ON DEFORMATION TYPES OF REAL ELLIPTIC SURFACES Alex Degtyarev, Ilia Itenberg, Viatcheslav Kharlamov LeYiJingn’estpasunlivre,untextequ’onlitdud´ebut`alafin,maisunouvrage quel’onconsultequandonenabesoin. Lorsqu’onh´esitesurunevoie`asuivre, uneattitude`aprendre,unchoix`afaire,undilemme`ar´esoudre,onpeutalors s’enservirpourcequ’ilestdanslapratique: unmanueld’aide`alad´ecision. CyrilleJavary,LesRouagesduYiJing,Ed. PhillipePicquier,2001 Abstract. We study real elliptic surfaces and trigonal curves (over a base of an arbitrary genus) and their equivariant deformations. We calculate the real Tate- Shafarevichgroupandreducethedeformationclassificationtothecombinatoricsof a real version of Grothendieck’s dessins d’enfants. As a consequence, we obtain an explicitdescriptionofthedeformationclassesofM-and(M−1)-(i.e.,maximaland submaximalinthesenseoftheSmithinequality)curvesandsurfaces. Contents 1. Introduction 2 1.1. Motivation and historical remarks 2 1.2. Subject of the paper 3 1.3. Tools and results 4 1.4. Contents of the paper 5 1991Mathematics Subject Classification. 14J27,14P25,05C90. Key words and phrases. Realellipticsurface,dessins d’enfants,Tate-Shafarevichgroup. The second and third authors were partially funded by the ANR-05-0053-01 grant of Agence NationaledelaRechercheandagrantofUniversit´eLouisPasteur,Strasbourg. TypesetbyAMS-TEX 1 2 ALEX DEGTYAREV, ILIA ITENBERG, VIATCHESLAV KHARLAMOV 1.5. Acknowledgements 6 2. Involutions and real structures 6 2.1. Real structures and real sheaves 6 2.2. Kalinin’s spectral sequence 8 3. Real elliptic surfaces 10 3.1. Elliptic surfaces 10 3.2. Jacobian surfaces 11 3.3. Trigonal curves and Weierstraß models 15 4. Real Tate-Shafarevich group 19 4.1. Topological invariance 19 4.2. The case of generic singular fibers 22 4.3. The geometric interpretation 24 4.4. Deformations 27 5. Real trigonal curves and dessins d’enfants 28 5.1. Trichotomic graphs 28 5.2. Deformations 31 5.3. Dessins 33 5.4. The oval count 35 5.5. Inner ◦- and •-vertices 36 5.6. Indecomposable dessins 40 5.7. Scraps 45 6. Applications: M- and (M −1)-cases 46 6.1. Junctions 47 6.2. Classification of trigonal M-curves 48 6.3. Classification of elliptic M-surfaces 49 6.4. (M −1)-curves and surfaces 51 6.5. Oval chains 54 6.6. Further generalizations and open questions 55 References 57 1. Introduction 1.1. Motivation and historical remarks. Ingeometryofnonsingularalgebraic surfaces, over the reals as well as over the complex numbers, there are two major equivalence relations: the first one, called deformation equivalence, is up to iso- morphism and deformation (of the complex structure), and the second one, called topological equivalence, is up to diffeomorphism (ignoring the complex structure). Certainly, deformation equivalence implies topological equivalence, and one of the principalquestionsinthesubjectistowhatextenttheconverseholds,i.e.,towhat extent is the deformation class of a surface controlled by its topology. Since we re- gardarealvarietyasacomplexvarietyequippedwitharealstructure(whichisan anti-holomorphic involution), by a deformation of real varieties we mean an equi- variant Kodaira-Spencer deformation, and by a diffeomorphism between two real varieties we mean an equivariant diffeomorphism. Therefore, the Dif = Def ques- tionabovestatedovertherealswouldinvolvethesamequestionfortheunderlying complex varieties. Luckily, due to Donaldson’s and Seiberg-Witten’s revolution ON DEFORMATION TYPES OF REAL ELLIPTIC SURFACES 3 in four dimensional topology (as well as the Enriques-Kodaira classification of al- gebraic surfaces), one does have an advanced level of control over the discrepancy betweenthedeformationclassofacompactcomplexsurfaceanditsdiffeomorphism class. This fact makes it reasonable to fix a deformation class of compact complex surfaces beforehand and to concentrate on the topology and deformations of the real structures that can appear on (some of) the surfaces in question. The problem of enumerating the equivariant deformation classes of real struc- tures within a fixed complex deformation class goes back at least to F. Klein [Kl], who studied the nonsingular real cubic surfaces in P3 (i.e., Del Pezzo surfaces of degree 3) from a similar pointof view. He provedthat the equivariant deformation class of such a surface is already determined by the topology of its real part, which is the real projective plane with up to three handles or up to one sphere (Schl¨afli’s famous five ‘species’ of nonsingular cubics). Further important steps in this direc- tion were made by A. Commessatti [Co1], [Co2], who found a classification of all real abelian surfaces and all R-minimal real rational surfaces, thus extending (at least implicitly) Klein’s result to these special classes. In general, we call a deformation class of complex varieties quasi-simple if a real variety within the complex class is determined up to equivariant deformation by the diffeomorphism type of the real structure. For curves, the problem was settled by F. Klein and G. Weichold (see, e.g., the survey [N1]) who proved that the family of compact curves of any given genus is indeed quasi-simple. Note that the equivariant deformation class of a real curve is no longer determined by its genus and real part; in addition, one should take into account the so called type of the curve, i.e., whether the real part does or does not divide the complexification. However, the type is certainly a topological invariant of the real structure. Further advance in the study of quasi-simplicity called for appropriate tools in complexalgebraicgeometry. Theirdevelopmenttookhalfacentury,anditwasnot until the late 70s that the study was resumed. Now, due to the results obtained in [Ni], [DK2], [DIK1], [We], [CF], [DIK2], we know that quasi-simplicity holds for any special (in the sense of the Enriques-Kodaira classification) class of C-minimal complexsurfacesexceptelliptic. (Forthesurfacesofgeneraltypetherearecounter- examples,see,e.g.,[KK].)Aslightlydifferentbutrelatedfiniteness statement,i.e., finitenessofthenumberofequivariantdeformationclassesofrealstructureswithin a given deformation class of complex varieties, is known to hold for all surfaces except elliptic or ruled with irrational base. For ruled surfaces, the statement is probably true and its proof should not be difficult, cf., e.g., [DK2], but it does not seem to appear in the literature. Thus, elliptic surfaces are essentially the last special class of surfaces for which the quasi-simplicity and finiteness questions are still open. It is worth mentioning that, in spite of noticeable activity in the theory of com- plex elliptic surfaces, literature dealing with the real case is scant. Among the few works that we know are [AMn], [Ba], [BMn], [DK1], [Fr], [GW], [Kh], [Mn], [Si], and [Wa]. 1.2. Subject of the paper. Inthispaper,ourgoalistostudyrelativelyminimal real elliptic surfaces without multiple fibers and, in particular, to understand the extent to which the equivariant deformation class of such a surface is controlled by the topology of its real structure. Recall that the complex deformation class of an ellipticsurfaceasaboveisdeterminedbythegenusgofthebasecurveandtheEuler 4 ALEX DEGTYAREV, ILIA ITENBERG, VIATCHESLAV KHARLAMOV characteristicχofthesurface,providedthatχispositive. (Thecaseg =0istreated inA.Kas[Ka],andthegeneralcase,inW.Seiler[Se],seealso[FM].)Notethat,ifχ issmall(foragivengenus),onedeformationclassmayconsistofseveralirreducible components: theprincipalcomponentformedbythenon-isotrivialsurfacesmaybe accompanied by few others, formed by the isotrivial ones. Each isotrivial surface can be deformed to a surface that perturbs to a non-isotrivial one. However, from theknownconstructionsitisnotimmediatelyobviousthatthedeformationcanbe chosen real. For this reason, we confine ourselves to the more topological study of non-isotrivialsurfaces, leavingthealgebro-geometricaspectstosubsequentpapers. Anellipticsurfacecomesequippedwithanellipticfibration. Moreover,formost surfaces, in particular, for all elliptic surfaces of Kodaira dimension 1, the elliptic fibration is unique. (In the case of relatively minimal surfaces without multiple fibers, the Kodaira dimension is known to be equal to 1 whenever g > 0, as well as when g = 0 and the Euler characteristic χ, which is divisible by 12, is > 24.) Thus, the elliptic fibration is an important part of the structure, and we include it into the setting of the problem, considering equivariant deformations of real elliptic fibrations (with no confluence of singular fibers allowed) on the one hand, and equivariant diffeo-/homeomorphisms on the other hand. Furthermore, as any non-isotrivial surface can be perturbed to an almost generic one, i.e., a surface withsimplestsingularfibersonly,weconsidersolelydeformationsofalmostgeneric surfaces. Here, ‘almost generic’ can be thought of as ‘topologically generic’, as opposed to ‘generic’, or ‘algebraically generic’, where one requires in addition that thefiberswithnontrivialcomplexmultiplicationshouldalsobesimple. Weusethe latter assumption when treating an individual surface via algebro-geometric tools. Note that during the deformation we never assume the base curve fixed; it is alsosubjecttoadeformation. Theclassificationofrealellipticsurfacesoverafixed base does not seem feasible; in general it may not even be possible to perturb a given surface to an almost generic one. 1.3. Tools and results. Asinthecomplexcase,thestudyofrealellipticsurfaces is based upon two major tools: the real version of the Tate-Shafarevich group, which enumerates all real surfaces with a given Jacobian, and a real version of the techniques of dessins d’enfants, which reduces the deformation classification of non-isotrivial Jacobian elliptic surfaces (or, more generally, trigonal curves on ruledsurfaces)toacombinatorialproblem. Wedevelopthetwotoolsand,asafirst application, obtain a rather explicit classification of the so called M- and (M−1)- surfaces (i.e., those maximal and submaximal in the sense of the Smith inequality) andM-and(M−1)-curves. Theprincipalresultsofthepaperarestatedin6.2–6.4. As a straightforward consequence of the description of deformation classes in terms of groups and graphs, the whole number of equivariant deformation classes of real elliptic fibrations with given numeric invariants is finite. This settles the finiteness problem stated above for non-isotrivial elliptic surfaces without multiple fibers. The real Tate-Shafarevich group RX(J) is defined as the set (with a certain group operation) of the isomorphism classes of all real elliptic fibrations with a given Jacobian J. Contrary to the complex case, RX(J) is usually disconnected, and we describe, in purely topological terms, its discrete part RXtop(J), which enumerates the deformation classes of real fibrations whose Jacobian is J. This description gives an explicit list of all modifications that a fibration may undergo, ON DEFORMATION TYPES OF REAL ELLIPTIC SURFACES 5 and the result shows that they can all be seen in the real part. As a consequence, we prove that, up to deformation, an elliptic surface is Jacobian if and only if the real part of the fibration admits a topological section (Proposition 4.3.5), each Betti number of an elliptic surface is bounded by the corresponding Betti number of its Jacobian, and each M- or (M −1)-surface is Jacobian up to deformation (Proposition 4.3.7). The real version of dessins d’enfants was first introduced by S. Orevkov [Or1], who used it to study real trigonal curves on C-minimal rational surfaces Σ . (A d similarobjectwasconsideredindependentlyin[SV]and[NSV]).Thecurvesconsid- ered by Orevkov do not intersect the ‘exceptional’ section of the surface; for even valuesofd,thesearethebranchcurvesoftheWeierstraßmodelsofJacobianelliptic surfaces. (In the Weierstraß model, the elliptic surface appears as the double cov- ering of Σ branched at the union of the exceptional section and a trigonal curve.) d Using the dessin techniques, Orevkov invented a kind of Viro-LEGO(cid:114) game: he introduced a few elementary pieces (which are the dessins of cubic curves), defined theoperationofconnecting‘freeends’oftwopieces,andusedthisproceduretocon- struct bigger curves, thus proving a number of existence statements. Orevkov also noticed (private communication [Or2]; cf. similar observations in V. Zvonilov [Z]) that, as long as almost generic M-curves over a rational base are concerned, this procedure is universal, i.e., there is a unique way to break any M-curve into ele- mentarypieces. Clearly,thisconstructiongivesadeformationclassificationofsuch M-curves. We extend Orevkov’s approach to trigonal curves over a base of an arbitrary genus and obtain similar results for M- and (M −1)-curves. We show that, as in the rational case, any M- or (M −1)- curve breaks into certain elementary pieces. (AnessentialingredienthereisTheorem5.7.6,whichstatesthatunbreakablecurves must be sufficiently ‘small’. Another decomposability statement, Theorem 5.6.1, is used to handle large pieces of (M −1)-curves.) In the M-case, this procedure is unique; in the (M −1)-case it is unique up to a few moves that are described explicitly. As a consequence, we obtain a deformation classification of M- and (M −1)-curves and, when combined with the results on RXtop, that of M- and (M−1)-surfaces(see6.3.1and6.3.2fortheM-caseand6.4.4and6.4.5forthe(M− 1)-case). A surprising by-product of the classification is the fact that, essentially, M- and (M −1)-surfaces and curves exist only over a base of genus g (cid:54)1. (Here, ‘essentially’ means that certain ‘trivial’ handles should be ignored. Without this convention the genus can be made arbitrary large.) 1.4. Contents of the paper. Sections 2 and 3 are introductory. In Section 2 we remind the reader a few basic facts concerning topology of involutions, and in Section 3 we discuss certain complex and real aspects of the theory of trigonal curves, elliptic surfaces, their Jacobians and Weierstraß models. In Section 4 we introduce a real version of the Tate-Shafarevich group, express it in cohomologi- cal terms, and study its discrete part. The main results here are Theorems 4.2.7 and 4.3.2, as well as their corollaries. Section 5 plays a central rˆole in the paper. Here we develop Orevkov’s results on real dessins d’enfants. After a brief intro- duction, we concentrate on a special class of dessins that represent meromorphic functions having generic branching behavior, i.e., j-invariants of generic trigonal curves. The principal results of Section 5 are the decomposability theorems 5.6.1 and5.7.6,whichassertthat,undercertainassumptions,adessinbreaksintosimple 6 ALEX DEGTYAREV, ILIA ITENBERG, VIATCHESLAV KHARLAMOV pieces. Finally, in Section 6 we apply the results obtained to the case of M- and (M−1)-curvesand surfaces. We prove the structure theorems, derive a few simple consequences, and discuss further generalizations and open problems. 1.5. Acknowledgements. OurthanksgotoStepanOrevkov,whoenthusiastical- lysharedhisobservationswithus,motivatingourinterestindessins d’enfants. We wouldalsoliketothankVictoriaDegtyarevaforcourageouslyreadingandpolishing a preliminary version of the text. We are grateful to the Max-Planck-Institut fu¨r Mathematik and to the Mathe- matisches Forschungsinstitut Oberwolfach anditsRiPprogramfortheirhospitality and excellent working conditions which helped us to complete this project. An es- sentialpartoftheworkwasdoneduringthefirstauthor’svisitstoUniversit´e Louis Pasteur, Strasbourg. 2. Involutions and real structures In this section we recall basic results concerning topology of involutions. Proofs and further details can be found in the monograph [Br1], which deals with general theory of compact transformation groups. A survey of sheaf theory, cohomology, and spectral sequences is found in [Br2]. For a self-contained exposition specially tailored for the needs of topology of real algebraic varieties, we refer to [DIK1]. 2.1. Real structures and real sheaves. Throughoutthissectionalltopological spaces are assumed paracompact and Hausdorff. 2.1.1. A real structure on a complex variety X (not necessarily connected or non- singular) is an anti-holomorphic involution c : X → X. Clearly, any two real X structuresdifferbyanautomorphismofX. BytheRiemannextensiontheorem,an (anti-)holomorphic endomorphism f of the smooth part of X extends to an (anti-) holomorphic endomorphism of X if and only if f admits a continuous extension. Areal variety isacomplexvarietyX equippedwitharealstructurec . (Some- X times it is convenient to refer to the pair (X,c ) as a real form of X.) The fixed X point set Fixc is called the real part of X and is denoted X . A holomorphic X R map f: X → Y between two real varieties (X,c ) and (Y,c ) is called real if it X Y commutes with the real structures: c ◦f =f ◦c . Y X Recall that for any continuous involution c on a finite dimensional topological X space X with finitely generated total cohomology group H∗(X;Z ) the following 2 Smith inequality holds: dimH∗(Fixc ;Z )(cid:54)dimH∗(X;Z ). X 2 2 Furthermore, the difference dimH∗(X;Z ) − dimH∗(Fixc ;Z ) is even. If the 2 X 2 difference is 2d, the involution c is called an (M −d)-involution. If c is the real X X structure of a real variety X, then X itself is called an (M −d)-variety. 2.1.2. Given an abelian group A with an involution c: A → A, we define the cohomology groups H∗(Z ;A) to be the cohomology of the complex 2 1−c 1+c 1−c 0 →A −−→A −−→A −−→... (the leftmost copy of A being of degree zero). Similarly, given a sheaf A with an involutive automorphism c: A → A, we define the cohomology sheaves H∗(Z ;A) 2 ON DEFORMATION TYPES OF REAL ELLIPTIC SURFACES 7 to be the cohomology of the complex 1−c 1+c 1−c 0 →A −−→A −−→A −−→... . (Certainly,theformerisnothingbutaspecializationofthegeneraldefinitionofthe cohomology of a discrete group G with coefficients in a G-module, see, e.g., [Bro], to the group G = Z and the simplest invariant cell decomposition of the space 2 S∞ =EZ . The latter is a straightforward sheaf version of the former.) 2 2.1.3. Let X be a topological space with an involution c : X → X. Denote by X π: X →X/c the projection. Given a sheaf A on X, any morphism c: A→c∗ A X X (over the identity of X) descends to a morphism π c: π A → π A. By a certain ∗ ∗ ∗ abuseofthelanguage,ciscalledaninvolution ifπ cisaninvolution. Thiscondition ∗ isequivalenttotherequirementc◦c∗ c=id,wherec∗ c: c∗ A→Aisthepull-back X X X of c. (Certainly, this definition is merely an attempt to refer to involutive lifts A→A of c to A in terms of sheaf morphisms identical on the base.) X The constant sheaf G (for any abelian group G) has a canonical involution, X which is the identity G =c∗ G . As a lift of c it is given by s(cid:55)→s◦c . X X X X X 2.1.4. Now, let X be a complex manifold and let c : X →X be a real structure. X Then the structure sheaf O has a canonical involution, called the canonical real X structure (defined by c ); it is given by the complex conjugation O =c∗ O , or, X X X X asaliftofc , bys(cid:55)→s◦c . IfAisacoherentsheafonX, thepull-backc∗ Aisa X X X (coherent,inasense)sheafofc∗ O -modules. Aninvolutionc: A→c∗ Aiscalleda X X X real structure on A if it is compatible with the module structure (via the canonical real structure on O ). A typical example is the canonical real structure on the X sheaf F of sections of a ‘Real’ vector bundle F (i.e., a holomorphic vector bundle on X supplied with an involution c covering c and anti-linear on the fibers, so F X that it is a real structure on the total space); it is given by s (cid:55)→ c ◦s◦c . This F X formula applies as well in a more general situation, when F →X is a holomorphic fibration with abelian groups as fibers (so that F is a sheaf of abelian groups) and c : F → F is a fiberwise additive real structure covering c . Although, in F X general, F is not a coherent sheaf, we will still refer to the result as the canonical real structure on F. ThecanonicalrealstructureonO definesinvolutionsontheothertwomembers X of the exponential sequence 0 −−→Z −2π−→i O −−→O∗ −−→0. X X X NotethattheresultinginvolutionontheconstantsheafZ differsfromthecanoni- X calinvolutionaboveby(−1). Inordertoemphasizethisnonstandardrealstructure, we will use the notation Z− (and, more generally, G−). X X 2.1.5. LetAbeasheafonX withaninvolutionc: A→c∗ A. Denotebyπ: X → X X/c the projection and consider the complex X π A∗ : 0 −−→π A 1−−−→c π A 1−+−→c π A 1−−−→c ... ∗ ∗ ∗ ∗ of sheaves on X/c (the leftmost copy of π A being of degree zero; for simplicity, X ∗ we use the same notation c for the automorphism π c: π A → π A). We will ∗ ∗ ∗ 8 ALEX DEGTYAREV, ILIA ITENBERG, VIATCHESLAV KHARLAMOV refer to the hypercohomology H∗(X/c ;π A∗) as the hypercohomology of (A,c) X ∗ and denote it H∗(A,c) (or just H∗(A), when c is understood). For the constant sheaf G with its canonical real structure we will also use the notation H∗(X;G). X Recall that there are natural spectral sequences (2.1.6) Hq(X/c ;Hp(Z ;A))=⇒Hp+q(A), X 2 where Hp(Z ;A) stand for the cohomology sheaves of π A∗, and 2 ∗ (2.1.7) Hp(Z ;Hq(X/c ;π A))=Hp(Z ;Hq(X;A))=⇒Hp+q(A). 2 X ∗ 2 (Since π is finite-to-one, the higher direct images Riπ , i > 0, vanish and one ∗ has Hq(X/c ;π A) = Hq(X;A).) Furthermore, since π is finite-to-one, one can X ∗ calculate H∗(A) using c -invariant Cˇech resolutions of A. More precisely, given a X c -invariant open covering U ={U } of X, one can consider the bi-complex X i (cid:77) (Cˇp,∗,d ,d )= (Cˇ∗(A),d ), d =1−(−1)pc: Cp,∗ →Cp+1,∗ U 1 2 U 2 1 U U p(cid:62)0 (direct sum of copies of the ordinary Cˇech complex with the first differential given above). Then Hn(A) is the limit, over all coverings, of the cohomology Hn(Cˇ∗,∗). U 2.2. Kalinin’s spectral sequence. LetX beaafinitedimensionalparacompact Hausdorff topological space with an involution c : X →X. X 2.2.1. The Borel construction over (X,c ) is the twisted product X X =X× S∞ =(X×S∞)/(x,r)∼(c (x),−r). c c X ThecohomologygroupsH∗(X;G)=H∗(X ;G)arecalledtheequivariantcohomol- c c ogy ofX (withcoefficientsinanabeliangroupG). Notethatthesubscriptcstands for the involution c=c ; as we never use cohomology with compact supports, we X hopethatthisnotationwillnotleadtoaconfusion. TheLerayspectralsequenceof the fibration X →Rp∞ =S∞/±id with fiber X is called the Borel-Serre spectral c sequence of (X,c ): X 2Epq(X;G)=Hp(Z ;Hq(X;G))=⇒Hp+q(X;G). 2 c Sometimesitisconvenienttostartthesequenceattheterm1Epq =Hq(X;G)with the differential 1dp∗ =1−(−1)pc∗ . X There is a canonical isomorphism Hp(X;G) = Hp(X;G), and the Borel-Serre c spectral sequence is isomorphic to the spectral sequence (2.1.7) for the constant sheaf A=G . If G is a commutative ring, then the Borel-Serre spectral sequence X is a spectral sequence of H∗(Rp∞;G)-algebras. 2.2.2. Let G = Z , and let (cid:126) ∈ H1(Rp∞;Z ) = Z be the generator. Assume, 2 2 2 in addition, that X is a CW-complex of finite dimension. Then the stabilization homomorphisms ∪(cid:126): rEpq(X;Z2)→rEp+1,q(X;Z2), ∪(cid:126): Hcn(X;Z2)→Hcn+1(X;Z2) ON DEFORMATION TYPES OF REAL ELLIPTIC SURFACES 9 are isomorphisms for p(cid:192)0 and one has lim Hn(X;Z )=Hn(cid:192)0(Fixc ×Rp∞;Z )=H∗(Fixc ;Z ). c 2 X 2 X 2 n→∞ Thus,oneobtainsaZ-gradedspectralsequencerHq(X;Z )=lim rEpq(X;Z ), 2 p→∞ 2 calledKalinin’s spectral sequence of(X,c ). Asabove, itisconvenienttostartthe X sequence at the term 1H∗(X;Z )=H∗(X;Z ) with the differential 1d∗ =1+c∗. 2 2 Kalinin’s spectral sequence converges to H∗(Fixc ;Z ). More precisely, there X 2 is an increasing filtration {F } = {F (X;Z )} on H∗(Fixc ;Z ), called Kalinin’s q q 2 X 2 filtration, and homomorphisms bvq: ∞Hq(X;Z ) → H∗(Fixc ;Z )/F , called 2 X 2 q−1 Viro homomorphisms, which establish isomorphisms of the graded groups. In gen- eral, the convergence does not respect the ordinary grading of H∗(Fixc ;Z ). X 2 The Smith inequality in 2.1.1 can be derived from Kalinin’s spectral sequence, and c is an M-involution if and only if the sequence degenerates at 1H. If the X sequence degenerates at 2H, the involution (real variety, etc.) is called Z -Galois 2 maximal. Asimilarconstructionappliestothehomology,producingaZ-gradedspectralse- quencerH (X;Z )startingfromH (X;Z )andconvergingtoH (Fixc ;Z ). The q 2 q 2 ∗ X 2 correspondingdecreasingfiltrationonH (Fixc ;Z )andVirohomomorphismsare ∗ X 2 denoted by {Fq(X;Z )} and bv : Fq →∞H , respectively. 2 q q 2.2.3. The cup-products in H∗(X;Z ) descend to a multiplicative structure in 2 rH∗(X;Z ), sothatrH∗ isa Z -algebraandthedifferentialsrd∗ aredifferentiations 2 2 for all r (cid:62) 2, i.e., rd∗(x∪y) = rd∗x∪y +x∪rd∗y. The filtration F∗ and Viro homomorphisms bv∗ are multiplicative, i.e., Fp ∪ Fq ⊂ Fp+q and bv∗(x ∪ y) = bv∗x∪bv∗y. 2.2.4. If X is a closed connected n-manifold and Fixc (cid:54)= ∅, Kalinin’s spectral X sequence inherits Poincar´e duality: for each r (cid:54) ∞ one has rHn(X;Z ) = Z , 2 2 the cup-product rHp(X;Z )⊗rHn−p(X;Z ) → Z is a perfect pairing, and the 2 2 2 differentials rdp and rdn−p−r+1, 1(cid:54)r <∞, are dual to each other. The last member Fn of the homological filtration is the group Z spanned by 2 the class w−1(ν)∩[XR], where ν is the normal bundle of XR in X and w(ν) is its total Stiefel-Whitney class. (Recall that, if X is a complex manifold and c is a real structure, the normal bundle ν is canonically isomorphic to the tangent bundle τ of X ; the isomorphism is given by the multiplication by i.) Hence, in terms of R the cohomology of Fixc the Poincar´e duality above can be stated as follows: the pairing (x,y) (cid:55)→ (cid:104)x∪y ∪w−1(ν),[XR](cid:105) ∈ Z2 is perfect and, with respect to this pairing, one has F =F⊥. n−q−1 q 2.2.5. Now, let G = Z, and let h ∈ H2(Rp∞;Z) = Z be the generator. Assume, 2 as above, that X is a CW-complex of finite dimension. Then the stabilization homomorphisms ∪h: rEpq(X;Z)→rEp+2,q(X;Z), ∪h: Hcn(X;Z)→Hcn+2(X;Z) are isomorphisms for p(cid:192)0 and one has lim H2k(X;Z)=H2k(cid:192)0(Fixc ×Rp∞;Z)=Heven(Fixc ;Z ), c X X 2 k→∞ lim H2k+1(X;Z)=H2k+1(cid:192)0(Fixc ×Rp∞;Z)=Hodd(Fixc ;Z ). c X X 2 k→∞ 10 ALEX DEGTYAREV, ILIA ITENBERG, VIATCHESLAV KHARLAMOV (cid:76) (WeusethenotationHpmod2 = Hi,Heven =H0mod2,Hodd =H1mod2.) i=pmod2 Thus, one obtains a (Z ×Z)-graded spectral sequence 2 rHpq(X;Z)= lim rE2k+p,q(X;Z), p∈Z , 2 k→∞ whichisalsocalledKalinin’sspectralsequence of(X,c )(withcoefficientsinZ). It X converges to Heven(Fixc ;Z )⊕Hodd(Fixc ;Z ), i.e., there are increasing filtra- X 2 X 2 tions{Fp}={Fp(X;Z)}onHpmod2(Fixc ;Z ),p∈Z ,calledKalinin’sfiltration, q q X 2 2 and homomorphisms bvpq: ∞Hpq(X;Z) → Hpmod2(Fixc ;Z )/Fp , called Viro X 2 q−1 homomorphisms, which establish isomorphisms of the graded groups. As in 2.2.2, one can start the sequence at the term 1Hpq(X;Z)=Hq(X;Z)with differential 1dpq = 1−(−1)p+qc∗. If the sequence rH∗∗(X;Z) degenerates at 2H, the involution (real variety, etc.) is called Z-Galois maximal. 2.2.6. Kalinin’s spectral sequence rH∗∗(X;Z) is multiplicative (in the same sense as in 2.2.3), the multiplicative structure inducing the product x⊗y (cid:55)→x∪y+Sq1x∪Sq1y in the limit term Heven(Fixc ;Z )⊕Hodd(Fixc ;Z ). (Here Sq1: Hp(·;Z ) → X 2 X 2 2 Hp+1(·;Z ) stands for the Bockstein homomorphism.) 2 2.2.7. Reduction modulo 2 induces a homomorphism rH0,q(X;Z)⊕rH1,q(X;Z)→rHq(X;Z ) 2 of Z-graded spectral sequences, which is compatible with the isomorphism Heven(Fixc ;Z )⊕Hodd(Fixc ;Z )=H∗(Fixc ;Z ) −1+−S−q−→1 H∗(Fixc ;Z ) X 2 X 2 X 2 X 2 of their limit terms. If H∗(X;Z) is free of 2-torsion, reduction modulo 2 is an isomorphism starting from the term 2H. 3. Real elliptic surfaces In what follows, a surface is a nonsingular complex manifold of complex di- mension two. In the few cases when singular surfaces are considered, it is specified explicitly. Proofs of most statements in this section are omitted. We refer the reader to the excellent founding paper by K. Kodaira [Ko], or to the more recent monographs [FM] and [BPV]. 3.1. Elliptic surfaces. 3.1.1. An elliptic surface is a surface E equipped with an elliptic fibration, i.e., a proper holomorphic map p: E → B to a nonsingular curve B (called the base of the fibration) such that for all but finitely many points b ∈ B the fiber p−1(b) is a nonsingular curve of genus 1. We will use the notation p: E| → U (or just U E| ) for the restriction of the fibration to a subset U ⊂ B. The restriction to the U subset B(cid:93) of the regular values of p is denoted by p(cid:93): E(cid:93) → B(cid:93). (In other words, E(cid:93) is formed by the nonsingular fibers of p.) Two fibrations p: E →B and p(cid:48): E →B(cid:48) on the same surface E are considered identical if there is an isomorphism b: B → B(cid:48) such that p(cid:48) = p◦b. A morphism