p-adic simpson correspondense.pdf

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1、AdvancesinMathematics198(2005)847–862www.elsevier.com/locate/aimAp-adicSimpsoncorrespondenceGerdFaltings∗Max-Planck-InstitutfürMathematik,Vivatsgasse7,53111Bonn,GermanyReceived28May2004;accepted25May2005CommunicatedbyJohanDeJongDedicatedtoM.Artinontheoccasionofhis70thbirthdayAb

2、stractForcurvesoverap-adicfieldweconstructanequivalencebetweenthecategoryofHiggs-bundlesandthatof“generalisedrepresentations”oftheétalefundamentalgroup.Thedefinitionof“generalisedrepresentations”usesp-adicHodgetheoryandalmostétalecoverings,anditincludesusualrepresentationswhichfo

3、rmafullsubcategory.Theequivalencedependsonthechoiceofanexponentialfunctionforthemultiplicativegroup.©2005ElsevierInc.Allrightsreserved.Keywords:Almostétaleextensions;Higgs-bundles;p-adicHodgetheory1.IntroductionThepurposeofthisnoteistoconstructHiggs-bundlesassociatedtorepresent

4、ationsofthegeometricfundamentalgroupofacurveoverap-adicfieldK.Itthuscanbeconsideredap-adicanalogueoftheresultsofSimpsonandCorlette(see[12]).Thefunctorisfullyfaithfulbutitisdifficulttocharacteriseitsimage:namelytheresultingHiggs-bundlesaresemistableofslopezero,butwedonotknowwhethe

5、ranysuchHiggs-bundleliesintheimage(thisistrueforline-bundlesoncurvesoverp-adiclocalfields).ConverselywecanconstructforallHiggs-bundlesso-called“generalisedrepresentations”,whichformacategorycontainingtheusualrepresentationsasfullsub-category.However,wedonotknowwhichofthosecomefr

6、omgenuinerepresentations.∗Fax:+49228402277.E-mailaddress:gerd@mpim-bonn.mpg.de.0001-8708/$-seefrontmatter©2005ElsevierInc.Allrightsreserved.doi:10.1016/j.aim.2005.05.026848G.Faltings/AdvancesinMathematics198(2005)847–862OveralocalfieldKandforline-bundlesonecancheckthatonegetsall

7、line-bundlesofdegreezero,soonecanhopethatoverlocalfieldsallsemistableHiggs-bundlesofdegreezerolieintheimage.RecallthataHiggs-bundleonanalgebraicmanifoldXisapair(E,
),whereEisavectorbundleonXand
aglobalsectionofEnd(E)⊗
Xsatisfying
∧
=0(thatisinlocalcoordinatesthecomponentsof
comm

8、ute).WealsousevariantswhereXisonlylogsmooth,orwhere
ha