Asymptotics for general connexions at infinity

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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

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AsymptoticsforgeneralconnexionsatinﬁnityCarlosSimpsonCNRS,LaboratoireJ.A.Dieudonne´,Universit´edeNice-SophiaAntipolisSeptember24th,2003:preliminarynotesformytalkinToulouseonSept.ht62IntroductionThisisapreliminarysetofnotesontheasymptoticbehaviorofthemon-odromyofconnexionsnearageneralpointat∞inthespaceMDRofconnex-ionsonacompactRiemannsurfaceX.Wewillconsiderapathofconnexionsoftheform(E,r+tθ)whichapproachestheboundarydivisortransversallyatthepointontheboundaryofMDRcorrespondingtoageneralHiggsbun-dle(E,θ).Thisisverysimilartothesituationconsideredin[1],andweimportthevastmajorityofourtechniquesdirectlyfromthere.However,ourpresentsituationisslightlymoregeneralwithregardtotheformofthefamilyofconnexions.Wereducetoacasesimilartothatwhichwastreatedin[1],ofafamilyofconnexionsoftheformd+B+tA,butthematrixBmayhavepoles.ThisdiﬃcultyofpolesinthematrixBisthenewphenomenonwhichistreatedhere.However,wearenotabletogetresultsasgoodasthepreciseasymptoticexpansionsof[1].OurresultofTheorem2(cfp.14)isthatifm(t)denotesthefamilyofmonodromyortransportmatricesforagivenpath,thentheLaplacetransformf(ζ)ofmhasananalyticcontinua-tionwithlocallyﬁnitesingularitiesoverthecomplexplane.Thesingularitiesarewhatdeterminetheasymptoticbehaviorofm(t).Whatwedon’tknowisthebehavioroff(ζ)nearthesingularities;themainquestionleftopeniswhetherfhaspolynomialgrowthatthesingularities,andifso,towhatextentthegeneralizedLaurentseriescanbecalculatedfromtheindividualtermsinourintegralexpressionforf.Eventhoughhedoesn’tappearinthereferencesof[1],theideasofJ.-P.Ramisindirectlyhadaprofoundinﬂuenceonthatwork(andhenceonthepresentnote).Thiscanbetracedtoatleasttwoinputsasfollows:(1)IhadpreviouslyfollowedG.Laumon’scourseabout`-adicFouriertrans-form,whichwaspartlyinspiredbythecorrespondingnotionsincomplexfunctiontheory,asubjectinwhichRamis(andothersinthesubjectof“resurgence”)hadagreatinﬂuence;and(2)atthetimeofwriting[1]IwasfollowingN.Katz’scourseaboutexpo-1

nentialsums,whereagainmuchoftheinspirationcamefromRamis’swork(whichKatzmentionnedveryoften)onirregularsingularities.ThecompactiﬁedmodulispaceofconnexionsLetXbeasmoothprojectivecurveoverthecomplexnumbersC.Fixr.RecallthatwehaveamodulispaceMDRofrankrvectorbundleswithintegrableconnexiononX.ThishasacompactiﬁcationMDR⊂MDRcon-structedasfollows.ThereisamodulispaceMHod→A1forvectorbundleswithλ-connexion,λ∈A1.Theﬁberoverλ=0isthemodulispaceMDolforsemistableHiggsbundlesofdegreezero.ThishasasubvarietyMDniollparametrizingtheHiggsbundles(E,θ)suchthatθisnilpotentasanΩ1X-valuedendomorphismofE.LetMD∗oldenotethecomplementofMDniollinMDolandletMH∗oddenotethecomplementofMDniollinMHod.ThenthealgebraicgroupGmactsonMHodpreservingalloftheabovesubvarieties,dnaMDR:=MH∗od/Gm.ThecomplementofMDolinMHod(whichisalsothecomplementofMD∗olinMH∗od)isisomorphictoMDR×GmandthisgivestheembeddingMDR,→MDR.ThecomplementarydivisorisgivenbyPDR=MD∗ol/Gm.Inconclusion,thismeansthatthepointsat∞inMDRcorrespondtoequiv-alenceclassesofsemistable,non-nilpotentHiggsbundles(E,θ)undertheequivalencerelation(E,θ)∼=(E,uθ)foranyu∈Gm.RecallthatthemodulispaceMDolisanirreduciblealgebraicvariety,soPDRisalsoirreducible.Thegeneralpointthereforecorrespondstoa“general”Higgsbundle(E,θ).Forageneralpoint,thespectralcurveofθisanirreduciblecurvewithramiﬁedmaptoX,suchthattheramiﬁcationpointsareallofthesimplesttype.CurvesgoingtoinﬁnityThemodulispacesconsideredabovearecoarseonly;however,inanetaleneighborhoodofthegenericpointtheyareﬁnemodulispaces,andalso2

smooth.AtageneralpointofPDR,bothMDRandthedivisorPDRaresmooth.ThuswecanlookforacurvecuttingPDRtransversallyatageneralpoint.SuchacurvemaybeobtainedbytakingtheprojectionofacurveinMHodcuttingMDolatageneralpoint.Inturn,thisamountstogivingafamily(Ec,rc)wherercisaλ(c)-connexion,parametrizedbyc∈CforsomecurveC.Alsoinanetaleneighborhoodofthepointλ=0,thefunctionλ(c)shouldbeetale.Notealsothat(E0,r0)shouldbeageneralsemistableHiggsbundleofdegreezero.Theeasiestwaytoobtainsuchacurveisasfollows:let(E,θ)beageneralHiggsbundle,stableofdegreezero.ThebundleEisstableasavectorbundle(sincestabilityisanopenconditionanditcertainlyholdsonthesubsetofHiggsbundleswithθ=0,soitholdsatgeneralpoints).InparticularEsupportsaconnexionrandwecansetrλ:=λr+θforλ∈A1.Heretheparameterisλitself.ThesubsetGm⊂A1correspondstopointswhicharemappedintoMDR,andindeedthevectorbundlewithconnexioncorrespondingtotheaboveλ-connexionis(E,r+tθ),t=λ−1.ThemapactuallyextendstoamapfromA1intoMDRfortheothercoor-dinatechartA1providinganeighborhoodat∞inP1.Inconclusion,thefamilyofconnexions{(E,r+tθ)}correspondstoamorphismP1→MDRsendingt∈A1intoMDR,sendingthepointt=∞toageneralpointinthedivisorPDR,andthecurveistransversetothedivisoratthatpoint.Wewillinvestigatetheasymptoticbehaviorofthemonodromyrepresen-tationsoftheconnexions(E,r+tθ)ast→∞.RecallthattheBettimodulispaceMBisthemodulispaceforrepresentationsofπ1(X)uptoconjugation,andwehaveananalyticisomorphismMDanR=∼MBansendingaconnexiontoitsmonodromyrepresentation.WewilllookattheasymptoticsoftheresultinganalyticcurveA1→MB.Inordertosetthingsupitwillbeusefultoﬁxabasepointp∈Xandatrivializationτ:Ep=∼Cr.Thenforanyγ∈π1(X,x)weobtainthemonodromymatrixρ(E,r+tθ,τ,γ)∈GL(r,C).3

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