CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS VIA NECKLACE DIAGRAMS

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CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS VIA NECKLACE DIAGRAMS NERMI˙N SALEPCI˙ Abstract. We show that totally real elliptic Lefschetz fibrations admitting a real section are classified by their “real loci” which can be encoded in terms of a combinatorial object that we call a necklace diagram. By means of necklace diagrams, we obtain an explicit list of certain classes of totally real elliptic Lefschetz fibrations. 1. Introduction As is well known, a Lefschetz fibration is a projection from an oriented connected smooth 4-manifold onto an oriented connected smooth surface such that there exist finitely many critical points around which one can choose complex charts on which the projection takes the form (z 1 , z 2 ) ? z 2 1 + z 2 2 . Regular fibers of a Lefschetz fibration are oriented closed smooth surfaces of genus g, while singular fibers have only nodes. In the present work, we consider only those fibrations whose fiber genus is 1. We call such fibrations elliptic Lefschetz fibrations. Without loss of generality, we assume that each singular fiber contains only one node and that no fiber contains a self intersection -1 sphere (fibrations with the latter property are called relatively minimal). We study real elliptic Lefschetz fibrations; that is to say, elliptic Lefschetz fibra- tions whose total and base spaces have real structures which are compatible with the fiber structure.

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CLASSIFICATIONOFTOTALLYREALELLIPTICLEFSCHETZFIBRATIONSVIANECKLACEDIAGRAMSNERMI˙NSALEPCI˙Abstract.WeshowthattotallyrealellipticLefschetzﬁbrationsadmittingarealsectionareclassiﬁedbytheir“realloci”whichcanbeencodedintermsofacombinatorialobjectthatwecallanecklacediagram.Bymeansofnecklacediagrams,weobtainanexplicitlistofcertainclassesoftotallyrealellipticLefschetzﬁbrations.1.IntroductionAsiswellknown,aLefschetzﬁbrationisaprojectionfromanorientedconnectedsmooth4-manifoldontoanorientedconnectedsmoothsurfacesuchthatthereexistﬁnitelymanycriticalpointsaroundwhichonecanchoosecomplexchartsonwhichtheprojectiontakestheform(z1,z2)→z12+z22.RegularﬁbersofaLefschetzﬁbrationareorientedclosedsmoothsurfacesofgenusg,whilesingularﬁbershaveonlynodes.Inthepresentwork,weconsideronlythoseﬁbrationswhoseﬁbergenusis1.WecallsuchﬁbrationsellipticLefschetzﬁbrations.Withoutlossofgenerality,weassumethateachsingularﬁbercontainsonlyonenodeandthatnoﬁbercontainsaselfintersection-1sphere(ﬁbrationswiththelatterpropertyarecalledrelativelyminimal).WestudyrealellipticLefschetzﬁbrations;thatistosay,ellipticLefschetzﬁbra-tionswhosetotalandbasespaceshaverealstructureswhicharecompatiblewiththeﬁberstructure.Arealstructureonanorientedsmooth4-manifoldisdeﬁnedasanorientationpreservinginvolutionwhoseﬁxedpointset(calledtherealpart)hasdimension2,ifitisnotempty.Tworealstructuresareconsideredthesameiftheydiﬀerbyaconjugationbyanorientationpreservingdiﬀeomorphism.Itisworthmentioningherethatnotevery4-manifoldadmitssuchaninvolution.Examplesof4-manifoldswhichdonotadmitrealstructurescanbefoundin[3].Likewise,arealstructureonasmoothorientedsurfaceisdeﬁnedasanorientationreversinginvo-lution.Obviouslyeverysurfaceadmitsarealstructure.Besides,theclassiﬁcation(uptoconjugationbyanorientationpreservingdiﬀeomorphism)ofrealstructuresonsurfacesisknown.Therearetwoinvariantsthatdeterminetheconjugacyclassofarealstructure:itstype(separating/non-separating)andthenumberofthecomponentsofitsrealpart.Arealstructureiscalledseparatingiftherealpartdividesthesurfaceintotwodisjointhalves;otherwise,itiscallednon-separating.Throughoutthepresentwork,wefocusonﬁbrationsoverS2,andtherealstructureconsideredonS2istheoneinducedfromthecomplexconjugation,denotedconj,11onCP.BydeﬁnitionofrealLefschetzﬁbrations,ﬁbersovertherealpartSofTheauthorwaspartiallysupportedbytheEuropeanCommunity’sSeventhFrameworkPro-gramme([FP7/2007-2013][FP7/2007-2011])undergrantagreementno[˙258204],aswellasbytheFrenchAgencenationaledelarecherchegrantANR-08-BLAN-0291-02.1

2NERMI˙NSALEPC˙Iconjinheritrealstructuresfromtherealstructureofthetotalspace.Suchﬁbersarecalledreal(ﬁbers).ArealellipticLefschetzﬁbrationhas3typesofrealregularﬁbers.Theyaredistinguishedbythenumberofrealcomponentsthatcanbe0,1,2(onlythestructurewith2realcomponentsisseparatingonT2).Forthesakeofsimplicity,mostofthetimeweassumethattherealpartS1of(S2,conj)isoriented.Fibrationswithsuchafeaturearecalleddirected.Moreover,weconsidermainlyﬁbrationswhichadmitarealsection(asectioncompatiblewiththerealstructures).Butthecasesofnon-directedﬁbrationsaswellasofﬁbrationswithoutarealsectionarealsocovered.Theonlyessentialconditionimposedonﬁbrationsisthatallthecriticalvaluesarereal.Fibrationswiththispropertyarecalledtotallyreal.OurmaininterestisthetopologicalclassiﬁcationoftotallyrealellipticLefschetzﬁbrations.TworealLefschetzﬁbrationswillbeconsideredisomorphiciftheydiﬀerbyorientationpreservingequivariantdiﬀeomorphisms.RecallthattheclassiﬁcationofellipticLefschetzﬁbrationsoverS2hasbeenknownforover30years.ItisduetoB.MoishezonandR.Livne´[4]that(non-real,relativelyminimal)ellipticLefschetz2ﬁbrationsoverSareclassiﬁed(uptoisomorphism)bythenumberofthecriticalvalues.Thelatterisdivisibleby12andtheclassofellipticLefschetzﬁbrationswith12ncriticalvaluesisdenotedbyE(n),n∈N.Furthermore,E(1)isisomorphictotheﬁbrationCP2#9CP2→CP1,obtainedbyblowingupapencilofcubicsinCP2,andE(n)=E(n−1)]FE(1)where]Fstandsfortheﬁbersum.Inthisnote,wegivetherealversionofthisresultfortotallyrealellipticLefschetzﬁbrations.Theclassiﬁcationisobtainedbymeansofcertaincombinatorialobjectsthatwecallnecklacediagrams.NecklacediagramsarecombinatorialcounterpartsofrealLefschetzchainsintroducedin[6].MainresultsofthisworkarepresentedasTheorem4.1andTheorem7.1inwhichwetreatthecasesofdirectedtotallyrealellipticLefschetzﬁbrationsadmittingarealsectionand,respectively,thoseﬁbrationspossiblywithoutarealsection.Asimmediatecorollaries(Corollary4.4,respectively,Corollary7.2)ofthesetheorems,weobtainthatnon-directedtotallyrealellipticLefschetzﬁbrationsadmittingasectionareclassiﬁedbytheirnecklacediagrams(deﬁneduptosymmetryandwiththeidentitymonodromy),whilethoseﬁbrationspossiblywithoutarealsectionareclassiﬁedbythesymmetryclassesoftheirreﬁnednecklacediagramswiththeidentitymonodromy.AsaconsequenceofCorollary4.4,weobtainanexplicitlistoftotallyrealE(1)andE(2)thatadmitarealsection.Weinvestigatethealgebraicrealizationsoftheseﬁbrationsandshowthatcertainﬁbrationsonthelistarenotalgebraicallyrealizable.Wealsoconsidersomeoperationsonthesetofnecklacediagrams:mild/harshsums,ﬂip-ﬂopsandmetamorphoses.Theseoperationsallowustoconstructnewnecklacediagramsfromthegivenones.Bymeansoftheseoperations,weconstructanexampleofarealLefschetzﬁbrationwhichcannotbewrittenasaﬁbersumoftworealﬁbrations(seeProposition6.5).Acknowledgements.Thematerialpresentedhereisextractedfrommythesis.IamdeeplyindebtedtomysupervisorsSergeyFinashinandViatcheslavKharlamovfortheirguidanceandlimitlesssupport.IowemanythankstoAndyWandwhowrotetheprogramtogettheexplicitlistofnecklacediagrams,whoalsoeditedmypresentandpastarticlesasanativeenglishspeaker.IthankAlexDegtyarevforhispreciouscommentsontheﬁrstmanuscriptandforproductivediscussions.

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