows that a is obtained (up to translation)

by integration: a(x) = ∫_{x_{o}}^{x} ω, where ω is the vector ^{t}(ω_{1}, … ω_{b}), ω_{1}, … ω_{b} being a basis

of f^{∗}H^{0}(B, Ω_{B}^{1}). Similarly, a′(x = ∫_{x_{o}}^{x} ω′, with ω′ the vector ^{t}(ω′_{1}, … ω′_{b}), ω′_{1}, … ω′_{b}

being a basis of f^{′∗}H^{0}(B′, Ω_{B′}^{1}), and also a′ factors, through f′. We can clearly assume that the cohomology class of ω_{i} in H^{1}(S, IC) equals the one of ω′_{i}, whence we can write ω′ = ω + dψ, with ψ: S → IC^{b} a differentiable function.

\underline{Step 1:} a is homotopic to a′ under a(x, t) = ∫_{x_{o}}^{x} ω + [tψ(x)], [v] denoting the class in

J = H^{0}(B, Ω_{B}^{1})/H^{1}(B, IZ) = IC^{b}/Λ of a vector v of IC^{b}. \underline{Claim 2:} The respective fibres F of f, F′ of f′, yield cohomology classes which are either equal or opposite. \underline{Proof of claim 2:} The cohomology class of F (i.e., Poincaré dual to F) is the pull back under a of the standard polarization λ in H^{2}(B, IZ) ⊂ H^{2}(J, IZ). Since a, a′ are homotopic, to show that F, F′ are cohomologous it suffices to show that if λ′ is defined analogously, then λ′ = λ. Now,

a^{∗}H^{2}(B, IZ) = f^{∗}H^{1}(B, IZ) ∧ f^{∗}H^{1}(B, IZ) = f^{′∗}H^{1}(B, IZ) ∧ f^{′∗}H^{1}(B, IZ) =
= a^{′∗}H^{2}(B′, IZ),

thus λ′ and λ are IZ proportional, whence, since both of them are unimodular in H^{2}(J, IZ), λ′ = ±λ. □ for claim 2. ¿From our assumption it follows that either S is ruled and g = 0, or S is elliptic and g = 1, or S is of general type and g ≥ 2. The three cases are distinguished by the sign of K^{2}, < 0, = 0, > 0. Since K^{2}, by the Index formula and Noether’s formula, is a topological invariant for orientation preserving homeomorphisms, we can directly reduce to consider the case where both S, S′ are of general type. But then one of the main results of Seiberg-Witten theory (cf. [Mor], cor. 7.4.3, page 123), still identifying S and S′ as differentiable manifolds, says that K_{S′} = ±K_{S}. Since 2g − 2 = K_{S}F, we get therefore (2g′ − 2) = ±(2g − 2). But since as we noticed g′, g ≥ 2, we conclude that g′ = g. □

(2.10) Remarks.

(1) Under the hypotheses of theorem 2.9 it follows that the fibres F, F′ are diffeomorphic (being g′ = g). This result is no longer true in higher dimensions, as remarked by Bogomolov and Kollar as a consequence of the s-cobordism theorem (cf. [Cat4], prop. 1.6, page 267 , where though , as pointed out by D. Kotschick , is incorrectly quoted the h-cobordism theorem instead of the s-cobordism theorem of [Maz] ).

(2) If we only assume φ to be an orientation preserving homeomorphism, the answer to the question g′ = g? is still unknown.

(3) Notice that the statement of theorem 2.9 cannot be strengthened to require that there is a diffeomorphism ψ: B′ → B such that fφ = ψf′. In fact, the topological structure of the singular fibres is not invariant by deformation.

We end this section by quoting the analogues of 2.6 , 2.8 for the logarithmic case , obtained by I. Bauer in [Ba],( theorems 2.1 and 3.1 ) and by D. Arapura in [Ara] ( corollary 1.9 ) i.e., for the case of a fibration f: Y → C where C is a curve and Y, C are only required to be quasiprojective (in this case the role of the genus of the base curve is taken by the logarithmic genus c of C, which can be defined as the difference between the first Betti number of C and the genus b of the compactification B of C). The following is a more detailed statement of theorem 2.1 of [Ba] .

(2.11) Theorem of the logarithmic isotropic subspace. Let Y be a quasi projective manifold Y = X − D, where X is smooth, projective, and D is a normal crossings divisor. Then every real maximal isotropic subspace V of H^{1}(Y, IR) either of dimension ≥ 3 or of dimension 2 but not coming from a non - isotropic subspace V′ of H^{1}(X, IR) (in this case the theorem does not hold ) determines a unique logarithmic irrational pencil f: Y → C onto a curve C with logarithmic genus at least 2. One has that C is projective iff V ⊂ H^{1}(X, IR),and is isotropic there , otherwise V = f^{∗}(H^{1}(C, IR)) and one says that the pencil is strictly logarithmic.

Proof. We indicate here how to modify the argument of Theorem 2.1 of [Ba] . One notices that we can assume that the elements α_{1}, ..α_{l} are IR- linearly independent in H^{1}(Y, IR) / H^{1}(X, IR) , whence they are also IC- linearly independent in H^{0}(X, Ω_{X}^{1}(logD)) / H^{0}(X, Ω_{X}^{1}) . The case k = 0 works as in [loc.cit] (except that it is better to consider m′ = rank_{I C}(η_{1}, ..η_{l}) , , instead of m = rank_{IR}(η_{1}, ..η_{l}) , and, in the case m’=1 , to replace α_{2} by α_{2} + iη_{1} ). The case k ≥ 1 is the one where one gets a difficulty to apply the Castelnuovo de Franchis type Theorem, which requires two IC- linearly independent forms . This occurs precisely when l = 0 and the ω′_{i}s are dependent , i.e. , in the case where the complex span of V is generated by a holomorphic 1-form ω and by its conjugate ω̄. This is exactly the case which is excluded by our assumption , since ω ∧ ω̄ is certainly non zero in the cohomology of X . The last assertion being easy to show , there remains to show an example where iω ∧ ω̄ is zero in H^{2}(Y, IR) but Y has no holomorphic map to a curve of genus at least 1 . Our example will be a principally polarized Abelian surface X with Picard number ρ = 3 , but not containing any elliptic curve . We take as period matrix in Siegel’s upper half space the matrix Z = iY with Y_{11} = 1, Y_{12} = 2^{1/2}, Y_{22} = 5 . For this choice , using the notation of [L-B] ,exercise 4, page 43 and exercise 10, page 319, the Neron Severi Group corresponds to the integral antisymmetric matrices for which f = b, a = 3c, e = 5d , and there are no elliptic curves since the following Diophantine equation has no solutions

3c^{2} + 5d^{2} = b^{2}

( one verifies that this is the case by looking at the congruence mod5 ) .

Finally, one looks at the antisymmetric matrices which correspond to elements, in the cohomology group H^{2}(X, IR), of the form iω ∧ ω̄ : these are alternating forms which can be written as

−L(x) ∧ L(ix)

where L is an arbitrary IR- linear form on the vector space underlying X. As in [Ba] , 2.6 , iω ∧ ω̄ is zero in a suitable Zariski open set Y of X if and only if −L(x) ∧ L(ix) lies in the real span of the Neron-Severi group . A direct calculation shows that a = 3c, e = 5d are automatically verified , whereas f = b boils down to the equation

3l_{1}^{2} + 5l_{3}^{2} = 5l_{2}^{2} + l_{4}^{2} ,

for the four coefficients of L : but this equation clearly has real solutions.

□

The following result is due to I. Bauer[Ba] under the restriction n ≥ m + 3 and to D. Arapura [Ara] in the general case n ≥ m + 2

(2.12) Theorem of the logarithmic few relations. Let Y be a quasi projective manifold Y = X − D, and assume there is a surjection π_{1}(Y) → Γ, where Γ has a presentation with n generators and m relations, where n ≥ m + 2. Then there is an integer β ≥ n−m, a quasiprojective curve C with first Betti number equal to β, and a fibration f: Y → C.

We finally mention a quite straightforward generalization of the Zeuthen-Segre inequality to the logarithmic case. It requires, firstly, the use of Borel Moore homology (cf. [B-M], [B-H]), thus it is phrased in terms of the Borel-Moore homology Euler number e_{BM}(Y); secondly, it requires also the appropriate

(2.13) Definition. Let Y be a quasi projective surface Y = S − D, where S is smooth, projective, and D is a normal crossings divisor. Consider a logarithmic pencil f: Y → C: then f is surjective with connected fibres, and we can assume that f extends to a holomorphic map f: S → B ⊃ C. f is said to be very good if moreover every component Δ of D which does not map to (the finite set) B −C maps onto B (in other words, we remove from S either entire fibres, or horizontal curves).

(2.14) Theorem. Let Y be a quasi projective surface Y, and f: Y → C a very good logarithmic pencil. Then e_{BM}(Y) ≥ e_{BM}(F) · e_{BM}(C), F being the general fibre of f.

Proof. Let the notation be as in def. 2.13. Let Z ⊂ S be f^{−1}(C). Then by the usual argument e_{BM}(Z) ≥ e_{BM}(Φ) · e_{BM}(C), where Φ is the complete fibre. Write Y = Z − D′: then D′ is a finite (proper) degree d covering of C, whence e_{BM}(D′) ≤ d e_{BM}(C). We conclude since e_{BM}(Y) = e_{BM}(Z) − e_{BM}(D′) ≥ (e_{BM}(Φ) − d) · e_{BM}(C) = e_{BM}(F) · e_{BM}(C). □

3. Surfaces isogenous to a product

(3.1) Definition. A surface S is said to be isogenous to a product if S admits a finite unramified covering which is isomorphic to a product of curves (u: C_{1}×C_{2} → S) of genera g_{i} = genus (C_{i}) ≥ 1. In the case where each g_{i} ≥ 2 we shall also say that S is isogenous to a higher product.

(3.2) Remark. If in the above definition we would allow g_{i} = 0, then we would have the case where S → B is a particular type of IP^{1}- bundle (this case is well understood, and if you view the condition upside down, you want the existence of an unramified cover C → B such that the pull back of the bundle is trivial; the condition is not satisfied, for a small deformation of S → B, in the case where b = genus(B) ≥ 1). Whereas in the case g_{1} = g_{2} = 1 we have a classical hyperelliptic surface or an algebraic torus. Observe that both g_{i}’s are ≥ 2 iff S is of general type. Moreover, a surface isogenous to a product is always minimal.

We shall denote for later use by Π_{g} the fundamental group of a compact Riemann surface of genus g. We shall give now a first topological characterization of surfaces of general type isogenous to a higher product. Note that in the remaining cases S is an elliptic surface, and, by surface classification, a topological characterization is possible except in the case where S is an algebraic torus (since the tori isogenous to a product of elliptic curves form a (complex) codimension 1 set). In fact e.g. from [B-P-V], table 10, page 188, follows easily

(3.3) Remark. S is a hyperelliptic surface iff e(S) = 0, b_{1}(S) = 2, and π_{1}(S) is not Abelian. S is isogenous to a product with g_{1} = 1, g_{2} > 1, iff e(S) = 0, and π_{1}(S) admits a finite index subgroup isomorphic to Π_{1} × Π_{g_{2}}.

(3.4) Theorem. A surface S is isogenous to a higher product if and only if π_{1}(S) admits a finite index subgroup Γ isomorphic to Π_{g_{1}} × Π_{g_{2}}, with g_{1}, g_{2} > 1, and moreover:

e) if d denotes the index of Γ, then e(S) = \frac{4(g_{1}−1)(g_{2}−1)}{d}. If S is minimal, hypothesis e) can be replaced by

k) K_{S}^{2} = \frac{8(g_{1}−1)(g_{2}−1)}{d}, or by

c) χ(O_{S}) = \frac{(g_{1}−1)(g_{2}−1)}{d}. Setting Π = Π_{g_{1}} × Π_{g_{2}} , one can indeed relax the hypothesis Γ ≅ Π to the weaker assumption that Γ/Γ_{3} ≅ Π/P i_{3} , where, for a group Γ , Γ_{3} is the third group in the nilpotent series ( i.e., Γ_{2} is the commutator subgroup [Γ, Γ] and Γ_{3} = [Γ, Γ_{2}]).

Proof. The “only if” part is obvious. For the other implication, assuming we are given such a subgroup Γ we take the associated unramified covering. Therefore it suffices to prove the theorem in the special case d = 1. The first step of the proof consists in showing that we have two distinct pencils on S , providing a surjective holomorphic mapto a product of curves f = (f_{1} × f_{2}): S → C_{1} × C_{2}.

For the first step we have two proofs, the first exploiting the isomorphism π_{1}(S) ≅ Π_{g_{1}} × Π_{g_{2}} which produces, by Gromov’s theorem 2.8, two fibrations f_{i}: S → B_{i} such that

f_{i}^{∗}(H^{1}(B_{i}, IC)) ⊃ H^{1}(Π_{g_{i}}, IC) ⊂ H^{1}(S, IC) = H^{1}(Π_{g_{1}}, IC) ⊕ H^{1}(Π_{g_{2}}, IC) .

We claim then that the two pencils are distinct. Were they equal, we would get a fibration f: S → B where B has genus g_{1} +g_{2}. Therefore, in the latter case we would get a corresponding surjection of fundamental groups Π_{g_{1}} × Π_{g_{2}} → Π_{g_{1}+g_{2}}, and a corresponding homomorphism of cohomology algebras inducing an isomorphism of first cohomology groups; we get a contradiction, since

Λ^{4}(H^{1}(Π_{g_{1}}, IC) ⊕ H^{1}(Π_{g_{2}}, IC)) → H^{4}(Π_{g_{1}} ⊕ Π_{g_{2}}, IC))

has non zero image, while

Λ^{3}(H^{1}(Π_{g_{1}+g_{2}}, IC)) → H^{3}(Π_{g_{1}+g_{2}}, IC) = 0.

The second proof uses the weaker assumption Γ/Γ_{3} ≅ Π/Π_{3} and the fact that, for a Kähler manifold X , the kernel of the cup product map

H^{1}(X, IC) ∧ H^{1}(X, IC) → H^{2}(X, IC) is dual to (Γ_{2}/Γ_{3}) ⊗ IR ( cf. [ABCKT] , cor. 3.14, page 33 and prop. 3.25 , page 38 ).

Applying this observation to X = S and X = C_{1} × C_{2} , we infer that H^{1}(X, IC) is the direct sum of two maximal isotropic subspaces , whence we obtain the two distinct pencils by the Isotropic Subspace Theorem 2.6.

Step II : since the pencils are distinct, we obtain a surjective holomorphic map f = (f_{1} × f_{2}): S → C_{1} × C_{2}. Consider now the fibration f_{2}, which factors through f. The genus g′_{1} of the fibres of f_{2} is therefore at least g_{1}. Since by assumption 4(g_{1} − 1)(g_{2} − 1) = e(S) which, by 2.1 (Zeuthen-Segre) is ≥ 4(g′_{1} − 1)(g_{2} − 1) ≥ 4(g_{1} − 1)(g_{2} − 1), equality holds throughout whence g′_{1} = g_{1}, and f_{2} is a topological bundle. Since the general fibre of f_{2} is isomorphic then to C_{1}, and f_{2} is a topological bundle, then f yields the desired isomorphism S ≅ C_{1} × C_{2}. In the case where we assume K_{S}^{2} = 8(g_{1} − 1)(g_{2} − 1), we invoke Arakelov’s theorem (2.2), and we obtain therefore that f is birational. Since moreover we assume S minimal, then f yields an isomorphism. Case c) follows immediately from (2.3). □

(3.5) Remark. As we saw in section 1 (Cor. 1.3), the property that a surface S is isogenous to a higher product cannot be detected solely by its fundamental group. From the geographical viewpoint, the previous theorem is rather interesting, because, in terms of the topological invariants of the minimal surface S, x = χ(O_{S}), y = K_{S}^{2}, it tells that on the lines x = constant, y = constant, 12x − y = constant, such a fundamental group commands a geometric property. It is an interesting question whether the condition of lying on the watershed line y = 8x (S has zero index) can replace in theorem 3.4 the conditions e), k) or c). One can further weaken the hypothesis on the fundamental group of S using the following

(3.6) Proposition. A fundamental group π_{1}(S) is \underline{commensurable} \underline{to} \underline{a} (\underline{direct}) \underline{product} Π_{g′_{1}} × Π_{g′_{2}} if and only if it contains a subgroup of finite index which is isomorphic to a group Π_{g_{1}} × Π_{g_{2}}.

Proof. Assume π_{1}(S) contains a finite index subgroup Γ which is isomorphic to a subgroup of Π_{g′_{1}} × Π_{g′_{2}} (again of finite index). Then Γ contains Γ′ normal and of finite index in Π_{g′_{1}} × Π_{g′_{2}}. Set Γ′_{1} = Γ′ ∩ Π_{g′_{1}}: it is normal in Π_{g′_{1}}, of finite index, and isomorphic to a Π_{g_{1}}. Define similarly Γ′_{2} ≅ Π_{g_{2}}. Then Π_{g_{1}} × Π_{g_{2}} is of finite index in Γ, whence also in π_{1}(S). □

In order to study the moduli space of surfaces isogenous to a product, we need to show that a stronger property is satisfied by these surfaces.

(3.7) Definition. A surface S is said to be strongly isogenous to a product if S is the quotient of the action of a group G acting freely on a product of curves (S = C_{1} × C_{2}/G).

In order to study this stronger property, we need some auxiliary results, of which the following is well known.

(3.8) Rigidity lemma. Let f: C_{1} × C_{2} → B_{1} × B_{2} be a surjective holomorphic map between products of curves. Assume that both B_{1}, B_{2} have genus ≥ 2. Then, after possibly exchanging B_{1} with B_{2}, there are holomorphic maps f_{i}: C_{i} → B_{i} such that f(x, y) = (f_{1}(x), f_{2}(y)).

Sketch of proof. Hol(C_{i}, B_{j}) is a discrete (indeed finite) set, since the normal bundle to the graph of a φ: C_{i} → B_{j} is isomorphic to the pull back of the tangent bundle to B_{j}. Thus, if a component of f, f_{i}(x, y) is not constant in x, it is costant in y and the statement follows then easily from the surjectivity of f. □

(3.9) Corollary. Assume that both C_{1}, C_{2} are curves of genus ≥ 2. Then the inclusion Aut (C_{1} ×C_{2}) ⊃ Aut (C_{1})×Aut (C_{2}) is an equality if C_{1} is not isomorphic to C_{2} , whereas Aut (C × C) is a semidirect product of Aut (C)^{2} with the IZ/2 given by the involution Φ exchanging the two coordinates.

(3.10) Remark. Assume that S can be represented as a quotient S = C_{1} × C_{2}/G by a free action. Then we can assume that G, or the subgroup G_{o} = G ∩ Aut (C)^{2} in the case C_{1} ≅ C_{2}, embeds in Aut (C_{i}) for i = 1, 2 (we shall then say that this quotient realization is minimal). Else, e.g. the kernel K_{1} acts trivially on C_{1} and freely on C_{2}; we can thus replace C_{1} × C_{2} by (C_{1}/K_{2}) × (C_{2}/K_{1}).

(3.11) Proposition. A surface S is strongly isogenous to a higher product if and only if it is isogenous to a higher product.

Proof. We clearly only need to show that if there is a finite unramified covering u: C_{1} × C_{2} → S, then S can be represented as a quotient S = C′_{1} × C′_{2}/G by a free action. Let Z be the Galois closure of u, and G′ the Galois group, so that we have another unramified covering ψ: Z → C_{1} × C_{2}. All the fibres of Z → C_{2} are unramified coverings of C_{1}, whence, taking the Stein factorization of Z → C_{2}, we get a holomorphic bundle Z → C′′_{2}. Finally, the canonical unramified and Galois base change C′_{2} → C′′_{2} associated to the kernel of the monodromy yields by pull-back a product C′_{1} × C′_{2}. To conclude, since S is a Galois unramified quotient of Z, and Z is a Galois unramified quotient of C′_{1} × C′_{2}, we need to show that every Galois automorphism τ of Z lifts to C′_{1} × C′_{2}. Let v: C′_{1} × C′_{2} → Z be the quotient map. Then, by the rigidity lemma, either ψτv, or possibly Φψτv is a product map of respective maps C′_{i} → C_{i}. Therefore either τ preserves the fibres of the map p′′_{2}: Z → C′′_{2}, or sends them to the fibres of p′′_{1}: Z → C′′_{1}. In both cases there is an isomorphism τ′′: C′′_{2} → C′′_{i} such that τ′′p′′_{2} = p′′_{i}τ. Clearly then τ′′ leaves invariant the kernel of the monodromy, whence we get a lift τ′: C′_{2} → C′_{i}, and, by taking the fibre product with τ: Z → Z, we obtain the desired lifting of τ to C′_{1} × C′_{2}. □ We can sharpen the previous result

(3.12) Corollary. Every minimal product cover of a surface S strongly isogenous to a higher product is Galois.

Proof. In the notation of 3.11, we have S = C′_{1}×C′_{2}/G and a factorization C′_{1}×C′_{2} → C_{1} × C_{2} → S , so that there is a subgroup G′ of G such that C_{1} × C_{2} = C′_{1} × C′_{2}/G′. But since G′ = π_{1}(C_{1} × C_{2})/π_{1}(C′_{1} × C′_{2}) and by the rigidity lemma we have a product map, G′ is a direct product G′_{1} × G′_{2}. Then G′_{1} ⊂ K′_{2}, G′_{2} ⊂ K′_{1} (cf. 3.10), thus C_{1} × C_{2} dominates (C′_{1}/K′_{2}) × (C′_{2}/K′_{1}). Finally, by the minimality assumption, we must have equality, whence C_{1} × C_{2} → S is Galois. □

We shall now show that a minimal realization is unique: by the above it follows that any minimal product covering is Galois and unique.

(3.13) Proposition. If S is isogenous to a higher product, then a minimal realization S = C_{1} × C_{2}/G is unique.

Proof. Assume that we have two such minimal realizations S = C_{1} × C_{2}/G, S = B_{1}×B_{2}/Γ. Then we can dominate both products C_{1}×C_{2}, B_{1}×B_{2} with a component Z of their fibre product over S. By looking at the holomorphic bundle induced on Z by the projection onto B_{2}, we obtain a Galois unramified covering D_{2} of B_{2} and a product D_{1} × D_{2} dominating Z compatibly with the projection onto B_{2}. By the rigidity lemma we know that D_{1} × D_{2} dominates C_{1} × C_{2} by a product map, and we can assume that it does with a map for which D_{1} surjects onto C_{1}, (resp.: D_{2} onto C_{2}). This shows that the two fibrations on S induced by the projections on C_{2}/G, resp. B_{2}/Γ, coincide. Since both products C_{1} × C_{2}, B_{1} × B_{2}, are obtained by the same canonical procedure (pull back under the Galois cover associated to the kernel of the monodromy), they are isomorphic. □

In most of the cases surfaces which are isogenous to a higher product are distinguished by the behaviour of the Albanese map. Observe preliminarly:

(3.14) Remark. A symmetric product C^{(2)} of a curve C of genus g ≥ 2 is never birational to a product of curves of respective genera a ≥ 2, b ≥ 1.

Proof. The invariants of C^{(2)} are q = g, p_{g} = \frac{g(g−1)}{2} ≥ 1, whereas the invariants of a product of curves of respective genera a, b are q = a + b, p_{g} = a b. If g = a + b, then p_{g} = \frac{g(g−1)}{2} = ab + \frac{1}{2}(a^{2} − a + b^{2} − b) and this expression equals ab (which has to be ≥ 1) iff a = b = 1. In this case (excluded by our assumptions) then g = 2 and C^{(2)} is the Jacobian of C, and it can indeed be isomorphic to a product of elliptic curves (cf. [L-B], pages 313-316, and ex. 11, 12 on page 319). □

(3.15) Proposition. Let S = C_{1} × C_{2}/G be the minimal realization of a surface S isogenous to a higher product, and let G^{o} be the subgroup of G given by the transformations which do not exchange the two factors C_{1}, C_{2}. We shall say that we have the \underline{mixed} case (denoted by M) when G ≠ G^{o}, and the \underline{double} case otherwise (denoted by D). Denote by C′_{i} the quotient C_{i}/G when G = G^{o}, and let g′_{i} = genus of C′_{i}. In the mixed case, G ≠ G^{o}, so we set C′ = C/G^{o} and we let g′ = genus of C′. Then for the Albanese image α(S) of S, occur the following possibilities: M^{++}) when G ≠ G^{o}, and g′ ≥ 2, then α(S) is birational to the symmetric product C^{′(2)} of C′. M^{+}) when G ≠ G^{o}, and g′ = 1, then α(S) is an elliptic curve. D^{++}) when G = G^{o}, and g′_{i} ≥ 1, then α(S) is isomorphic to the product C′_{1} × C′_{2}. GH) (Generalized Hyperelliptic) is defined to be the case when G = G^{o}, G operates freely on C_{1} and g′_{2} = 0: then α(S) is isomorphic to C′_{1}, which has genus at least 2. In the case D^{+}), when g′_{1} ≥ 1, g′_{2} = 0 , but G does not act freely on C_{1}, then α(S) is isomorphic to C′_{1}. In the cases M^{0}), when g′ = 0, or D^{0}) (g′_{1} = g′_{2} = 0), then α(S) is a point (S has irregularity q(S) = 0).

Proof. Since S = C_{1} × C_{2}/G we have H^{0}(Ω_{S}^{1}) = (H^{0}(Ω_{C_{1}}^{1}) ⊕ H^{0}(Ω_{C_{2}}^{1}))^{G} ⊂ H^{0}(Ω_{C_{1}}^{1})^{G^{o}} ⊕H^{0}(Ω_{C_{2}}^{1})^{G^{o}} = H^{0}(Ω_{C′′_{1}}^{1})⊕H^{0}(Ω_{C′′_{2}}^{1}). In case M the quotient IZ/2 ≅ G/G^{o} exchanges the two last addenda, whence H^{0}(Ω_{S}^{1}) ≅ H^{0}(Ω_{C′′}^{1}). Therefore α: S → α(S) factors as the projection to C^{′(2)} composed with the Albanese map a of C^{′(2)}, which is just the Abel Jacobi map on degree 2 divisors. Whence when g′ ≥ 2 a is birational to its image, and indeed an embedding unless C′ is hyperelliptic. The cases g′ = 0, 1 are obvious. In the non mixed cases, α is the composition of the projection to C′_{1} × C′_{2} with the product of the respective Albanese maps. □

We have therefore divided the surfaces S isogenous to a higher product into 7 classes, D^{++}), D^{+}) and D^{0}) = of double type, GH) = Generalized Hyperelliptic surfaces, plus those of mixed type. We want to describe in greater detail the latter.

(3.16) Proposition. Surfaces S isogenous to a higher product and of mixed type are obtained as follows. There is a (faithful) action of a finite group G^{o} on a curve C of genus at least 2 and a non split extension 1 → G^{o} → G → IZ/2 → 1, yielding a class [φ] in Out (G^{o}) = Aut(G^{o})/Int(G^{o}), which is of order 2. Once we fix a representative φ of the above class, there exists an element τ = λ(φ) in G^{o} such that

i) φφ(γ) = τγτ^{−1} ii) G^{o} acts, under a suitable isomorphism of C_{1} with C_{2}, by the formulae:

γ(x, y) = (γx, (φγ)y) for γ in G^{o}; whereas the lateral class of G^{o} consists of the transformations

τ′γ(x, y) = ((φγ)y, τγx).

Let Σ be the subset of G^{o} consisting of the transformations having some fixed point. Then the condition that G acts freely amounts to:

A) Σ ∩ φ(Σ) = {identity} B) there is no γ in G^{o} such that φ(γ)τγ is in Σ.

Indeed if we pick another automorphism φ of G^{o} in the given class [φ], then the choice of τ has to vary according to the following cocycle-like condition:

iii) if φ̂ = φ ∘ Int (γ′′), then τ̂ = φ(γ′′)τγ′′.

Proof. G^{o} is the subgroup of the transformations not exchanging the two factors of C ×C, while we let τ′ be a transformation not in G^{o}. Then, fixed some isomorphism C_{1} × C_{2} ≅ C × C, τ′(x, y′) = (τ_{2}y′, τ_{1}x). We choose then y = τ_{2}y′ as new coordinate on the second factor. Then τ′(x, y) = (y, τx), where τ = τ_{2}τ_{1}x. Since τ′τ′(x, y) = (τx, τy), we see that τ belongs to G^{o}. Since the realization is minimal, we see G^{o} as ⊂ Aut (C) under the action on the first factor, and we write γ(x, y) = (γx, γ_{2}y), so that γ → γ_{2} gives the other embedding of G^{0} in Aut (C) provided by the action on the second factor. The condition that τ′ normalizes G^{o} means that for every γ there is a γ′′ such that τ′γ = γ′′τ′. This more precisely means: (γ_{2}y, τγx) = (γ′′y, γ′′_{2}τx). Thus γ_{2} = γ′′, γ′′_{2} = τγτ^{−1}. The first equality tells us that the image of G^{o} in Aut (C) is equal for the action on both factors, whence exactly that there is an automorphism φ of G^{o} such that γ_{2} = φ(γ); the second equality says exactly that φφ(γ) = τγτ^{−1}. There remains to ensure the condition that G acts freely: for G^{o}, this means that either γ or (φγ) acts freely on C, and this is just condition A). For τ′γ, it acts freely if there is no pair (x, y) = (φ(γ)y, τγx), i.e., φ(γ)τγ (or, equivalently, τγφ(γ)) has no fixed points on C. Finally, our calculations above indeed show that, should we replace τ′ by τ′′ = τ′ · γ′′, then we would have to take as new variable y′′ = φ(γ′′)y so that τ′′(x, y′′) = (y′′, φ(γ′′)τγ′′x). Thus τ gets replaced by τ̂ = φ(γ′′)τγ′′, and φ by φ̂ = φ∘Int (γ′′), where Int (γ′′)(γ) = γ′′γγ^{′′−1}; and again φ̂ ∘ φ̂ = Int (τ̂). The extension 1 → G^{o} → G → IZ/2 → 1 is non split since τγφ(γ) is different always from the identity. Moreover such an extension gives (by conjugation of a lift of the generator of IZ/2) an element [φ] of order 2 in Out (G^{o}) = Aut (G^{o})/Int (G^{o}). Observe that such a [φ], in general, only provides an extension

1 → Int (G^{o}) → Γ → IZ/2 → 1.  □

(3.17) Remarks.

1) A trivial solution for finding surfaces of type M), is to take an unramified Abelian Galois covering C of C′, of order divisible by 2, let φ be the identity, and take τ an element in the Galois group G^{o} which is not a square.

2) The easiest way to construct surfaces of the other types, is to take a quotient C_{2} → C_{2}/G, and then, for any curve C′_{1} of genus ≥ 2, take a surjection π_{1}(C′_{1}) → G.

We shall return more amply to these issues in the next section. We end this section by giving other different characterizations of Generalized Hyperelliptic surfaces, again using the standard inequalities which we recalled in section 2.

(3.18) Theorem. Let S be a surface such that o) K_{S}^{2} = 8χ(S) > 0 (S has index 0 but e(S) > 0)

i) S has irregularity q ≥ 2 and the Albanese map is a pencil (i.e.,

Λ^{3}(H^{1}(S, IC)) → H^{3}(S, IC) has zero image). Then, letting g be the genus of the Albanese fibres, we have: (g − 1) ≤ \frac{χ(S)}{(q−1)}. A surface S is a Generalized Hyperelliptic surface if and only if o), i) hold and g = \frac{1+χ(S)}{(q−1)}. In particular, every S satisfying o), i), and with p_{g} = 2q − 2 is a Generalized Hyperelliptic surface where G is a group of automorphisms of a curve C_{2} of genus 2 with C_{2}/G ≅ IP^{1}.

Proof. If o) holds and S is fibred over a curve B of genus b, note that the inequalities of Zeuthen-Segre, Arakelov, Beauville, are all equivalent to each other. Since by i) we have a pencil with base curve B of genus q, the first assertions are simply a restatement of the aforementioned theorems; for instance, if equality holds, we have that the Albanese map α yields a holomorphic bundle, whence S is a GH surface. Finally, p_{g} = 2q − 2, since χ(S) = 1 − q + p_{g}, is equivalent to χ(S) = (q − 1).Thus g ≤ 2 and since χ(S) > 0, g = 2, therefore we have again a holomorphic bundle and S is a GH surface. □

(3.19) Remark. The GH surfaces with g = 2 can be explicitly classified. In fact, all the groups G acting effectively as a group of automorphisms of a curve C_{2} of genus 2 with C_{2}/G ≅ IP^{1}, are explicitly classified by Bolza ([Bo], cf. also the Pisa 1995 Ph.D. Thesis by F. Zucconi). One can easily verify that for every such group G there is a surjection Π_{2} → G. It is a trivial remark that, for each k ≥ 2, there is a surjection Π_{k} → Π_{2}. We obtain therefore the following

(3.20) Corollary. Assume χ(S) = q − 1, K_{S}^{2} = 8χ(S) > 0, and that the Albanese map is a pencil. Then S is a Generalized Hyperelliptic surface with Albanese fibre of genus 2. These exist for each integral value of χ(S).

We end this section by describing surfaces in the classes D^{0}) and D^{+}), following some examples given by Beauville ([Bea1], exercise 4, page 159).

(3.21) Lemma. Let G = (IZ/n)^{2} and Σ = {(a, b) | a = 0, or b = 0, or a = b}. Then there is an automorphism φ of G with φ(Σ) ∩ Σ = {0} if and only if 6 and n are relatively prime.

Proof. If m divides n , then (IZ/m)^{2} ⊂ G and is characteristic. Since such a φ clearly does not exist if n = 2 or n = 3 because in these cases card Σ−1 = 3n−3 ≥ \frac{(card G)}{2}, it follows that 2,3 cannot divide n. But if 2,3 do not divide n, the map φ sending e_{1} = (1, 0) to (1, −2), and e_{2} = (0, 1) to (1, −1) has determinant 1, whence it is an isomorphism. φ sends e_{0} = (1, 1) to (2, −3) and all determinants formed with two vectors e_{i}, φ(e_{j}) are invertible in IZ/n, whence the assertion follows. □

(3.22) Beauville’s Examples. Let C be the Fermat curve of degree n in the projective plane. C = {(z_{0}, z_{1}, z_{2}) | z_{0}^{n} + z_{1}^{n} + z_{2}^{n} = 0} is a Galois cover of the line IP^{1} = {(x_{0}, x_{1}, x_{2}) | x_{0} + x_{1} + x_{2} = 0} under the map ψ: IP^{2} → IP^{2} sending (z_{0}, z_{1}, z_{2}) to (z_{0}^{n}, z_{1}^{n}, z_{2}^{n}). ψ is branched in 3 points, and is Galois with Galois group G = (IZ/n)^{2}. In this example the 3 stabilizers are the diagonal, and the two coordinate subgroups of (IZ/n)^{2}. Thus lemma 3.21 applies if n is odd and not divisible by 3. We let S_{n} = C_{n} × C_{n}/(IZ/n)^{2}, where G = (IZ/n)^{2} acts by g(x, y) = (gx, φ(g)y). So the action is free and S_{n} is a surfaces of type D^{0}) (i.e., isogenous to a product and with q = 0), with invariants K^{2} = 2(n − 3)^{2}, χ = \frac{(n−3)^{2}}{4} . (The most famous of these examples occurs for n = 5, yielding χ = 1). We obtain also surfaces of class D^{+}), simply by letting C_{1} = C, and C_{2} be the fibre product of C with an elliptic curve E mapping to IP^{1} with degree 2 and branched in 4 general points. Again by twisting the action on the second factor, we get a free action of G such that S = C_{1} × C_{2}/G has invariants

K^{2} = 4(n − 3)(3n − 3), χ = \frac{(n − 3)(3n − 3)}{2}, q(S) = 1 .

The final remark we want to make is that, in view of the results we shall show in section 4 ( esp.: Thm. 4.13 and its proof) the above examples are strongly rigid; in the sense that any surface S with isomorphic fundamental group and the same Euler number as one of Beauville’s examples is exactly one of those examples. This motivates the following

(3.23) Definition. A Beauville surface is a rigid surface which is isogenous to a product.

Classifying all the Beauville surfaces is then a problem in group theory. That is, we give a finite group G, representable as a quotient of a triangle group ⟨a, b, c | abc = 1 = a^{p} = b^{q} = c^{r}⟩, with p, q, r minimal. The question is to classify all those G for which there is an automorphism φ for which φ(Σ) ∩ Σ = {1}, Σ being the union of the 3 subgroups generated by a, b, c, respectively.

4. The fundamental group of surfaces isogeneous to a product, and the irreducibility of their moduli space

Let S be a surface isogenous to a product, and let as usual S = C_{1} × C_{2}/G be its minimal realization. Let us assume for the time being that S is not of mixed type. Therefore we have an exact sequence

(4.1) 1 → Π_{g_{1}} × Π_{g_{2}} → π_{1}(S) → G → 1,

where the two factors Π_{g_{1}}, Π_{g_{2}}, are both normal. We can therefore take the quotient (i = 1, 2 has to be understood as an element of IZ/2)

(4.2) Π(i + 1) := π_{1}(S)/Π_{g_{i}},

which sits naturally into an exact sequence

(4.3) 1 → Π_{g_{i}} → Π(i) → G → 1.

If G operated freely e.g. on C_{1}, we would then have a bundle S → C′_{1}, with fibre C_{2}, and the homotopy exact sequence of the bundle would tell us immediately that Π(1) ≅ π_{1}(C′_{1}); what we are going to see next is that (4.3-i) is the “orbifold” exact sequence associated to the possibly ramified Galois covering C_{i} → C′_{i}, so that Π(i) is the “orbifold fundamental group of the covering”. So, let us recall the notion of orbifold fundamental group.

(4.4) Definition. Let X be a complex manifold and p: X → Y = X/G the quotient by a finite group of automorphisms. Then Y is a normal complex space, and we denote by B the branching locus of p (B = p({x | x has a non trivial stabilizer G_{x}})). Let B_{1}, … B_{r} be the divisorial irreducible components of B. Let γ_{1}, … γ_{r} in π_{1}(Y − B) be simple geometric loops around the components B_{1}, … B_{r}. Then, since we have an exact sequence

1 → π_{1}(X − p^{−1}(B)) → π_{1}(Y − B) → G → 1

we get that each γ_{i} maps to an element g_{i}, whose order will be denoted by m_{i}. Let m′′ be the vector ^{t}(m_{1}, … m_{r}). Then the orbifold fundamental group πorb_{1}(Y −B|m′′) is defined to be the quotient of π_{1}(Y − B) by the minimal normal subgroup containing the elements (γ_{i})^{m_{i}}.

Notice that p^{−1}(B_{i}) is a divisor = m_{i}R_{i}, where R_{i} (not necessarily connected) is a covering of B_{i} of degree \frac{d}{m_{i}} (d being the order of G); (γ_{i})^{m_{i}} lifts to a simple geometric loop in X − p^{−1}(B) around a component of R_{i}, whence it belongs to the kernel of π_{1}(X − p^{−1}(B)) → π_{1}(X). On the other hand, the above kernel is normally generated by a set of simple geometric loops around each component of each R_{i}. But, varying the components of R_{i}, we get such loops simply as conjugates of (γ_{i})^{m_{i}}.

We have therefore shown

(4.5) Proposition. Let, as in 4.4, X be a complex manifold and p: X → Y = X/G a finite quotient map. Then we have an exact sequence

1 → π_{1}(X) → πorb_{1}(Y − B|m′′) → G → 1.

(4.6) Proposition. The exact sequence (4.3-i)

1 → Π_{g_{i}} → Π(i) → G → 1,

is the orbifold exact sequence for the quotient map C_{i} → C′_{i}.

Proof. Let C_{2}^{∗} be equal to the inverse image of C_{2}^{′∗} = C′_{2} −B, B being the branching divisor. Denoting by S^{∗} = (C_{1} × C_{2}^{∗})/G, we get a covering space exact sequence, fitting into a diagram (4.7)

Π_{g_{1}}
↘
1 → Π_{g_{1}} × π_{1}(C_{2}^{∗}) → π_{1}(S^{∗}) → G  →  1
↓  ↓ ↘ ↗
1 → Π_{g_{1}} × π_{1}(C_{2}) →  π_{1}(S)  π_{1}(C_{2}^{′∗}) → πorb_{1}(C_{2}^{′∗}|m′′) → G

where the vertical maps are surjective, and the oblique sequence is the exact homotopy sequence of the bundle S^{∗} → C_{2}^{′∗}.

Notice that S − S^{∗} is the union of the fibres F_{b} for b in B, which are multiples of multiplicity m_{b}; thus the kernel of π_{1}(S^{∗}) → π_{1}(S) is normally generated by simple geometric loops δ_{b}, mapping to (γ_{b})^{m_{b}} down in π_{1}(C_{2}^{′∗}).

Therefore π_{1}(S) surjects onto πorb_{1}(C_{2}^{′∗}|m′′), and indeed the surjection π_{1}(S) → G factors through πorb_{1}(C_{2}^{′∗}|m′′) → G. There remains to show that Π_{g_{1}}, which is contained in the kernel, is indeed precisely the kernel of the above surjection. Please note that π_{1}(C_{2}^{′∗}) → πorb_{1}(C_{2}^{′∗}|m′′) → G is a sequence of surjections (it is not exact!).

After this remark, we observe that the kernel of π_{1}(S^{∗}) → πorb_{1}(C_{2}^{′∗}|m′′) is normally generated by the δ_{b}’s, and by Π_{g_{1}}, whence it is precisely the pull back of the normal subgroup Π_{g_{1}} of π_{1}(S). The assertion for Π(1) follows by identical proof. □

The proof of the following corollary is now obvious

(4.7) Corollary. Let the surface S be isogenous to a product and not of mixed type. Then we have an exact sequence

1 → Π_{g_{1}} × Π_{g_{2}} → Π(1) × Π(2) → G × G → 1

(where Π(1) × Π(2) ≅ πorb_{1}(C_{1}^{′∗}|m′′(1)) × πorb_{1}(C_{2}^{′∗}|m′′(2))) such that π(S) is the inverse image of G diagonally embedded in G × G.

(4.8) Remark. Although 4.7 does not yet give a presentation of π_{1}(S), it is possible to get one using the Reidemeister Schreier process. Let us give the classical example of the general hyperelliptic surface, where G = IZ/2. We get 1 → IZ^{4} → IZ^{2} ⊕ G(2, 2, 2, 2) → IZ/2 ⊕ IZ/2 → 1. G(2, 2, 2, 2) has generators a_{1}, … a_{4} such that a_{1} · . . . · a_{4} = 1, a_{i}^{2} = 1, IZ^{2} has generators E_{1}, E_{2}, IZ^{4} has generators e_{1} = 2E_{1}, e_{2} = E_{2}, e_{3} = a_{1} · a_{2}, e_{4} = a_{2} · a_{3}, and each a_{i} maps to (0, 1), while E_{1} maps to (1, 0), E_{2} to (0, 0). Let a = (E_{1}, a_{1}): then π_{1}(S) is generated by e_{1}, … e_{4}, a, where the e_{i}’s commute, a^{2} = e_{1}, and ae_{j}a^{−1} = δ_{j}e_{j}, with δ_{j} = 1 for j = 1, 2, δ_{j} = −1 for j = 3, 4. Therefore π_{1}(S) is a semidirect product of the normal subgroup IZ^{3} generated by e_{2}, e_{3}, e_{4}, with the cyclic group IZ generated by a. (4.9) Remarks.

A) The exact sequence (4.3-i) is also important from the point of view of uniformization. Viewing C_{i} = H/Π_{g_{i}}, where H is the upper half plane, one can extend the embedding of Π_{g_{i}} → P SL(2, IR) to an embedding of Π(i) such that C′_{i} = H/Π(i).

B) The same exact sequence gives a homomorphism ρ: G → Out (Π_{g_{i}}) (to an element g of G we associate the action on Π_{g_{i}} obtained by conjugation with a lifting of g: this action is well defined then only up to inner automorphisms). ρ is injective because every complex automorphism which acts as the identity on the first homology group has to be the identity. Conversely, we claim that the above extension is completely classified by the injective homomorphism ρ. In fact, the inclusion G ⊂ Out (Π_{g_{i}}) provides an exact sequence 1 → Int (Π_{g_{i}}) → Π → G → 1. Observe that the center of Π_{g_{i}} is trivial (two commuting hyperbolic transformations in P SL(2, IR) have the same fixed points, whence they lie in a maximal torus IR^{∗}; thus, if they lie in a discrete subgroup, then they generate a cyclic subgroup). Therefore Π_{g_{i}}) ≅ Int (Π_{g_{i}}), and we recover the extension (4.3.i).

What in fact proposition 4.6 really tells is that the geometry of the covering C_{i} → C′_{i} is encoded in the exact sequence (4.1). Indeed, (take e.g. i = 2) from it we recover firstly the orbifold exact sequence

1 → π_{1}(C_{2}) = Π_{g_{2}} → πorb_{1}(C_{2}^{′∗}|m′′) = Π(2) → G → 1.

Then, by remark 4.9, it follows that every branch point b corresponds exactly to a Π(2)-conjugacy class of a maximal cyclic subgroup of Π(2) (every finite order element in P SL(2, IR) is elliptic, and has precisely one fixed point in H), whose order is the multiplicity m_{b}. The ramification points above correspond instead to the Π_{g_{2}} conjugacy classes of such cyclic subgroups.

Moreover, since the epimorphism πorb_{1}(C_{2}^{′∗}|m′′) → π_{1}(C′_{2}) has a kernel which is generated precisely by the elements of finite order, we likewise recover π_{1}(C′_{2}). The irreducibility of the moduli space of surfaces isogenous to a product with given Euler number and given fundamental group would follow at once from a generalization of the classical results of Fricke-Teichml̈ler-theory, namely that the embeddings of Π(i) in P SL(2, IR) form an irreducible variety. But we are not aware of such results in the literature, therefore we shall resort in the general case to the solution of the Nielsen realization problem (cf. [Ker], [Wol], [Tro]). In fact, the classical theory suffices when the action of G is free on both factors (4.10) Proposition. The moduli space 𝔐^{is}(Γ) of the surfaces S isogenous to a higher product not of mixed type, with the same Euler number e(S) and with fundamental group π_{1}(S) isomorphic to a fixed group Γ, is irreducible in the case where G acts freely on both factors.

(4.11) Remark. Γ being isomorphic to π_{1}(S) it contains, by 3.13, a unique maximal subgroup Γ′′ of finite index isomorphic to Π_{g_{1}} × Π_{g_{2}}, such that e(S) · (index(Γ′′)) = 4(g_{1} − 1)(g_{2} − 1). The property that S is of mixed type can be read off Γ and the Euler number e(S) of S. In fact S is of mixed type exactly when, although Π_{g_{1}} × Π_{g_{2}} is, as we know, normal, the individual factors Π_{g_{1}} and Π_{g_{2}} are not normal. Also the property that G acts freely on both factors can be read off Γ, since it means that the quotients Γ(i) sitting in the exact sequences 1 → Π_{g_{i}} → Γ(i) → G → 1 have no element of finite order.

Proof of 4.10. The minimal realization S = C_{1} × C_{2}/G is given by an action of G on C_{1}, and another action on C_{2}. Notice that we obtain a deformation of S by deforming C_{1}, C_{2} together with the action of G. Consider now an isomorphism φ of π_{1}(S) with Γ, and notice that Γ comes equipped with an exact sequence

1 → Π_{g_{1}} × Π_{g_{2}} → Γ → G → 1,

inducing

1 → Π_{g_{i}} → Γ(i) → G → 1.

¿From now on we will fix Γ(1), Γ(2) as the fundamental groups of curves of respective genera g′_{1}, g′_{2}. Notice that φ induces then isomorphisms π_{1}(C′_{i}) ≅ Γ(i). Given another such surface Ŝ, and another isomorphism φ̂ of π_{1}(Ŝ) with Γ, using the connectedness of the Teichmüller space for C′_{2}, we can find a real analytic path deforming Ĉ′_{2} to C′_{2}, and preserving the isomorphism with Γ(2), and we can similarly deform Ĉ′_{1} to C′_{1} preserving the isomorphism with Γ(1). Since the action of G on a curve C_{i} is given by the surjection of π_{1}(C′_{i}) onto G obtained composing the isomorphism π_{1}(C′_{i}) ≅ Γ(i) with the given surjection of Γ(i) onto G, our assertion is proven. □

In the general case, observe that the orbifold fundamental group extension

1 → π_{1}(C_{i}) = Π_{g_{i}} → πorb_{1}(C_{i}^{′∗}|m′′(i)) → G → 1

gives an action on C_{i} which is topologically classified by the associated injective homomorphism G → Out (Π_{g_{i}}) that we mentioned earlier. More precisely, Out (Π_{g}) = Π_{0}(Diff (M_{g})) , M_{g} being a standard differentiable model for Riemann surfaces of genus g. The modular group Mod_{g} = Π_{0}(Diff (M_{g})) acts properly discontinuously on the Teichmüller space T_{g} parametrizing the complex structures on M_{g} (modulo diffeomorphisms isotopic to the identity). Once a finite subgroup G of Π_{0}(Diff (M_{g})) is given, the fixed locus of the action on T_{g} corresponds to the isomorphism classes of curves of genus g with a holomorphic action by G which is differentially equivalent to the given one on M_{g}. The following is a slight generalization of one of the solutions ([Tro]) of the Nielsen realization problem.

(4.12) Lemma. Given a finite subgroup G of Π_{0}(Diff (M_{g})), the fixed locus Fix(G) of G on T_{g} is non empty and connected (indeed, diffeomorphic to an euclidean space).

Proof. Teichmüller space T_{g} is diffeomorphic to an Euclidean space of dimension 6g − 6, and it admits a Riemannian metric, the Weil-Petersson metric. Wolpert and Tromba proved:

i) there exists a C^{2}-function f on T_{g} which is proper, G-invariant, non negative (f ≥ 0), and finally f is strictly convex for the given metric (i.e., strictly convex along the W-P geodesics). We remark that, G being a finite group, its action can be linearized at the fixed points, in particular

ii) Fix(G) is a smooth (non a priori connected) submanifold.

Sublemma. There is a unique critical point x_{o} for f, which is an absolute minimum on T_{g}. Every connected component M of Fix(G) contains x_{o}.

Proof of the sublemma. Let M′ be either the submanifold M, or T_{g} . Since f is proper and ≥ 0, there is a constant c such that f^{−1}([0, c]) ≠ ∅, and clearly compact. Thus there is an absolute minimum x_{o} on T_{g}, and an absolute minimum y_{o} for f|_{M}. Since f is strictly convex on T_{g}, every critical point for f is isolated and a local minimum. f behaves like a Morse function, therefore, T_{g} being homeomorphic to an Euclidean space IR^{6g−6}, it follows that x_{o} is the only critical point of f on T_{g}. It suffices therefore to show that y_{o} is a critical point for f on T_{g}, because then y_{o} = x_{o} and M contains x_{o}. Let us look at the tangent space V to T_{g} at y_{o}: this is a G- representation and the subspace of invariants V^{G} is just the tangent space to M at y_{o}. We look at the differential of f evaluated at y_{o}: this gives a linear form df in V^{∨} which is invariant since f is G-invariant. We can write V = V^{G} ⊕ W, where W is a sum of irreducible non trivial representations. Therefore V^{∨} = (V^{G})^{∨} ⊕ W^{∨}, and (V^{∨})^{G} = (V^{G})^{∨}. Since df belongs to (V^{∨})^{G}, and it vanishes on (V^{G}), df is = 0, therefore we have shown that y_{0} is a critical point on T_{g}. The same argument gives that there is exactly one critical point for f|_{M}, whence M is homeomorphic to an Euclidean space. □ We can easily conclude the proof of the lemma: x_{0}, being the unique minimum, belongs to Fix(G). Moreover, every connected component of Fix(G) contains x_{0}, thus Fix(G) is a non empty connected submanifold. □ ¿From the above, we derive also an existence theorem for moduli spaces

(4.13) Theorem. Let Γ be a group fitting into an exact sequence 1 → Π_{g_{1}} ×Π_{g_{2}} → Γ → G → 1 where moreover:

a) G is a finite group,

b) the individual factors Π_{g_{1}}, Π_{g_{2}} are normal in Γ,

c) the induced quotients 1 → Π_{g_{i}} → Γ(i) → G → 1 are such that c1) there is no element of Γ mapping in each Γ(i) to an element of finite order, c’) by the above exact sequences G embeds in Out (Π_{g_{i}}).

Let d be the order of the group G. Then the moduli space of the surfaces S with π_{1}(S) ≅ Γ, and with Euler number e(S) = \frac{4(g_{1}−1)(g_{2}−1)}{d} is non empty and irreducible.

Remark. As in theorem 3.4, the assumption on e(S) can be replaced by′′S minimal and K_{S}^{2} = \frac{8(g_{1}−1)(g_{2}−1)}{d}, or χ(O_{S}) = \frac{(g_{1}−1)(g_{2}−1)}{d}.

Proof of 4.13. The proof is entirely analogous to the one of prop. 4.10, except that the sequence 1 → Π_{g_{2}} → Γ(2) → G → 1, determines a fixed topological type of action for G on the curve C_{2}. According to lemma 4.11, then the pairs (curve C_{2}, action of G on C_{2} with the fixed topological type) are parametrized by an irreducible and non empty moduli space. The same holds for (curve C_{1}, action of G on C_{1} with the fixed topological type). The conclusion is then the same as in 4.10, and we have shown that the moduli space is non empty and irreducible. □ There finally remains the case of the surfaces of mixed type, for which we shall make use of the description given in corollary 3.16.

(4.14) Theorem. The moduli space 𝔐^{is}(Γ) of the surfaces S isogenous to a higher product of mixed type, with the same Euler number e(S) and with fundamental group π_{1}(S) isomorphic to a fixed group Γ, is irreducible. In the case where G^{o} operates freely on C, it is indeed a quotient of the Teichmüller space T_{g′} by a subgroup of Mod_{g′}, whence a finite covering of the moduli space 𝔐_{g′} of curves of genus g′.

Proof. Γ comes equipped with a unique maximal subgroup isomorphic to Π_{g} × Π_{g}, of index \frac{4(g−1)^{2}}{e(S)} . We get an exact sequence

1 → Π_{g} × Π_{g} → Γ → G → 1,

and Γ contains an index 2 subgroup Γ^{o} containing both subgroups Π_{g} as normal subgroups. The exact sequence 1 → Π_{g} × Π_{g} → Γ^{o} → G^{o} → 1 provide two embeddings ε_{i} of Γ^{o}(i) in Aut (Π_{g}) (where 1 → Π_{g_{i}} → Γ^{o}(i) → G^{o} → 1).

Let t′ be an element in Γ − Γ^{o}, so that conjugating by t′ yields an automorphism of Π_{g} × Π_{g} which exchanges the two factors. Therefore we can write t′(σ_{1}, σ_{2})t^{′−1} = (ϑσ_{2}, ψσ_{1}). We change the isomorphism of the second factor with Π_{g} by setting ϑσ_{1} = σ′_{2}. Applying the square of the transformation, we get an element of Γ^{o}, τ′′ = ϑψ. In fact t^{′2}(σ_{1}, σ′_{2})t^{′−2} = (ϑψσ_{1}, ϑψσ′_{2}). For γ in Γ^{o}, which we identify to the element of Aut (Π_{g} × Π_{g}) given by inner conjugation, we associate an element φ′′(γ) simply by the rule Int (t′) ∘ Int (γ) = Int (φ′′(γ)) ∘ Int (t′). Thus we obtain an automorphism φ′′ of Γ^{o}, and an easy calculation yields φ′′ · φ′′ = τ′′. Our proof is over: since again using the Teichmüller space of the curves C′ with π_{1}(C′) ≅ Γ^{o}(1) we can find an irreducibile parametrization of all the curve C with an action of G induced by the surjective homomorphism ν: Γ^{o}(1) → G → 1. Then the data ([φ], τ) are clearly recovered by the classes induced by τ′′, φ′′. Finally, when the action is free, we have to see when another different point of Teichmüller space gives an isomorphic surface to the one associated to (C′, β: π_{1}(C′) ≅ Γ^{o}(1)). The condition is that we get an isomorphic curve, and that under this isomorphism the given subgroups (pull back of ker ν) correspond to each other. This means that we get the orbit of the subgroup of the Teichmüller modular group, Mod_{g′}, consisting of the automorphisms μ which leave the subgroup (ker ν) invariant. □

(4.15) Remark. One can then ask the natural question: when is the above moduli space a variety of general type? (cf. e.g. [Har]) We will end this section by showing how to recover the “branching data” (e.g. the isotropy subgroups,...) for the covering C = C_{i} → C′ = C′_{i} (in terms of the orbifold fundamental group extension

1 → π_{1}(C_{i}) = Π_{g_{i}} → πorb_{1}(C_{i}^{′∗}|m′′(i)) = Π(i) → G → 1)

in a more computable way. Recall that, for x in C, the subgroup Stab (x) is cyclic, and therefore we will restrict ourselves to considering cyclic subgroups H of G.

(4.16) Definition. Given a cyclic subgroup H = ⟨g⟩ of G, define σ(H) = card {x|H = Stab (x)}, s(H) = card {x|H ⊂ Stab (x)} = card (Fix (g)).

Our goal is to determine the function σ, since then, working out the action of the finite group G, via inner automorphisms, on the set of its cyclic subgroups, we shall recover branching indices and a picture of the covering (of course, once we will know σ(H), to identify explicitly the corresponding σ(H) ramification points, it will in general be necessary to look at liftings H → Π(i) of the inclusion H ⊂ G). To this purpose, we shall first determine the function s. We already noticed in fact that we have an injective homomorphism of G to Out (π_{1}(C_{i})) = Out (Π_{g_{i}}), inducing an injective homomorphism G → Aut (H_{1}(C_{i}, IZ)). By Lefschetz’ fixed point formula we then get:

(4.17) s(⟨g⟩) = card (Fix (g)) = 2 − Tr (g^{∗}) (g^{∗} being the action of g on H^{1}(C_{i}, IZ)). We observe that

(4.18) σ(H) = s(H) if H is a maximal cyclic subgroup of G.

(4.19) Proposition. The function σ(H) is effectively computable from the function s(H).

Proof. Define the coheight of a cyclic subgroup to be k if k = max{m | there exists a proper chain of length m of cyclic subgroups H = H_{m} ⊂ H_{m−1} ⊂ H_{m−2} … ⊂ H_{0}}. Thus, the maximal cyclic subgroups are exactly those with coheight 0. We can thus proceed by induction on the coheight k of H. In fact, {x | H = Stab (x)} ⊂ {x | H ⊂ Stab (x)} = (Fix (H)), moreover, for each y in (Fix (H)) − {x | H = Stab (x)}, Stab (y) is a cyclic subgroup H′ ⊃ H, and different from H. Obviously such a subgroup H′ has coheight ≤ (k − 1). We finally observe that ∑_{{H′≠H′ \atop H′⊃H}} σ(H′) .

A closed ‘inversion’ formula to recover σ from s, involves the geometry of the directed graph whose vertices are the cyclic subgroups H of G. □

5. A topological characterization of isotrivially fibred surfaces

Let f: S → B be a constant moduli (relatively) minimal fibration. Taking the canonical Galois cover associated to the kernel of the monodromy of the associated holomorphic bundle f^{∗}: S^{∗} → B^{∗}, we obtain a minimal birational realization of S as a quotient of a product of two curves, ε: S ≅ Y = C_{1} × C_{2}/G (B = C_{1}/G). Recall (cf. 2.4) that ε is not necessarily a morphism, whence the other fibration S → C_{2}/G can be only a rational map, when C′_{2} = C_{2}/G has genus 0. This motivates the asymmetry of the following

(5.1) Definition. Let f: S → B be a constant moduli minimal fibration. Then the good locus U of S is the complement of the inverse image under the rational map ε : S → Y = C_{1} × C_{2}/G of the singular locus of Y. Note that the good locus U maps isomorphically to an open set Y^{o} in Y and that, in view of 2.4, when ε is not a morphism, the full fibres corresponding to the points where ε is not defined are being removed, both in S and in Y.

The good locus Y^{o} coincides, when ε is a morphism, with the locus where C_{1} ×C_{2} → Y = C_{1} × C_{2}/G is unramified, so we get an unramified covering of Y^{o} given by (C_{1} × C_{2}) − Σ, Σ being the finite set of points which have a non trivial stabilizer. When ε is not a morphism, we are also removing from C_{1} × C_{2} a finite (G-invariant) union of fibres.

Thus Y^{o} has an unramified covering of finite degree which has the fundamental group of a product of projective curves (case: ε a morphism) or the product of a projective curve with a quasi- projective one. In order to generalize to constant moduli fibrations the topological characterization of surfaces isogenous to a product, the most natural way is to recall that for a smooth complete surface S the Euler number e(S) = c_{2}(Ω_{S}^{1}). Then, for a quasi projective manifold U = X − D (X is smooth, proj., and D is a normal crossings divisor), the natural object to consider is Ω_{X}^{1}(logD) in the place of Ω_{X}^{1}; for the top Chern class c_{n}(Ω_{X}^{1}(log D)), which we identify with the number it gives by integration on X, holds:

(5.2) c_{n}(Ω_{X}^{1}(log D)) = e_{BM}(U) = e(X) − e(D)

where e_{BM}(U) is the Borel-Moore homology Euler number (cf. [B-M], [B-H]). If we have an unramified covering Z → U of degree d, then (5.3) e_{BM}(Z) = d e_{BM}(U).

Since removing points makes e_{BM}(U) become smaller, going back to the proof of 3.4 we see immediately that the results are bound to be a little weaker in the quasiprojective case. We have two such generalizations of 3.4.

(5.4) Theorem. Let U = S−D be a quasi projective surface, and assume that U is proper homotopically equivalent to the good locus Y^{o} of a constant moduli fibration. Then S carries a constant moduli fibration with the same invariants as the relatively minimal fibration associated to the projection Y → C_{1}/G. Moreover, all such surfaces S form an irreducible subvariety of the moduli space.

Proof. We proceed as in theorem 3.4, replacing S by U, using Arapura and Bauer’s logarithmic versions 2.11, 2.12 of the fibration theorems. We thus get, for a suitable Galois unramified cover U′ of U, a surjective holomorphic map f to a product of logarithmic curves C_{1}^{∗} × C_{2}^{∗}. We get a holomorphic extension f: S′ → C_{1} × C_{2}, by choosing S′ to be a suitable completion of U′. There remains to show that f has degree 1. This amounts to f^{∗}: H^{4}(C_{1} ×C_{2}, IZ) → H^{4}(S′, IZ) being an isomorphism. But, on the other hand, H^{4}(C_{1} × C_{2}, IZ) = Λ^{4}H^{1}(C_{1} × C_{2}, IZ), so we only need to show that there

are 4 elements η_{1}, η_{2}, η_{3}, η_{4} in f^{∗}(H^{1}(C_{1} × C_{2}, IZ)) such that ∫_{S′}(η_{1} ∧ η_{2} ∧ η_{3} ∧ η_{4}) = 1

(integration here stands for evaluation on the fundamental class). We want to transform this calculation into a calculation on U′, and if the hypothesis would be of a diffeomorphism between Y^{o} and U, then we could represent the η_{i}’s by differential forms and use that ∫_{S′} = ∫_{U′}.

We can accomplish the same purpose by using the fundamental class of U′ in Borel-Moore homology H_{4}^{BM}(U′, IZ), and its natural pairing with cohomology with compact supports. In our situation we have natural isomorphisms H_{4}^{BM}(U′, IZ) ≅ H_{4}(S′, IZ), H^{4}(S′, IZ) ≅ H_{c}^{4}(U′, IZ), compatible with the 2 natural pairings into IZ. We have natural maps H_{c}^{i}(U′, IZ) → H^{i}(S′, IZ) → H^{i}(U′, IZ) and we observe that the classes η_{j} in H^{i}(U′, IZ) come from classes in H_{c}^{i}(U′, IZ) which we shall denote by η′_{j}. More precisely, the exact sequence for Borel-Moore homology (which coincides for compact spaces with singular homology) yields

H_{i}(S′ − U′, IZ) → H_{i}(S′, IZ) → H_{i}^{BM}(U′, IZ) → H_{i−1}(S′ − U′, IZ),

thus H_{i}(S′, IZ) ≅ H_{i}^{BM}(U′, IZ) for i = 3, 4, and by duality we thus get H_{c}^{i}(U′, IZ) ≅ H^{i}(S′, IZ) for i = 0, 1. Thus

∫_{S′} (η_{1} ∧ η_{2} ∧ η_{3} ∧ η_{4}) = [U′], (η′_{1} ∧ η′_{2} ∧ η′_{3} ∧ η′_{4}).

The last integer can now be calculated on the corresponding unramified cover Y^{o′} of Y^{o}, using that the groups H_{i}^{BM}(U′, IZ) are invariant by proper homotopy equivalences. Reversing the chain of arguments, since Y^{o′} is a Zariski open set in a product of two curves, we get that

[U′], (η′_{1} ∧ η′_{2} ∧ η′_{3} ∧ η′_{4}) = 1,

as we wanted. The last assertion concerning the irreducibility of the family of such surfaces follows with the same proof as in 4.11, 4.12. □

(5.5) Remark. From the proof of 5.4 we see immediately that an entirely analogous result holds for quasiprojective varieties of higher dimension. Moreover , one can clearly weaken the assumption on U to :

we have an isomorphism of fundamental groups π_{1}(U) ≅ π_{1}(Y^{0}) and an siomorphism of cohomology algebras with compact supports

H_{c}^{⋆}(U′, IZ) ≅H_{c}^{⋆}(Y^{0}′, IZ) .

We use now the logarithmic version (thm. 2.14) of the Zeuthen-Segre inequality to obtain a sharper result in the 2-dimensional case. But to this purpose we must recall some notion.

(5.6) Remarks.

i) A manifold U is said to have only one end if there exists an increasing sequence (K_{n}) of compacts whose union is U, and such that U − K_{n} is connected. In the particular case of a quasi projective surface U = S − D, U has only one end iff D is connected.

ii) Recall that π_{1}^{∞}(U), for a manifold with one end, is the limit of the funda-

mental groups π_{1}(U − K_{n}) where K_{n} is a sequence of compacts as in i). iii) If U = S − D, then π_{1}^{∞}(U) is the fundamental group of the boundary Σ of a

good tubular neighbourhood T of D (or, equivalently, of T − D). iv) If Y has several ends (in our case, they correspond to the connected compo-

nents of D), we can define π_{1}^{∞}(U) as the disjoint union of the corresponding groups.

(5.7) Theorem. Let U = S − D be a quasi projective surface. Assume that π_{1}(U) ≅ π_{1}(Y^{0}), where Y^{0} is the good locus of a relatively minimal constant moduli fibration and that π_{1}^{∞}(U) ≅ π_{1}^{∞}(Y^{0}), compatibly with the natural homomorphisms π_{1}^{∞} → π_{1}. Assume further thate(U) = e(Y^{0}) ( by Poincare’ duality this is equivalent to e_{BM}(U) = e_{BM}(Y^{0}). Then S also carries a constant moduli fibration which, in case S is minimal,is a deformation of the given one .

Proof. Let Γ = Γ′ × Γ′′ be the subgroup of π_{1}(Y^{0}) corresponding to the unramified covering birational to a product, and where Γ′ is the factor which corresponds to the fibration. Observe that the pull back of Γ within the groups of π_{1}^{∞}(Y) determines π_{1}^{∞}(U′) for the corresponding covering U′ of U (by assumption, the number of ends and their fundamental groups at infinity are the same for both surfaces). We have that each group in π_{1}^{∞}(U′) is trivial if Γ′ is not a free group. And that, in case Γ′ is free, there are as many nontrivial ones as there are ends of the target quasiprojective curve, each of them mapping onto an infinite cyclic subgroup of Γ′. By theorem 2.12, the unramified covering U′ associated to Γ admits a mapping to a product of two curves B_{1} × B_{2}, where B_{2} is compact, and B_{1} = C_{1} − Z, where Z has cardinality k = number of the nontrivial groups in π_{1}^{∞}(U′)’s. By our assumption, every connected component D′′ of S′ − U′ maps to a point in B_{1}. Moreover,either D” is a full fibre and then we have a corresponding non trivial group in π_{1}^{∞}(U′) , or D” is not a full fibre and , by the well known Zariski’s lemma, the intersection matrix for the irreducible components of D” is negative definite . We can then apply Mumford’s Main Result of [Mum], and conclude that the trivial groups in π_{1}^{∞}(U′) correspond to the case where D′′ is a point ( here, S′ is a minimal completion , whence D” need not be a normal crossings divisor ). By adding these, we obtain a very good fibration. In fact,the remaining components D′′ are contained in a fibre over Z, and we claim that they must be a full fibre . In fact, these fibres are missing in U′: since their number is the same(= k) as the number of non trivial groups in π_{1}^{∞}(U′), the only missing curves are full fibres. We can thus conclude as in the proof of 3.4, in view of thm. 2.14. □

(5.8) Remark. In fact, one sees from the above proof that a weaker assumption is really sufficient on π_{1}^{∞}(U). Namely, that Γ is such to determine on U′ the right number a + k of ends, k whose corresponding group in π_{1}^{∞}(U′) maps onto a cyclic nontrivial subgroup of Γ′, and a yielding a trivial group of π_{1}^{∞}(U′).

6. The fundamental group and moduli spaces

We have seen in section 4 that surfaces isogenous to a product yield an irreducible moduli space, once one fixes e.g. the Euler number and the fundamental group. On the other hand, there are several examples (cf. [Cat2], [Man1]) of surfaces of general type which are simply connected and for which the moduli space, even if we fix the oriented topological type (and also the canonical class, see [Man2]) still has a lot of connected components. The idea is to marry together these two types of moduli spaces by using a small variation of the construction of lemma 1.1 (the construction of 1.1, when applied to irregular surfaces, gives rise to deformations which are not complete intersections any longer, but just zero loci of a section of a rank 2 bundle). Unfortunately, we are not yet able to say whether we get in this way different connected components of moduli spaces. Nor can we in this way, for each group Γ which is a fundamental group of an algebraic surface, produce everywhere non reduced moduli spaces of surfaces having Γ as fundamental group (using e.g. the examples of [Cat7]). As in section 1, we shall take S ⊂ X × Y = Z as a smooth complete intersection of two hypersurfaces where

(6.1) i) X, Y are smooth complete surfaces of general type with ample canonical bundle; (6.1) ii) S is a complete intersection of two divisors in Z of type (A + B) and (2A′ + 2B′), where we let A, A′ vary among the divisors algebraically equivalent to rK_{X}, with r ≥ 2, and likewise we let B, B′ vary among the divisors algebraically equivalent to rK_{Y}.

(6.2) Therefore (cf. [Bo]), all the cohomology groups H^{i}(X, mA) = 0 except for i = 0, m ≥ 0, i = 2, m ≤ 0, and likewise for the cohomology groups H^{i}(X, mA′), H^{i}(X, mB), H^{i}(Y, mB′). We have the following exact sequence on S:

(6.3) 0 → T_{S} → T_{X} ⊕ T_{Y} |_{S} → O_{S}(A + B) ⊕ O_{S}(2A′ + 2B′) → 0,

where we identify T_{X}, resp. T_{Y}, to their pull backs to Z. Since X has ample canonical bundle we can take r ≥ 12 (cf. [Cor], lemma 4.1, page 120) so that

(6.4) H^{i}(X, T_{X}(−mA)) = 0 for i ≤ 1, m > 0,

and similarly with Y in the place of X, B or B′ instead of A. (Note that the above vanishing is indeed false for large r if X contains (-2)-curves). It follows then that

(6.5) H^{i}(Z, T_{X} ⊕ T_{Y} (−mA − nA′ − mB − nB′)) = 0 for i < 4, m + n > 0,

and similarly H^{i}(Z, O_{Z}(−mA − nA′ − mB − nB′)) = 0. We finally have the Koszul sequence

(6.6) 0 → O_{Z}(−A − 2A′ − B − 2B′) → O_{Z}(−A − B) ⊕ O_{Z}(−2A′ − 2B′) → I_{S} → 0

and the standard sequence

(6.7) 0 → I_{S} → O_{Z} → O_{S} → 0,

both of which we will tensor by vector bundles on Z.

(6.8) Lemma. H^{i}(Z, T_{X} ⊕ T_{Y}) ≅ H^{i}(Z, T_{X} ⊕ T_{Y} |_{S}) for i = 0, 1. Moreover H^{1}(Z, T_{X} ⊕ T_{Y}) ≅ H^{1}(X, T_{X}) ⊕ H^{1}(Y, T_{Y}) (for i = 0 we get 0).

Proof. Tensor (6.6) and (6.7) by T_{X} ⊕ T_{Y}, use (6.5). The rest is straightforward. □

We obtain therefore an exact sequence

(6.9) \begin{matrix}H^{0}(O_{S}(A + B)) ⊕ H^{0}(O_{S}(2A′ + 2B′)) & → & H^{1}(S, T_{S}) → \\ & → & H^{1}(X, T_{X}) ⊕ H^{1}(Y, T_{Y}). \end{matrix}

Moreover, from (6.2), (6.6), (6.7) we also have that there is an exact sequence

(6.10) \begin{matrix}H^{0}(O_{Z}(A + B)) ⊕ H^{0}(O_{Z}(2A′ + 2B′)) → \\ H^{0}(O_{S}(A + B)) ⊕ H^{0}(O_{S}(2A′ + 2B′) → H^{1}(O_{Z}) ⊕ H^{1}(O_{Z}) = \\ [H^{1}(O_{X}) ⊕ H^{1}(O_{Y})] ⊕ [H^{1}(O_{X}) ⊕ H^{1}(O_{Y})] → 0. \end{matrix}

The last two sequences tell us about the surjectivity of the Kodaira Spencer map

(6.11) Proposition. Assume that X, Y belong to respective irreducible families X , Y, with surjective Kodaira Spencer maps. Take r so large that (6.4) holds for each X in X, Y in Y (e.g. r ≥ 12). Then the family of smooth surfaces S obtained as above as complete intersection of two divisors in X, Y = Z of type (A+B) and (2A′+2B′) (with A, A′ any divisors alg. equiv. to rK_{X}, B, B′ any divisors alg. equiv. to rK_{Y}) is irreducible with surjective Kodaira Spencer map.

Proof. Since r ≥ 2, the family of effective divisors A algebraically equivalent to rK_{X} forms a projective bundle over the relative Picard scheme of X, Pic^{0}(X, and similarly for 2A′, B, 2B′. Since e.g. H^{0}(Z, O_{Z}(A+B)) = H^{0}(X, O_{X}(A))⊗H^{0}(Y, O_{Y} (B)), we conclude likewise that the family of effective divisors on Z algebraically equivalent to A + B form a projective bundle over Pic^{0}(K) × Pic^{0}(Y). Whence, by taking a fibre product we obtain an irreducible family whose Kodaira Spencer map is clearly surjective at each point. □

7. Results in the higher dimensional case

As already remarked in the introduction, Mok ([Mok1,2]) extended to the case of a compact quotient H^{n}/Γ of a polydisc (H being the unit disc ≅ Poincaré’s upper half plane) the result of Jost and Yau (including strong rigidity for the case where Γ is irreducible). We can partially further generalize the result of Jost and Yau (cf. thm. 3.4) to higher dimensions:

(7.1) Theorem. Let X be a compact Kähler manifold of dimension n, and assume that

i) Γ = π_{1}(X) admits a finite index subgroup Γ′ isomorphic to Π_{g_{1}} × Π_{g_{2}} × … × Π_{g_{n}}, with g_{1}, … g_{n} > 1, and that moreover

ii) H^{2n}(X, IZ) ≅ IZ has index d = index (Γ′) inside the image of H^{2n}(Γ′, IZ) in

H^{2n}(X′, IZ) (X′ being the covering associated to Γ′) Then X is a blow up of a variety isogenous to a higher product.

Proof. The beginning of the proof runs exactly as in 3.4. That is, we have a degree d unramified covering π: X′ → X, and therefore H^{2n}(X, IZ) ≅ IZ has index d in H^{2n}(X′, IZ) ≅ IZ. Now, X′ admits a holomorphic map f to a product W = C_{g_{1}} × C_{g_{2}} × … × C_{g_{n}} of curves of respective genera g_{1}, … g_{n} > 1. By our assumption f^{∗}H^{2n}(W, IZ) ≠ 0, and f is surjective. Moreover, H^{2n}(Γ′, IZ) ≅ f^{∗}H^{2n}(W, IZ) ⊂ H^{2n}(X′, IZ) ≅ IZ is therefore a chain of equalities, and therefore deg(f) = 1, so f is a birational morphism. If f is not an isomorphism, then there exists a f-exceptional curve C′ which has negative intersection with K_{X′}, whence also π^{−1}(π(C)) is f-exceptional, since K_{X′} = π^{∗}(K_{X}). We conclude, by induction on the number of elementary contractions, that the f-exceptional lows os π-invariant, whence also X is a blow up of a manifold Y having C_{g_{2}} × … × C_{g_{n}} as an unramified covering. □

(7.2) Example. In the previous theorem one cannot replace condition ii) by the analogous condition of condition e) of 3.4:

e(X) = 2^{n}(g_{1} − 1)(g_{2} − 1) … (g_{n} − 1)/d .

In fact, let W = C_{g_{1}} × C_{g_{2}} × C_{g_{3}} and let T be a double cover branched on a smooth ample surface B ⊂ W. Argueing e.g. as in [Cat1], section 1, we see that π_{1}(T) ≅ π_{1}(W). Whereas, again a Borel Moore homology calculation shows

(7.3) e(T) = 2e(W) − e(B) < 0

(indeed both terms are negative, the second term by Castelnuovo’s theorem (cf.[Bo]) since B is of general type). We obtain X by blowing up 1/2(e(W) − e(B)) points; thus, π_{1}(X) ≅ π_{1}(W) and e(X) = e(W), but the classifying map f: X → W has degree 2.

The above example shows that some minimality assumption is needed in dimension 3, when one can increase the Euler number by blowing up points, and decrease it by blowing up curves of genus ≥ 2.

(7.4) Remark. It is rather clear that condition ii) is also crucial in thm. 7.1 in order to have the surjectivity of f, which, in dimension n = 2, follows automatically from the fact that the pencils are distinct. In higher dimensions, by Lefschetz, we can always take a surface S with given fundamental group Γ and cross it with any simply connected manifold: this shows that condition ii) can be weakened, but not completely removed, e.g. we have also

(7.5) Theorem. Let X be a compact complex manifold of dimension n with ample canonical bundle and assume that

i) Γ = π_{1}(X) admits a finite index subgroup Γ′ isomorphic to

Π_{g_{1}} × Π_{g_{2}} × … × Π_{g_{1}},

with g_{1}, … g_{n} > 1,

i) the image of H^{2n}(Γ′, IZ) in H^{2n}(X′, IZ) is non zero.

k) K_{X}^{n} = n! 2^{n}(g_{1} − 1)(g_{2} − 1) … (g_{n} − 1)/d. Then X is isogenous to a higher product, and the moduli space of such varieties is irreducible (weak rigidity).

Proof. As in 7.1 we get f: X′ → W, where W is a product of curves, and we need to show that f is an isomorphis. W.l.o.g. let us assume X = X′ (d = 1).

Then K_{X}^{n} = K_{W}^{n} by our assumption. On the other hand, if R is the ramification divisor, then K_{X} = f^{∗}K_{W} + R. By a usual trick we calculate

K_{X}^{n} = (f^{∗}K_{W} + R)^{n} = K_{X}^{n−1} · (f^{∗}K_{W} + R) ≥ K_{X}^{n−1} · f^{∗}K_{W} ,

equality holding, since K_{X} is ample, if and only if dim R < n − 1. Then K_{X}^{n−1} · f^{∗}K_{W} = K_{X}^{n−2} · (f^{∗}K_{W} + R) · f^{∗}K_{W} ≥ K_{X}^{n−2} · (f^{∗}K_{W}^{2} ), since (K_{X}^{n−2} · R)f^{∗}K_{W} by the projection formula equals f_{∗}(K_{X}^{n−2} · R) · K_{W} and K_{W} is ample. Continuing in this way, K_{X}^{n} ≥ (f^{∗}K_{W})^{n} = deg(f)K_{W}^{n}. Therefore deg(f) = 1, and dim(R) < n − 1. Therefore, by normality, f is an isomorphism.

(7.6) Remark. It is rather clear that the classification of surfaces isogenous to a product which we carried through in section 3 carries over without any major alteration for higher dimensional varieties isogenous to a product (except of course the more complicated algebraic treatment of the varieties of mixed type). We remark now that the same ideas lead to a minor improvement of the results of Mok ([Mok1,2]).

(7.7) Theorem. Let X be a compact manifold of dimension n with ample canonical bundle, and assume that

i’) π_{1}(X) ≅ Γ, where Γ is a cocompact torsion free subgroup of Aut (H^{n}) (Γ ≅

π_{1}(W′), where H^{n}/Γ = W′ is a compact manifold), ii’) H^{2n}(Γ, IZ) → H^{2n}(X, IZ) is an isomorphism, or ii”) H^{2n}(Γ, IZ) → H^{2n}(X, IZ) is non zero and K_{X}^{n} = K_{W′}^{n}.

Then also X is biholomorphic to a quotient W = H^{n}/Γ.

Idea of Proof. By topology, since W′ is a K(π, 1), we get a mapping f′′: X → H^{n}/Γ = W′ which can be deformed to a harmonic map by the theorem of Eells and Sampson ([E-L]). Conditions ii) or ii”) ensure that f is surjective. That the universal covering X^{∧} of X has a holomorphic mapping f^{∧} to H^{n} which is Γ-equivariant was proven by Mok in [Mok2]. Then the final argument goes exactly as in 7.1. or in 7.5. □

(7.8) Remark. In the 2-dimensional case, the hypothesis K_{X}^{2} = K_{W′}^{2} can be replaced by χ(O_{X}) = χ(O_{W′}) or by e(X) = e(W′), without then assuming X to be minimal. In fact, assume in 7.5, 7.7, χ(O_{X}) = χ(O_{W′}). Since K_{W′}^{2} = 8 χ(O_{W′}) = 8χ(O_{X}), if f is not birational then K_{X}^{2} ≥ 2K_{W}^{2} = 16χ(O_{X}), contradicting the Miyaoka-Yau inequality ([Ya1,2], [Miy1,2]). Similarly if e(X) = e(W′), and S is the minimal model of X, then K_{S}^{2} ≥ 2K_{W}^{2} = 4e(X) ≥ 4e(S) again contradicts the same inequality. Then f is birational and an isomorphism since e(X) = e(W). We conjecture that the hypothesis on χ(O_{X}) (and of minimality) should be also sufficient in higher dimension (cf. the work of Green-Lazarsfeld [G-L1,2] and Ein-Lazarsfeld [E-L]) and one can ask whether in 7.5 the hypothesis e(X) = 2n(g_{1} − 1)(g_{2} − 1) ⋯ (g_{n} − 1)/d would be sufficient, assuming the ampleness of K_{X}.

Cf. also recent work of Kollar ([Kol1,2]) where instead more emphasis is set on the role of the plurigenera of X.

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AUTHORS’ ADDRESS F. Catanese, Mathematisches Institut der Georg-August Universität Göttingen, Bunsenstrasse 3-5 D-37073 Göttingen, Germany.

e-mail: catanese@uni-math.gwdg.de