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.

• necklace diagrams


• totally


• lefschetz fibrations


• real elliptic


• has been known


• around


• directed

Sujets

• Degtyarev
• Totally
• Around

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 32
• Langue: English

Extrait

CLASSIFICATIONOFTOTALLYREALELLIPTICLEFSCHETZ FIBRATIONSVIANECKLACEDIAGRAMS NERMI ˙ NSALEPCI ˙ Abstract. WeshowthattotallyrealellipticLefschetzfibrationsadmittinga realsectionareclassifiedbytheir“realloci”whichcanbeencodedintermsof acombinatorialobjectthatwecallanecklacediagram.Bymeansofnecklace diagrams,weobtainanexplicitlistofcertainclassesoftotallyrealelliptic Lefschetzfibrations. 1. Introduction Asiswellknown,aLefschetzfibrationisaprojectionfromanorientedconnected smooth4-manifoldontoanorientedconnectedsmoothsurfacesuchthatthereexist finitelymanycriticalpointsaroundwhichonecanchoosecomplexchartsonwhich theprojectiontakestheform(z 1 ,z 2 )→z 1 2 +z 2 2 .RegularfibersofaLefschetz fibrationareorientedclosedsmoothsurfacesofgenusg,whilesingularfibershave onlynodes.Inthepresentwork,weconsideronlythosefibrationswhosefibergenus is1.Wecallsuchfibrations ellipticLefschetzfibrations .Withoutlossofgenerality, weassumethateachsingularfibercontainsonlyonenodeandthatnofibercontains aselfintersection-1sphere(fibrationswiththelatterpropertyarecalled relatively minimal ). Westudy realellipticLefschetzfibrations ;thatistosay,ellipticLefschetzfibra- tionswhosetotalandbasespaceshave realstructures whicharecompatiblewith thefiberstructure.A realstructure onanorientedsmooth4-manifoldisdefinedas anorientationpreservinginvolutionwhosefixedpointset(calledthe realpart )has dimension2,ifitisnotempty.Tworealstructuresareconsideredthesameifthey differbyaconjugationbyanorientationpreservingdiffeomorphism.Itisworth mentioningherethatnotevery4-manifoldadmitssuchaninvolution.Examplesof 4-manifoldswhichdonotadmitrealstructurescanbefoundin[3].Likewise,a real structure onasmoothorientedsurfaceisdefinedasanorientationreversinginvo- lution.Obviouslyeverysurfaceadmitsarealstructure.Besides,theclassification (uptoconjugationbyanorientationpreservingdiffeomorphism)ofrealstructures onsurfacesisknown.Therearetwoinvariantsthatdeterminetheconjugacyclass ofarealstructure:itstype(separating/non-separating)andthenumberofthe componentsofitsrealpart.Arealstructureiscalled separating iftherealpart dividesthesurfaceintotwodisjointhalves;otherwise,itiscalled non-separating . Throughoutthepresentwork,wefocusonfibrationsoverS 2 ,andtherealstructure consideredonS 2 istheoneinducedfromthecomplexconjugation,denoted conj , 11 on C P.BydefinitionofrealLefschetzfibrations,fibersovertherealpartSof TheauthorwaspartiallysupportedbytheEuropeanCommunity’sSeventhFrameworkPro- gramme([FP7/2007-2013][FP7/2007-2011])undergrantagreementno[ ˙ 258204],aswellasbythe FrenchAgencenationaledelarecherchegrantANR-08-BLAN-0291-02. 1

2NERMI ˙ NSALEPC ˙ I conj inheritrealstructuresfromtherealstructureofthetotalspace.Suchfibers arecalled real(fibers) .ArealellipticLefschetzfibrationhas3typesofrealregular fibers.Theyaredistinguishedbythenumberofrealcomponentsthatcanbe0,1,2 (onlythestructurewith2realcomponentsisseparatingonT 2 ). Forthesakeofsimplicity,mostofthetimeweassumethattherealpartS 1 of (S 2 , conj )isoriented.Fibrationswithsuchafeaturearecalled directed .Moreover, weconsidermainlyfibrationswhichadmita realsection (asectioncompatiblewith therealstructures).Butthecasesofnon-directedfibrationsaswellasoffibrations withoutarealsectionarealsocovered.Theonlyessentialconditionimposedon fibrationsisthatallthecriticalvaluesarereal.Fibrationswiththispropertyare called totallyreal . OurmaininterestisthetopologicalclassificationoftotallyrealellipticLefschetz fibrations.TworealLefschetzfibrationswillbeconsidered isomorphic iftheydiffer byorientationpreservingequivariantdiffeomorphisms.Recallthattheclassification ofellipticLefschetzfibrationsoverS 2 hasbeenknownforover30years.Itisdueto B.MoishezonandR.Livne´[4]that(non-real,relativelyminimal)ellipticLefschetz 2 fibrationsoverSareclassified(uptoisomorphism)bythenumberofthecritical values.Thelatterisdivisibleby12andtheclassofellipticLefschetzfibrationswith 12ncriticalvaluesisdenotedbyE(n),n ∈ N .Furthermore,E(1)isisomorphicto thefibration C P 2 #9 C P 2 → C P 1 ,obtainedbyblowingupapencilofcubicsin C P 2 , andE(n)=E(n − 1)] F E(1)where] F standsforthefibersum. Inthisnote,wegivetherealversionofthisresultfortotallyrealellipticLefschetz fibrations.Theclassificationisobtainedbymeansofcertaincombinatorialobjects thatwecall necklacediagrams .Necklacediagramsarecombinatorialcounterparts of realLefschetzchains introducedin[6]. MainresultsofthisworkarepresentedasTheorem4.1andTheorem7.1inwhich wetreatthecasesofdirectedtotallyrealellipticLefschetzfibrationsadmittinga realsectionand,respectively,thosefibrationspossiblywithoutarealsection.As immediatecorollaries(Corollary4.4,respectively,Corollary7.2)ofthesetheorems, weobtainthatnon-directedtotallyrealellipticLefschetzfibrationsadmittinga sectionareclassifiedbytheirnecklacediagrams(defineduptosymmetryandwith theidentitymonodromy),whilethosefibrationspossiblywithoutarealsection areclassifiedbythesymmetryclassesoftheir refined necklacediagramswiththe identitymonodromy.AsaconsequenceofCorollary4.4,weobtainanexplicitlist oftotallyrealE(1)andE(2)thatadmitarealsection.Weinvestigatethealgebraic realizationsofthesefibrationsandshowthatcertainfibrationsonthelistarenot algebraicallyrealizable.Wealsoconsidersomeoperationsonthesetofnecklace diagrams: mild/harshsums , flip-flops and metamorphoses .Theseoperationsallow ustoconstructnewnecklacediagramsfromthegivenones.Bymeansofthese operations,weconstructanexampleofarealLefschetzfibrationwhichcannotbe writtenasafibersumoftworealfibrations(seeProposition6.5). Acknowledgements. Thematerialpresentedhereisextractedfrommythesis. IamdeeplyindebtedtomysupervisorsSergeyFinashinandViatcheslavKharlamov fortheirguidanceandlimitlesssupport.IowemanythankstoAndyWandwho wrotetheprogramtogettheexplicitlistofnecklacediagrams,whoalsoeditedmy presentandpastarticlesasanativeenglishspeaker.IthankAlexDegtyarevfor hispreciouscommentsonthefirstmanuscriptandforproductivediscussions.