Asymptotics for general connexions at infinity

Carlos Simpson

Description

Asymptotics for general connexions at infinity Carlos Simpson CNRS, Laboratoire J. A. Dieudonne, Universite de Nice-Sophia Antipolis September 24th, 2003: preliminary notes for my talk in Toulouse on Sept. 26th Introduction This is a preliminary set of notes on the asymptotic behavior of the mon- odromy of connexions near a general point at ∞ in the spaceMDR of connex- ions on a compact Riemann surface X. We will consider a path of connexions of the form (E,?+ t?) which approaches the boundary divisor transversally at the point on the boundary of MDR corresponding to a general Higgs bun- dle (E, ?). This is very similar to the situation considered in [1], and we import the vast majority of our techniques directly from there. However, our present situation is slightly more general with regard to the form of the family of connexions. We reduce to a case similar to that which was treated in [1], of a family of connexions of the form d+B+tA, but the matrix B may have poles. This difficulty of poles in the matrix B is the new phenomenon which is treated here. However, we are not able to get results as good as the precise asymptotic expansions of [1]. Our result of Theorem 2 (cf p. 14) is that if m(t) denotes the family of monodromy or transport matrices for a given path, then the Laplace transform f(?) of m has an analytic continua- tion with locally finite singularities

• give functions


• monodromy represen- tations


• betti moduli space


• mdr


• resulting analytic curve


• higgs bundle


• full results


• complex function


• nilpotent higgs

Sujets

• Antipolis
• Simpson
• Toulouse
• Complex analysis

Informations

• Publié par: pefav
• Nombre de lectures: 13
• Langue: English

Extrait

Asymptoticsforgeneralconnexionsatinfinity CarlosSimpson CNRS,LaboratoireJ.A.Dieudonne´,Universit´edeNice-SophiaAntipolis September24th,2003:preliminarynotesformytalkinToulouseonSept. ht62 Introduction Thisisapreliminarysetofnotesontheasymptoticbehaviorofthemon- odromyofconnexionsnearageneralpointat ∞ inthespaceM DR ofconnex- ionsonacompactRiemannsurfaceX.Wewillconsiderapathofconnexions oftheform(E, r +tθ)whichapproachestheboundarydivisortransversally atthepointontheboundaryofM DR correspondingtoageneralHiggsbun- dle(E,θ).Thisisverysimilartothesituationconsideredin[1],andwe importthevastmajorityofourtechniquesdirectlyfromthere.However, ourpresentsituationisslightlymoregeneralwithregardtotheformofthe familyofconnexions.Wereducetoacasesimilartothatwhichwastreated in[1],ofafamilyofconnexionsoftheformd+B+tA,butthematrixBmay havepoles.ThisdifficultyofpolesinthematrixBisthenewphenomenon whichistreatedhere.However,wearenotabletogetresultsasgoodas thepreciseasymptoticexpansionsof[1].OurresultofTheorem2(cfp.14) isthatifm(t)denotesthefamilyofmonodromyortransportmatricesfora givenpath,thentheLaplacetransformf(ζ)ofmhasananalyticcontinua- tionwithlocallyfinitesingularitiesoverthecomplexplane.Thesingularities arewhatdeterminetheasymptoticbehaviorofm(t).Whatwedon’tknow isthebehavioroff(ζ)nearthesingularities;themainquestionleftopen iswhetherfhaspolynomialgrowthatthesingularities,andifso,towhat extentthegeneralizedLaurentseriescanbecalculatedfromtheindividual termsinourintegralexpressionforf. Eventhoughhedoesn’tappearinthereferencesof[1],theideasofJ.-P. Ramisindirectlyhadaprofoundinfluenceonthatwork(andhenceonthe presentnote).Thiscanbetracedtoatleasttwoinputsasfollows: (1)IhadpreviouslyfollowedG.Laumon’scourseabout`-adicFouriertrans- form,whichwaspartlyinspiredbythecorrespondingnotionsincomplex functiontheory,asubjectinwhichRamis(andothersinthesubjectof “resurgence”)hadagreatinfluence;and (2)atthetimeofwriting[1]IwasfollowingN.Katz’scourseaboutexpo- 1

nentialsums,whereagainmuchoftheinspirationcamefromRamis’swork (whichKatzmentionnedveryoften)onirregularsingularities. Thecompactifiedmodulispaceofconnexions LetXbeasmoothprojectivecurveoverthecomplexnumbersC.Fixr. RecallthatwehaveamodulispaceM DR ofrankrvectorbundleswith integrableconnexiononX.ThishasacompactificationM DR ⊂ M DR con- structedasfollows.ThereisamodulispaceM Hod → A 1 forvectorbundles withλ-connexion,λ ∈ A 1 .Thefiberoverλ=0isthemodulispaceM Dol forsemistableHiggsbundlesofdegreezero.ThishasasubvarietyM D ni o l l parametrizingtheHiggsbundles(E,θ)suchthatθisnilpotentasanΩ 1 X - valuedendomorphismofE.LetM D ∗ ol denotethecomplementofM D ni o l l in M Dol andletM H ∗ od denotethecomplementofM D ni o l l inM Hod .Thenthe algebraicgroupG m actsonM Hod preservingalloftheabovesubvarieties, dna M DR :=M H ∗ od /G m . ThecomplementofM Dol inM Hod (whichisalsothecomplementofM D ∗ ol in M H ∗ od )isisomorphictoM DR × G m andthisgivestheembeddingM DR , → M DR .Thecomplementarydivisorisgivenby P DR =M D ∗ ol /G m . Inconclusion,thismeansthatthepointsat ∞ inM DR correspondtoequiv- alenceclassesofsemistable,non-nilpotentHiggsbundles(E,θ)underthe equivalencerelation (E,θ) ∼ = (E,uθ) foranyu ∈ G m . RecallthatthemodulispaceM Dol isanirreduciblealgebraicvariety, soP DR isalsoirreducible.Thegeneralpointthereforecorrespondstoa “general”Higgsbundle(E,θ).Forageneralpoint,thespectralcurveofθ isanirreduciblecurvewithramifiedmaptoX,suchthattheramification pointsareallofthesimplesttype. Curvesgoingtoinfinity Themodulispacesconsideredabovearecoarseonly;however,inanetale neighborhoodofthegenericpointtheyarefinemodulispaces,andalso 2

smooth.AtageneralpointofP DR ,bothM DR andthedivisorP DR are smooth.ThuswecanlookforacurvecuttingP DR transversallyatageneral point.Suchacurvemaybeobtainedbytakingtheprojectionofacurve inM Hod cuttingM Dol atageneralpoint.Inturn,thisamountstogiving afamily(E c , r c )where r c isaλ(c)-connexion,parametrizedbyc ∈ Cfor somecurveC.Alsoinanetaleneighborhoodofthepointλ=0,thefunction λ(c)shouldbeetale.Notealsothat(E 0 , r 0 )shouldbeageneralsemistable Higgsbundleofdegreezero. Theeasiestwaytoobtainsuchacurveisasfollows:let(E,θ)beageneral Higgsbundle,stableofdegreezero.ThebundleEisstableasavectorbundle (sincestabilityisanopenconditionanditcertainlyholdsonthesubsetof Higgsbundleswithθ=0,soitholdsatgeneralpoints).InparticularE supportsaconnexion r andwecanset r λ :=λ r +θ forλ ∈ A 1 .Heretheparameterisλitself.ThesubsetG m ⊂ A 1 corresponds topointswhicharemappedintoM DR ,andindeedthevectorbundlewith connexioncorrespondingtotheaboveλ-connexionis (E, r +tθ),t=λ − 1 . ThemapactuallyextendstoamapfromA 1 intoM DR fortheothercoor- dinatechartA 1 providinganeighborhoodat ∞ inP 1 .Inconclusion,the familyofconnexions { (E, r +tθ) } correspondstoamorphism P 1 → M DR sendingt ∈ A 1 intoM DR ,sendingthepointt= ∞ toageneralpointinthe divisorP DR ,andthecurveistransversetothedivisoratthatpoint. Wewillinvestigatetheasymptoticbehaviorofthemonodromyrepresen- tationsoftheconnexions(E, r +tθ)ast →∞ .RecallthattheBettimoduli spaceM B isthemodulispaceforrepresentationsofπ 1 (X)uptoconjugation, andwehaveananalyticisomorphismM D an R = ∼ M B an sendingaconnexiontoits monodromyrepresentation.Wewilllookattheasymptoticsoftheresulting analyticcurveA 1 → M B . Inordertosetthingsupitwillbeusefultofixabasepointp ∈ Xand atrivializationτ:E p = ∼ C r .Thenforanyγ ∈ π 1 (X,x)weobtainthe monodromymatrix ρ(E, r +tθ,τ,γ) ∈ GL(r,C). 3