---

On the fundamental group of an abelian cover ∗†

arXiv:alg-geom/9305011v1 20 May 1993On the fundamental group of an abelian cover ∗†Rita Pardini‡, Francesca TovenaR´esum´e. Soient X et Y deux vari´et´es complexes projectives et lisses de dimension n ≥2 etsoit f : Y →X un revˆetement abelien totalement ramifi´e.

Alors l’application f∗: π1(Y ) →π1(X)est surjective et donne une extension centrale:0 →K →π1(Y ) →π1(X) →1(1)o`u K est un groupe fini.Nous montrons comment le noyau K et la classe de cohomologie c(f) ∈H2(π1(X), K) de(1) peuvent ˆetre calcul´es en termes de classes de Chern des composantes du diviseur critique def et des sous-faisceaux inversibles de f∗OY stables sous l’action du groupe de Galois.Abstract. Let X, Y be smooth complex projective varieties of dimension n ≥2 and letf : Y →X be a totally ramified abelian cover.

Then the map f∗: π1(Y ) →π1(X) is surjectiveand gives rise to a central extension:0 →K →π1(Y ) →π1(X) →1(2)where K is a finite group.Here we show how the kernel K and the cohomology class c(f) ∈H2(π1(X), K) of (2) canbe computed in terms of the Chern classes of the components of the branch divisor of f and ofthe eigensheaves of f∗OY under the action of the Galois group.1Introduction.This work generalizes a result of Catanese and the second author, who analyze in [CT] thefundamental group of a special type of covering f : Y →X, with Galois group (Z/mZ)2, of acomplex smooth projective surface X, the so-called ”m-th root extraction” of a divisor D on X.By means of standard topological methods, the fundamental group π1(Y ) can be describedin that case as a central extension of the group π1(X), as follows:0 −→Z/rZ −→π1(Y ) −→π1(X) −→1,(3)r being a divisor of m which depends only on the divisibility of π∗(D) in H2( ˜X, Z), whereπ : ˜X →X is the universal covering of X.The main result of [CT] (see Thm.2.16) is that the group cohomology class correspondingto the extension (3) can be explicitly computed in terms of the first Chern class of D.This is an instance of a more general philosophy: in principle, it should be possible to recoverall the information about an abelian cover f : Y →X from the ”building data” of the cover,∗Both authors were supported by the SCIENCE Program (Contract n. SCI-0398-C(A)).†A.M.S. classification 14E20‡A member of G.N.S.A.G.A.

of C.N.R.1

i.e., from the Galois group G, the components of the branch locus, the inertia subgroups andthe eigensheaves of f∗OY under the natural action of G (see section 2 or [Pa] for more details).Actually, the description of the general abelian cover given in [Pa] enables us to treat (undersome mild assumptions on the components D1, . .

. Dk of the branch locus) the case of any totallyramified abelian covering f : Y →X, with X a complex projective variety of dimension at least2 (cf.

section 2 for the definition of a totally ramified abelian cover).Using the same methods as in [CT], we show that π1(Y ) is a central extension as before:0 −→K −→π1(Y ) −→π1(X) −→1 ,(4)where K is a finite abelian group which is determined by the building data of the cover and thecohomology classes of π∗(D1), . .

. π∗(Dk) (cf.

Prop.3.2).Consistently with the above ”philosophy”, a statement analogous to Thm.2.16 of [CT] actu-ally holds in the general case: our main result (Thm.4.4, Rem.4.5) can be summarized by sayingthat the class of the extension (4) can be recovered from the Chern classes of the Dj’s and of theeigensheaves of f∗OY ; in some special case, this relation can be described in a particularly simpleway (Thm.4.6, Cor.4.7, Rem.4.8). Moreover, one can construct examples of not homeomorphicvarieties realized as covers of a projective variety X with the same Galois group, branch locusand inertia subgroups (cf.

Rem.4.5).The idea of the proof is to exploit a natural representation of π1(Y ) on a vector bundle on theuniversal covering ˜X of X and the spectral sequence describing the cohomology of a quotient, inorder to relate the group cohomology class of the extension (4) to the geometry of the covering.These are basically the same ingredients as in the proof of [CT], but we think that we havereached here a more conceptual and clearer understanding of the argument.Acknowledgements: we wish to express our heartfelt thanks to Fabrizio Catanese, who sug-gested that the result of [CT] was susceptible of generalization and encouraged us to investigatethis problem.2A brief review of abelian covers.In this section we set the notation and, for the reader’s convenience, we collect here thedefinitions and the notions concerning abelian covers that will be needed later.For furtherdetails and proofs, we refer to [Pa], sections 1 and 2.Let X, Y be complex algebraic varieties of dimension at least 2, smooth and projective, andlet f : Y →X be a finite abelian cover, i.e. a Galois cover with finite abelian Galois group G.The bundle f∗(OY ) splits as a sum of one dimensional eigensheaves under the action of G,so that one has:f∗(OY ) =Mχ∈G∗L−1χ= OX ⊕(Mχ∈G∗\{1}L−1χ )(5)where G∗denotes the group of characters of G and G acts on L−1χvia the character χ.We warn the reader that the notation here and in the next section is dual to the one adoptedin [Pa]; however this does not affect the formulas quoted from there.Under our assumptions, the ramification locus of f is a divisor.Let D1, .

. .

Dk be theirreducible components of the branch locus D and let Rj = f −1(Dj), j = 1, . .

. k. For everyj = 1, .

. .

k, one defines the inertia subgroup Gj = {g ∈G|g(y) = y for each y ∈Rj}. Given anypoint y0 ∈Rj, one obtains a natural representation of Gj on the normal space to Rj at y0 bytaking differentials.

The corresponding character, that we denote by ψj, is independent of the2

choice of the point y0 ∈Rj. By standard results, the subgroup Gj is cyclic and the character ψjgenerates the group G∗j of the characters of Gj.

We denote by mj the order of Gj, by m the leastcommon multiple of the mj’s and by gj the generator of Gj such that ψj(gj) = exp(2π√−1mj).In what follows we will always assume that the cover f : Y →X is totally ramified, i.e., thatthe subgroups Gj generate G; then the group of characters G∗injects in Lkj=1 G∗j and everyχ ∈G∗may be written uniquely as:χ =kXj=1aχ,j ψj,0 ≤aχ,j < mj for every j . (6)In particular, let χ1, .

. .

χn ∈G∗be such that G∗is the direct sum of the cyclic subgroupsgenerated by the χ’s, and let di be the order of χi, i = 1, . .

. n. Write:χi =kXj=1aij ψj,0 ≤aij < mj ,i = 1, .

. .

n . (7)Then one has ([Pa], Prop.2.1):diLχi ≡kXj=1diaijmjDji = 1, .

. .

n(8)the corresponding isomorphism of line bundles being induced by multiplication in the OX-algebraf∗OY . More generally, if χ = Pni=1 bχ,iχi, with 0 ≤bχ,i < di ∀i, one has ([Pa], Prop.2.1):Lχ ≡nXi=1bχ,iLχi −kXj=1qχj Dj .

(9)where qχj is the integral part of the rational number Pni=1bχ,iaijmj , j = 1, . .

. k.Equations (8) are the characteristic relations of an abelian cover.Actually, since X iscomplete, for assigned G, Dj, Gj, ψj, j = 1, .

. .

k, to each set of line bundles Lχi, i = 1, . .

. n,satisfying (8) there corresponds a unique, up to isomorphism, G-cover of X, branched on theDj’s and such that Gj is the inertia subgroup of Dj and ψj is the corresponding character ([Pa],Thm.2.1).

Moreover, the cover is actually smooth under suitable assumptions on the buildingdata.3The fundamental group and the universal covering of Y .We keep the notation introduced in the previous section.Definition 3.1 ([MM], pag.218) A smooth divisor ∆on a variety X is called flexible if thereexists a smooth divisor ∆′ ≡∆such that ∆′ ∩∆̸= ∅and ∆and ∆′ meet transversely.We recall that a flexible divisor on a projective surface is connected (see [Ca], Remark 1.5).Hence, by considering a general linear section, one deduces easily that a flexible divisor on aprojective variety of dimension ≥2 is connected.Proposition 3.2 Let X, Y be smooth projective varieties over C of dimension n at least 2. Letf : Y →X be a totally ramified abelian cover branched on irreducible, flexible and ample divisors{Dj}j=1,...k. Then:3

a) The natural map f∗: π1(Y ) →π1(X) is surjective.b) Let K = ker(f∗); then K is finite and0 −→K −→π1(Y ) −→π1(X) −→1. (10)is a central group extension.c) Let π : ˜X →X be the universal covering of X and ˜D = π−1(D); then ˜Dj = π−1(Dj)is connected, j = 1, .

. .

k. Denote by Hic the cohomology with compact supports and byρ : H2n−2c( ˜X) →H2n−2c( ˜D) ∼=Lkj=1 Z ˜Dj the restriction map. Finally, let σ be the mapdefined by:σ :H2n−2c( ˜D) ∼=Lkj=1 Z ˜Dj→Lkj=1 Gj˜Dj7→gj.

(11)Then N = ker(L Gj →G) contains Im(σ ◦ρ) and K is isomorphic to the quotient groupN/Im(σ ◦ρ).Proof. a) and the fact that the extension (10) is central can be proven exactly as in [Ca],Thm.1.6 and in [CT], Lemma 2.1.For the proof of c) (that implies that K is finite), we refer the reader to [Ca], Prop.1.8 andto [CT], proof of Thm.2.16, Step I.

One only has to notice that, by Lefschetz theorem (cf. [Bo],Cor.

of Thm.1), π1(Dj) surjects onto π1(X), hence ˜Dj = π−1(Dj) is connected and smooth forevery j = 1, . .

. k.Remark 3.3a) From Prop.3.2, c), it follows in particular that the kernel K of the surjec-tion f∗: π1(Y ) →π1(X) does not depend on the choice of the solution Lχ of (8), once G,the gj’s and the class of the ˜Dj’s in H2( ˜X, Z/mjZ), j = 1, .

. .

k, are fixed.b) If f : Y →X is an abelian cover as in the hypotheses of Prop.3.2, then H1(Y, OY ) ∼=H1(X, OX) by (5) and the Kodaira Vanishing Theorem. Moreover, according to Prop.3.2,a) the map f∗: H1(Y, Z) →H1(X, Z) is surjective; thus the map f∗: alb(Y ) →alb(X)between the Albanese varieties is an isomorphism.Proposition 3.4 In the same hypotheses as in Prop.3.2, let q : ˜Y →Y be the universal coverof Y and let ˜f : ˜Y →˜X be the map lifting f : Y →X.

Then ˜f is a totally ramified abeliancover of ˜X with group ˜G = (Lkj=1 Gj)/Im(σ ◦ρ), branched on ˜D.Proof. By diagram chasing, it is easy to show that π1( ˜X \ ˜D) is isomorphic to the kernelV of the surjection π1(X \ D) →π1(X) induced by the inclusion X \ D ⊂X.

Since the Di’sare flexible, one proves as in ([CT], Lemma 2.1) that V is an abelian group. It follows that ˜f ,being branched on ˜D, is an abelian cover.Consider now the fiber product Y ′ of f : Y →X and π : ˜X →X, with the natural mapsf ′ : Y ′ →˜X and q′ : Y ′ →Y ; f ′ is a G-cover ramified on ˜D and q′ is unramified.

According4

to Prop.3.2, b), the universal covering q : ˜Y →Y of Y factors as q = q′ ◦q′′, for a suitableunramified cover q′′ : ˜Y →Y ′ with group K, giving a commutative diagram as follows:✲❅❅❘❄❄❄✲Y ′˜X˜YYX˜ff′q′′fq′π(12)In particular, K ∼= π1(Y ′) and ˜f = f ′ ◦q′′.Hence, the Galois group ˜G of ˜f is given as anextension:0 −→K −→˜G −→G −→0(13)Moreover, if one denotes by ˜Gj the inertia subgroup of ˜Dj with respect to ˜f , then ˜Gj mapsisomorphically onto Gj for every j = 1, . .

. k. The isomorphism ˜G = (L Gj)/Im(σ ◦ρ) can beobtained by computing the fundamental group of ˜Y as in [CT], proof of Thm.2.16.The following lemma will be used in the next section.Lemma 3.5 Consider the subgroups π1(Y ) and ˜G of Aut( ˜Y ); then one has:βg = gβ∀g ∈˜G, ∀β ∈π1(Y ).(14)Proof.

Since the cover ˜f is totally ramified, it is enough to show that all the elements ofπ1(Y ) commute with gj, j = 1, . .

. k.Let β ∈π1(Y ) and fix j = 1, .

. .

k. We remark firstly that βgjβ−1 is actually an elementof ˜G ⊂Aut( ˜Y ). In fact, consider the classes represented by βgj and gjβ modulo K: they docoincide as automorphisms of Y ′ ⊆Y × ˜X, since the group G × π1(X) acts there via the naturalaction on the components.

So, βgjβ−1g−1j∈K and βgjβ−1 ∈gjK ⊆˜G, as desired.By diagram (12), we have ˜Rj = ˜f−1( ˜Dj) = q−1(Rj) ∀j = 1, . .

. k. Since Rj = f −1(Dj) isample and connected, the same argument as in the proof of Lemma 3.2, c) shows that ˜Rj isconnected.

So, β ˜Rj = ˜Rj and βgjβ−1 fixes ˜Rj pointwise, namely βgjβ−1 ∈˜Gj.Finally, recalling the definition of the character ψj ∈G∗j introduced in section 2, one checksimmediately that ψj(βgjβ−1) = ψj(gj). The conclusion now follows from the faithfulness of ψj.4Computing the cohomology class of the central extension 0 →K →π1(Y ) →π1(X) →1.We keep the notation and the assumptions introduced in the previous sections, unless thecontrary is explicitly stated.

We need two technical Lemmas in order to state the main resultof this paper.Lemma 4.1 Let H be a finite abelian group and ζ1,. .

. ζm ∈H be such that H = Lmj=1 < ζj >is the direct sum of the cyclic subgroups generated by ζj, j=1, .

. .

m; denote by hj the order of5

ζj. Let p ∈Z be a prime and Hp be the p-torsion subgroup of H. Let χ1, .

. .

χt ∈Hp such that< χ1, . .

. χt >= Lti=1 < χi >.

Finally, let di be the order of χi and write χi = Phj=1 aijζj with0 ≤aij < hj.Then, ∀x1, . .

. xt ∈Z and ∀γ ≥1, the system:mXj=1diaijhjsj ≡ximod pγi = 1, .

. .

t(15)admits a solution (s1, . .

. sm) ∈Zm.Proof.

We set cij = diaijhjand, for x ∈Z, we denote by x the class of x in Z/pZ. Weproceed by induction on γ.Let γ = 1.

We show that the matrix (cij) has rank t.Let y1, . .

. ym ∈Z and assume that:Xicijyi = 0∀j = 1, .

. .

m . (16)This implies that:tXi=1cijyi ≡0mod p∀j = 1, .

. .

m(17)so that:tXi=1yidipaijhj∈Z∀j = 1, . .

. m .

(18)Recalling that p divides di ∀i, we deduce that:tXi=1yidipaij ≡0mod hj∀j = 1, . .

. m ,(19)so that Piyidip χi is the zero element in H. By the hypothesis on the χi’s, it follows that:yidip≡0mod di(20)and finally:yi ≡0mod p(21)showing, as desired, that the rows of the matrix (cij) are linearly independent over Z/pZ.Let now γ > 1 and assume by inductive hypothesis that (s1, .

. .

sm) ∈Zm is a solution of thesystem (15).We set s′j = sj + δjpγ and we look for a suitable choice of the integers δj. We have:Pj cijs′j=Pj cijsj + pγ Pj cijδj=xi + pγyi + pγ Pj cijδj∃yi ∈Z,(22)so that:Xjcijs′j ≡xi mod pγ+1⇐⇒Xjcijδj ≡−yimod p(23)and the latter system has a solution, by the case γ = 1.

This conclude the proof.We come back to the study of the cover f:6

Lemma 4.2 Let A be the subgroup of Pic(X) generated by D1, . .

. Dk and Lχ, χ ∈G∗.

Thenthere exist M1, . .

. Mq ∈Pic(X) such that A = Lql=1 < Ml > andD1...Dk≡CM1...Mq(24)where C = (cjl) is a matrix with integral coefficients such that each column (cjl)j=1,...k representsan element of N = ker (Lkj=1 Gj →G).Proof.

A is a finitely generated abelian group, so one can write A = F L T, where T isthe torsion part of A and F is free.Denote by {ξl}l a set of free generators of F and by {ηl}l a set of generators of T such thatT = L < ηl > and the order o(ηl) of ηl is the power of a prime, ∀l. Let finally χi be generatorsof G∗such that G∗= Lni=1 < χi > and the order o(χi) of χi is the power of a prime, ∀i.One can write:Lχi ≡Xlλilηl +Xlλ′ilξl∀i = 1, .

. .

n,(25)Dj ≡Xlcjlηl +Xlc′jlξl∀j = 1, . .

. k,(26)where the coefficients λ′il and c′jl are uniquely determined, whereas λil and cjl are determinedonly up to a multiple of o(ηl).We can apply the analysis of section 2 to the cover f.We write χi = Pkj=1 aijψj, with0 ≤aij < mj, and we set di = o(χi) as in the previous Lemma; the equations (8) become here:diLχi ≡kXj=1diaijmjDji = 1, .

. .

n,(27)so that we must have:diλ′il =kXj=1diaijmjc′jli = 1, . .

. n,(28)showing that (c′jl)j=1,...k represents an element of N, ∀l: in fact, by duality, (t1, .

. .

, tk) ∈Zkrepresents an element of N if and only if it satisfies the relations:kXj=1aijmjtj ∈Z∀i = 1, . .

. n.(29)For the coefficients of the torsion part, we have:diλil ηl =kXj=1diaijmjcjlηl(30)so that:diλil ≡kXj=1diaijmjcjlmod o(ηl) .

(31)7

We fix an index l. Let p be a prime such that o(ηl) = pα. We want to show that, for asuitable choice of the cjl, the following relation holds ∀i = 1, .

. .

n:diλil ≡kXj=1diaijmjcjlmod di . (32)Let χi be a generator such that di ≡0 mod p and set di = pαi.

By (31), it is enough to considerthe case in which α < αi.Setting c′′jl = cjl + pαsj and recalling (31), one has:Pkj=1diaijmj c′′jl=Pkj=1diaijmj cjl + pα Pkj=1diaijmj sj=diλil −pαxi + pα Pkj=1diaijmj sj(33)for a suitable choice of integers xi. One concludes that the relation (32) holds if and only if:kXj=1diaijmjsj ≡ximod pαi−α .

(34)Let β = max {αi −α}i. The system of congruences:kXj=1diaijmjsj ≡ximod pβ∀i such that di ≡0 mod p(35)admits a solution by Lemma 4.1.

So, we can assume that the coefficients (cjl)j=1,...k in (26)satisfy (32) for every i such that di ≡0 mod p.To complete the proof, let γ be an integer ≫0; we can still modify the coefficients asc′′jl = cjl + pγtj. It is enough to notice that, setting d = lcm{di | di ̸≡0 mod p}, then d and pare coprime and the system of congruences:cjl + pγtj ≡0mod d∀j(36)admits a solution.

So we can assume that c′′jl ≡0 mod d, and the proof is complete.To any decomposition (41) as in Lemma 4.2, we associate a cohomology class in H2(X, K):Definition 4.3 Given a decomposition (41) as in Lemma 4.2, consider the map:Zq→N(x1, . .

. xq)7→Pql=1 xlcl = (Pql=1 xlcjl)j=1,...k(37)and denote by Θ : Zq →K its composition with the projection N →K (cf.

Prop.3.2, c)). Then,set:ξ = Θ∗([M1], .

. .

[Mq]) ,(38)where Θ∗: H2(X, Zq) ∼=Lq H2(X, Z) →H2(X, K) is the map induced in cohomology by Θ and[M] is the Chern class of a divisor M on X.8

We briefly recall some facts about quotients by a properly discontinuous group action (seefor instance [Mu], Appendix to section 1, [Gr], ch. 5).Let ˜X be a simply connected variety, let Γ be a group acting properly and discontinuouslyon ˜X and let p : ˜X →X = ˜X/Γ be the projection onto the quotient.

Consider the following twofunctors:MF−→MΓ, for M a Γ-moduleFH−→H0( ˜X, p∗F), for F a locally constant sheaf on X .The spectral sequence associated to the functor F ◦H yields in this case the exact sequence ofcohomology group:0 −→H2(Γ, H0( ˜X, p∗F)) −→H2(X, F) −→H2( ˜X, p∗F)Γ(39)that will be used several times in the following and it is natural with respect to the sheaf mapson X.Theorem 4.4 Let X, Y be smooth projective varieties over C of dimension at least 2. Letf : Y →X be a totally ramified finite abelian cover branched on a divisor with flexible and amplecomponents {Dj}j=1,...k. According to Prop.3.2, b), the map f induces a central extension:0 −→K −→π1(Y )f∗−→π1(X) −→1(40)Denote by c(f) ∈H2(π1(X), K) ⊆H2(X, K) the cohomology class classifying the extension(40).Let:D1...Dk≡CM1...Mq(41)be a decomposition as in Lemma 4.2 and let ξ ∈H2(X, K) be the class defined in Def.4.3.In this notation, one has:c(f) = ξ;(42)in particular, the class ξ does not depend on the chosen decomposition.Proof.

It is enough to show that ξ and c(f) admit cohomologous representatives. This canbe done in three steps.Step I: we compute a cocycle representing c(f) ∈H2(X, K).We start by choosing suitable trivializations of the line bundles that appear in the compu-tation.Set Γ = π1(X) and ˜Γ = π1(Y ).

Let {Ur} be a sufficiently fine cover of X such that Γ actstransitively on the set of connected components of π−1(Ur), ∀r. If we fix a component Vr ofπ−1(Ur), then π−1(Ur) = ∪γ∈Γ γ(Vr); for every γ ∈Γ we write:γ(Vr) = V(γ,r)(43)and, in particular: V(1,r) = Vr.Such a covering has the following properties:9

a) For every (r, s) such that Ur ∩Us ̸= ∅, there exists a unique element β(r, s) ∈Γ such that:V(1,r) ∩V(β(r,s),s) ̸= ∅. (44)b) If Ur ∩Us ̸= ∅, then V(γ,r) and V(γβ(r,s),s) have nonempty intersection.c) Since π is a local homeomorphism, if Ur ∩Us ∩Ut ̸= ∅, then:∅̸= V(β(r,s),s) ∩V(β(r,t),t).

(45)Hence the following relation is satisfied for every Ur ∩Us ∩Ut ̸= ∅:β(r, t) = β(r, s)β(s, t) . (46)In particular: β(s, r) = β(r, s)−1.For later use, we set:V(α,r,s) = α(V(1,r) ∩V(β(r,s),s)) = V(α,r) ∩V(αβ(r,s),s)(47)for every α ∈Γ and for every (r, s) such that Ur ∩Us ̸= ∅.For every r and for every j = 1, .

. .

k, we choose a local generator wjr for OX(−Dj) on Ur(we ask that wjr is a local equation for Dj) and for every pair (r, s) such that Ur ∩Us ̸= ∅wewrite:wjr = kj(r,s) wjs on Ur ∩Us . (48)Now we apply to ˜f : ˜Y →˜X the analysis of section 2 , most of which can be easily extendedto the case of analytic spaces.

One has:˜f ∗(O ˜Y ) =M˜χ∈˜G∗L−1˜χ(49)Each element of the group ˜G can be interpreted as an automorphism of the sheaf ˜f ∗(O ˜Y ).In particular, by duality, the elements of K ⊆˜G are characterized by the property that theyinduce the identity on the subsheaf Lχ for every χ ∈G∗⊆˜G∗.Let ˜χ1, . .

. ˜χh ∈˜G∗be such that ˜G∗is isomorphic to the direct sum of the cyclic subgroupsgenerated by the ˜χi’s, and let ˜di be the order of ˜χi, i = 1, .

. .

h. Let ˜Di be the inverse image ofDi via the universal covering map π : ˜X →X, as before. If ˜χi = Pkj=1 ˜aijψj, with 0 ≤˜aij < mj,the system (8) yields in this case:˜diL˜χi ≡kXj=1˜di˜aijmj˜Dji = 1, .

. .

h . (50)So it is possible to choose local generators ˜zi(α,r) for L−1˜χi on V(α,r) such that for every α ∈Γ andfor every pair (r, s) with Ur ∩Us ̸= ∅one has:˜zi(α,r) ˜di =kYj=1wjr ˜di˜aijmj.

(51)10

Writing:˜zi(α,r) = ˜hi(α,r,s) ˜zi(αβ(r,s),s)on V(α,r,s)(52)we have:(˜hi(α,r,s))˜di =kYj=1(kj(r,s))˜di˜aijmj. (53)and the cocycle condition for ˜hi(α,r,s), that will be often used later on, yields the relation:1 = ˜hi(α,r,s)˜hi(αβ(r,s),s,t)˜hi(αβ(r,t),t,r)∀α ∈Γ, ∀i, ∀r, s, t with Ur ∩Us ∩Ut ̸= ∅.

(54)We observe that the generator ˜zi(α,r) is determined by (51) only up to a constant of theform exp2π√−1 (ui(α,r)/ ˜di) with ui(α,r) ∈Z. Moreover, according to (9), every choice of localgenerators wjr for OX(−Dj) and ˜zi(α,r) for L−1˜χi induces a choice of local generators for L−1˜χ ,∀˜χ ∈˜G∗, by the rule:˜z ˜χ(α,r) =nYi=1˜zi(α,r)b˜χ,ikYj=1(wjr)−q ˜χjif ˜χ =hXi=1b˜χ,i ˜χi, 0 ≤b˜χ,i < ˜di .

(55)where q ˜χj denotes the integral part of the real number Phi=1 b˜χ,i ˜aij.Let now χ1, . .

. χn be a set of generators for G∗such that G∗is the direct sum of the cyclicsubgroups generated by the χv’s and the order dv of χv is a power of a prime number, v = 1, .

. .

n.We recall that G∗⊆˜G∗and, ∀χ ∈G∗, the corresponding eigensheaf Lχ is a pullback from X.We write χv = Pni=1 bvi ˜χi ∈˜G∗(0 ≤bvi < ˜di) and qχvj= qvj ; the corresponding local generatorfor L−1χv chosen in (55) is:zv(α,r) =hYi=1˜zi(α,r)bvikYj=1(wjr)−qvj ;(56)we show that, for a suitable choice of the ˜zi(α,r), we can assume that the expression in (56) isindependent from α. In fact, using the characteristic equations of the cover f, one can choose alocal base yvr of L−1χv on V(α,r) that does not depend on α and satisfies the relation:(yvr)dv =kYj=1wjr avjdvmj.

(57)Since Phi=1 bvi˜aij = qvj mj + avj ∀j = 1, . .

. k, the two local generators yvr and zv(α,r) on V(α,r)differ by a dv-th root of unity, that we denote by exp(2π√−1 (xv(α,r)/dv)).

If we multiply ˜zi(α,r)by exp(2π√−1(ui(α,r)/ ˜di)), then (xv(α,r)/dv) becomes (xv(α,r)/dv) + Phi=1(bvi/ ˜di) ui(α,r). Hence, weonly need to solve the linear system of congruences, ∀(α, r):hXi=1dvbvi˜diui(α,r) ≡xv(α,r)mod dvv = 1, .

. .

n;(58)since we assume that dv is a power of a prime number, this system admits a solution accordingto the Chinese Remainder’s Theorem and Lemma 4.1.11

So we can assume that the expression zv(α,r) in (56) does not depend on α and it is thepullback of a local generator of the corresponding eigensheaf on X: we write zvr = zv(α,r). Forlater use, we define the corresponding cocycle hv(r,s) (v = 1, .

. .

n) by the rule:zvr = hv(r,s)zvson Ur ∩Us(59)and we observe that, according to (57), the following relation holds:(hv(r,s))dv =kYj=1(kj(r,s))avjdvmjif χv =kXj=1aijψj, 0 ≤aij < mj. (60)In order to compute the class of the extension (40), for every γ ∈Γ we choose a lifting˜γ ∈˜Γ.By Lemma 3.5, the induced map ˜γ∗: ˜f ∗O ˜Y →˜f ∗O ˜Y is a O ˜X-algebra isomorphismlifting γ : ˜X →˜X; in terms of the chosen trivializations we may write:˜zi(α,r)˜γ∗7→σi,γ(α,r)˜zi(γα,r)∀γ, α ∈Γ, i = 1, .

. .

h(61)for a suitable choice of a ˜di-th root of unity σi,γ(α,r), i = 1, . .

. h.For later use, we write down the transition relations for the constants σi,γ(α,r).

Let s, t be suchthat Us ∩Ut ̸= ∅and let α, γ ∈Γ; then, for i = 1, . .

. h:σi,γ(α,s)˜hi(γα,s,t) = σi,γ(αβ(s,t),t)˜hi(α,s,t) ◦γ−1on V(γα,s,t).

(62)We now exploit the action of the chosen elements in ˜Γ on ˜f ∗O ˜Y in order to compute theclass c(f) ∈H2(Γ, K) associated to the extension (40), and its image c(f) ∈H2(X, K).For any given δ, γ ∈˜Γ, the action of (fδγ)−1∗˜δ∗˜γ∗on L−1˜χi , i = 1, . .

. h, is described with respectto the chosen trivializations by:˜zi(α,r) 7→σi,(δγ)(α,r)−1 σi,δ(γα,r)σi,γ(α,r)˜zi(α,r)∀r, ∀α ∈Γ, i = 1, .

. .

h.(63)Since (63) represents a line bundle automorphism given by a root of the unity, the expressiondoes not depend on (α, r) by the connectedness of ˜X: therefore, we may set α = 1. So, the classc(f) ∈H2(Γ, K) is represented by the cocycle:c(f)(δ, γ) =σi,(δγ)(1,r)−1 σi,δ(γ,r)σi,γ(1,r)i=1,...h∀r,(64)where an element of K ⊆˜G is represented by its coordinates with respect to the basis dual to{˜χ1, .

. .

˜χh}.According to ([Mu], page 23), the class c(f) ∈H2(X, K) is represented on V(1,r,s) ∩V(1,r,t)by the cocycle:c(f)r,s,t = c(f)(β(r, s), β(s, t)) =σi,β(r,t)(1,p)−1 σi,β(r,s)(β(s,t),p)σi,β(s,t)(1,p)i=1,...h∀p(65)for r, s, t such that Ur ∩Us ∩Ut ̸= ∅.12

We set p = t and, by the relation (62), we rewrite (65) as follows:c(f)r,s,t =σi,β(r,t)(1,p)−1 σi,β(t,r)(1,r)σi,β(s,t)(1,t)˜hi(β(r,s),s,t)˜hi(1,s,t) ◦β(s, r)−1i=1,...h;(66)this shows that c(f)r,s,t differs from the following cocycle (that we still denote by c(f)r,s,t byabuse of notation):c(f)r,s,t =˜hi(β(r,s),s,t)˜hi(1,s,t) ◦β(s, r)−1i=1,...h(67)by the coboundary of the cochain:gr,t =σi,β(r,t)(1,t)i=1,...h . (68)The cochain gr,t actually takes values in K: in fact, it is enough to check gr,t acts trivially onthe eigensheaves corresponding to the chosen generators χv of G∗.

This follows easily by theprevous choices since the action of β(r, t) on L−1χv is given locally by Qhi=1σi,β(r,t)(1,t)bvi.Step II: we compute a cocycle representing ξ ∈H2(X, K).For every r and for every l = 1, . .

. q, we choose a local generator ylr for OX(−Ml) on Ur; ifMl has finite order e, then we require:ylre = 1 .

(69)We set m = lcm {mj}j=1,...k. For every pair of indices (r, s) such that Ur ∩Us ̸= ∅we write:ylr = µl(r,s) yls on Ur ∩Us(70)and we choose a m-th root ˆµl(r,s) of µl(r,s) in such a way that ˆµl(s,r) = (ˆµl(r,s))−1. Then, as in[CT], (2.45), one sees that the image of the class of −Ml in H2(X, Z/mlZ) is represented onUr ∩Us ∩Ut by the cocycleˆµl(r,s)ˆµl(s,t)(ˆµl(r,t))−1m/ml, l = 1, .

. .

q. We conclude that the classξ = Θ∗([M1], .

. .

[Mq]) is represented on Ur ∩Us ∩Ut by:ξr,s,t =kYj=1qYl=1ˆµl(r,s)ˆµl(s,t)ˆµl(t,r)−(m/mj)cjl˜aiji=1,...h(71)Step III: we show that ξ = c(f).We remark that, according to (41), the cocycle kj(r,s) in (48) representing OX(−Dj) (j =1, . .

. k) and the cocycles µl(r,s) representing −Ml (l = 1, .

. .

q) are related as follows:kj(r,s) =qYl=1µl(r,s)cjl f jrf js(72)for suitable nowhere vanishing holomorphic functions fr on Ur. For every j = 1, .

. .

k and everyr, we choose a m-th root ˆf jr of f jr on Ur; then, the expression:ˆkj(r,s) =qYl=1ˆµl(r,s)cjl(m/mj)ˆfjrˆfjs(m/mj)(73)13

is a mj-th root of the cocycle kj(r,s) and, as before, the product Qql=1ˆµl(r,s)ˆµl(s,t)ˆµl(t,r)cjl(m/mj)yields a cocycle representing the image of the class of −Dj in H2(X, Z/mjZ). In this notation,by (69), we rewrite as follows the cocycle in (71) representing the class ξ:ξr,s,t =kYj=1ˆkj(r,s)ˆkj(s,t)ˆkj(t,r)−˜aiji=1,...,h.(74)Let ǫ = exp(2π√−1m).

Then, by (53), one has:˜hi(α,r,s) =kYj=1(ˆkj(r,s))˜aij ǫ−qi(α,r,s)(75)where qi(α,r,s) is an integer, multiple of m/ ˜di, and (67) may be rewritten as:c(f)r,s,t = (ǫqi(1,s,t)−qi(β(r,s),s,t))i=1,...h . (76)From the cocycle condition (54) for ˜hi(α,r,s), it follows:ξr,s,t = (ǫ−qi(α,r,s)−qi(αβ(r,s),s,t)−qi(αβ(r,t),t,r))i=1,...h∀r, s, t, ∀α ∈Γ .

(77)In particular, for α = 1, one gets:ξr,s,t = (ǫ−qi(1,r,s)−qi(β(r,s),s,t)−qi(β(r,t),t,r))i=1,...h . (78)So, one has:c(f)r,s,t = ξr,s,t(ǫqi(1,r,s)+qi(1,s,t)+qi(β(r,t),t,r))i=1,...h .

(79)By the definition of qi(α,r,s), this equality can be rewritten as follows:c(f)r,s,t = ξr,s,t(ǫqi(1,r,s)−qi(1,r,t)+qi(1,s,t))i=1,...h . (80)To complete the proof of the theorem, we show that we can choose the m-th root ˆf j of fj(j = 1, .

. .

k) so that:q(1,r,s) =ǫqi(1,r,s)i=1,...his an element of K, ∀(r, s). (81)Let χv one of the chosen generators of G∗.

According to (56), (60) and (75), the action of q(1,r,s)on L−1χv is given by a dv-th root of unity, that we denote by exp(2π√−1 xvdv ) (for a suitable integerxv). We want to show that we can assume that xv ≡0 mod dv, v = 1, .

. .

n.We observe that:exp(2π√−1 xvdv) =hYi=1ǫqi(1,r,s)bvi = (hv(1,r,s))−1kYj=1(ˆkj(r,s))avj(82)14

and we compute the right-hand side of (82). By (73), one must have:hv(1,r,s)= exp(−2π√−1 xvdv ) Qkj=1(ˆkj(r,s))avj= exp(−2π√−1 xvdv ) Qql=1(ˆµl(r,s))m Pkj=1cjlavjmjQkj=1ˆfjrˆfjsavj(m/mj)= exp(−2π√−1 xvdv ) Qql=1(ˆµl(r,s))mdvPkj=1cjldvavjmjQkj=1ˆfjrˆfjsavj(m/mj).

(83)On the other hand, as in Lemma 4.2, we write Lχv ≡Pql=1 λvlMl and we get the followingrelation form of cocycles on V(1,r,s):hv(1,r,s) =qYl=1(µl(r,s))λvl ϕvrϕvs(84)for suitable nowhere vanishing holomorphic functions ϕvr on Ur. According to Lemma 4.1 andto (69), we can then assume that in the previous equations one has:qYl=1(ˆµl(r,s))mdvPkj=1cjldvavjmj−dvλvl = 1(85)so that one gets:exp(2π√−1 xvdv) =kYj=1ˆfjrˆfjsavj(m/mj)ϕvsϕvr.

(86)We observe that we may assume that:ϕvr = exp(2π√−1 tvrdv)kYj=1(ˆfjr)avj(m/mj)(87)for suitable integers tvr; hence the equation (86) gives:xvdv−tvsdv+ tvrdv∈Z. (88)If we replace ˆfjr by exp(2π√−1 sjrm)ˆfjr, then turdv is replaced by turdv + Pkj=1avjsjmj .

Therefore, weneed to solve the system:kXj=1avjsjrmj≡tvrmod dvv = 1, . .

. n.(89)Since this is possible according to Lemma 4.1 and the Chinese Remainder’s Theorem, the proofis complete.Remark 4.5 The cohomology class of the extension (40) of the fundamental groups depends onthe choice of the solution {Lχ} of the characteristic relations (8) for the covering f. Moreover,covers corresponding to different solutions {Lχ} may not be homeomorphic.This is shown, for instance, by the following class of examples.

Denote by ei the standardgenerators of the group (Z/4Z)3 and let G be the quotient of (Z/4Z)3 by the subgroup generated15

by 2e1 + 2e2 + 2e3. Let now X be a smooth projective surface such that H2(π1(X), Z/2Z) ̸= 0and Pic(X) has a 2-torsion element η whose class in H2(X, Z/2Z) is non zero.

Fix a very ampledivisor H on X and choose suitable divisors Di (i = 1, 2, 3) such that Di ≡4H and the Di’s arein general position. Then there exists a smooth abelian G-cover f : Y →X ramified on the Di’s(i = 1, 2, 3), with inertia subgroup Gi =< ei >= Z/4Z and character ψi dual to ei, respectively.In fact, taking the characters χ1 = ψ1 + 3ψ3, χ2 = ψ2 + 3ψ3, χ3 = 2ψ3 as generators of G∗, thecharacteristic relations (8) of the cover f are:4L1≡D1 + 3D34L2≡D2 + 3D32L3≡D3(90)and admit, in particular, the solution L1 = L2 = L3 = 2H.Under these hypotheses, L3generates the subgroup < Di, Lχ > (i = 1, 2, 3, χ ∈G∗) of Pic(X) and the decompositionDi ≡2L3 has the properties requested in Prop.4.2.

According to Prop.3.2 and Thm.4.4, sincethe pull back ˜Di of Di under the universal cover ˜X of X is 2-divisible, ∀i, then the map finduces a central extension of the form:0 →Z/2Z →π1(Y ) →π1(X) →1(91)and the cohomology class of this extension in H2(π1(X), Z/2Z) ⊆H2(X, Z/2Z) is the imageΨ∗([L3]) of the Chern class of L3 under the map induced in cohomology by the standard pro-jection Ψ : Z →Z/2Z: so, this class is trivial.Let now Y be the G-cover of X corresponding to the solution Li = 2H + η (i = 1, 2, 3) of(90); in this case the cohomology class describing π1(Y ) is given by Ψ∗([L3 + η]) and, by thehypotheses made, it is not trivial.In particular, when X is a projective variety with π1(X) = Z/2Z, the previous constructionyields two non homeomorphic G-covers Y , Y of X, branched on the same divisor, with the sameinertia subgroups and characters, such that:π1(Y ) = (Z/2Z)2π1(Y ) = Z/4Z . (92)The following theorem is an attempt to determine to what extent the class c(f) depends onthe choice of the Lχ’s, once the branch divisor and the covering structure are fixed.Theorem 4.6 Same hypotheses and notation as in the statement of Thm.4.4.

Consider theclass c(f) ∈H2(π1(X), K) associated to the central extension (40) given by the fundamentalgroups and denote by i(c(f)) ∈H2(π1(X), ˜G) ⊆H2(X, ˜G) its image via the map induced incohomology by the inclusion (13) K ⊆˜G.Denote by Φ the group homomorphism defined as follows:Φ :Zk→˜G(x1, . .

. xk)→gx11 · · · gxkk(93)and by Φ∗: H2(X, Zk) →H2(X, ˜G) the map induced by Φ in cohomology.Then:i(c(f)) = Φ∗([D1], .

. .

[Dk])(94)where [∆] denotes the class of a divisor ∆on X in H2(X, Z).16

Corollary 4.7 Same hypotheses and notation as in Thm.4.4. Assume moreover that the naturalmorphism Hom(π1(X), ˜G) →Hom(π1(X), G), induced by the surjection ˜G →G, is surjective.Then the map i : H2(π1(X), K) →H2(π1(X), ˜G) is injective and the class Φ∗([D1], .

. .

[Dk]) in(94) determines uniquely the class c(f) ∈H2(π1(X), K) of the extension (40) of the fundamen-tal groups.This happens, in particular, if π1(X) is torsion free or Hom(π1(X), G) = 0 (e.g., if π1(X)is finite with order coprime to the order of G), or the sequence 0 →K →˜G →G →0 splits.Proof of Thm.4.6. We keep the notation and the results in Step I of the proof of Thm.4.4,noticing that the cocycle c(f)r,s,t in (67) also represents the class i(c(f)) in H2(X, ˜G).We want to write down a cocycle representing the class Φ∗([D1], .

. .

[Dk]) ∈H2(X, ˜G) andto show that it represents the same cohomology class as the cocycle in (67).We consider as before the cocycle kj(r,s) representing OX(−Dj) in the choosen covering Ur ofX. For every pair of indices r, s with Ur ∩Us ̸= ∅and for every j = 1, .

. .

k, we choose a mj-throot ˆkj(r,s) of kj(r,s) on Ur ∩Us in such a way that ˆkj(s,r) = (ˆkj(r,s))−1. As before, the image ofthe class of −Dj in H2(X, Z/mjZ) is represented on Ur ∩Us ∩Ut by the cocycle ˆkj(r,s)ˆkj(s,t)ˆkj(t,r),j = 1, .

. .

k. Then the class −Φ∗([D1], . .

. [Dk]) ∈H2(X, ˜G) is represented on Ur ∩Us ∩Ut by:br,s,t = (kYj=1(ˆkj(r,s)ˆkj(s,t)ˆkj(t,r))˜aij)i=1,...h .

(95)and we have shown in the equality (80) in the proof of Thm.4.4, Step III, that this cocyclerepresents the same class then c(f)r,s,t in H2(X, ˜G).Remark 4.8 From Thm.4.6 it follows in particular that the class i(c(f)) ∈H2(Γ, ˜G) dependsonly on the class of the Dj’s in H2(X, Z/mjZ) (j = 1, . .

. k), once G and the gj’s are fixed.

Inparticular, if Dj is mj-divisible on X (∀j = 1, . .

. k), then i(c(f)) = 0.References[Bo]Bott R., On a theorem of Lefschetz, Mich.

Math. J.

6 (1969), 211–216. [Ca]Catanese F., On the moduli spaces of surfaces of general type, J. Differential Geometry19 (1984), 483–515.

[CT]Catanese F., Tovena F., Vector bundles, linear systems and extensions of π1, Proceedingsof the Conference on Complex Algebraic Varieties, Bayreuth 1990, Lecture Notes inMathematics 1507, Springer, 51–70. [Gr]Grothendieck A., Sur quelques points d’algebre homologique, Tohoku Math.

J. 9 (1957),119–221.

[MM] Mandelbaum R., Moishezon B., On the topology of algebraic surfaces, Trans. Amer.

Math.Soc. 260 (1980), 195–222.

[Mu]Mumford D., Abelian Varieties, Tata Institute of Fundamental Research, Bombay, OxfordUniversity Press, 1974.17

[Pa]Pardini R., Abelian covers of algebraic varieties, J. reine angew. Math.

417 (1991), 191–213.Rita PardiniDipartimento di Matematica, Universit`a di PisaVia Buonarroti 256127 Pisa, ItalyFrancesca TovenaDipartimento di Matematica, Universit`a di BolognaPiazza di Porta S.Donato 540127 Bologna, Italy18

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