on semi-positivity and filtered frobenius crystals

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1、SEMI-POSITIVITYANDFROBENIUSCRYSTALSOnSemi-PositivityandFilteredFrobeniusCrystalsbyShinichiMOCHIZUKI*§0.IntroductionThepurposeofthispaperistoprovethatcertainsubquotientsoftheMF∇-objectsofFaltings([1])aresemi-positive.Inparticular,weobtainanalgebraicproofofaresult(generalizingthoseof[5],[

2、9])onthesemi-positivityofthehigherdirectimagesofcertainkindsofsheavesforsemistablefamiliesofalgebraicvarieties.OurMainTheorem,provenin§3,isasfollows:Theorem3.4:Letf:(X,E)→(S,D)beasemistablefamilyofvarietiesofrelativedimensiond,withSasmooth,properschemeoverafieldLofcharacteristiczero.Let(

3、A,∇,Fi(A))beagloballycrystallinefilteredvectorbundlewithconnectionon(X,E).AThenforanynonnegativeintegerα,thecoherentsheafofOS-modulesRαf(ω⊗(A/F1(A))∨)∗X/SOXisasemi-positivevectorbundle,asareallofitstensorpowers.(Theterm“semistable”(respectively,“globallycrystalline”)isdefinedin§1(respecti

4、vely,§3).)Inparticular,thisTheoremimpliesthefollowing:Corollary3.5:Letf:(X,E)→(S,D)beasabove.Thenforanynonnegativeintegerα,thecoherentsheafofOS-moduleslogαf∗((ΩX/S))isasemi-positivevectorbundle,asareallofitstensorpowers.Thus,FujitaandKawamataprovedtheaboveCorollaryinthecaseα=d.ReceivedF

5、ebruary24,1994.RevisedJuly20,1994.1991MathematicsSubjectClassification:14F30*ResearchInstituteforMathematicalSciences,KyotoUniversity,Kyoto606,Japan1Roughlyspeaking,theideaoftheproofisasfollows.Werestricttoacurve,andconsiderthepossibilityofquotientsofthebundleinquestionthathavenegativede

6、gree.Ifsuchaquotientdidexist,itwouldmeanthatwecouldconstructalargenumberofsectionsofaspaceofboundeddimension.Thisconcludestheproof.Sincetheproofissubstantiallysimplerandlesshigh-poweredinthecasewhenthefibresoff,aswellasthebaseS,areone-dimensionalandthefilteredvectorbundleistrivial,weprese

7、nttheproofinthatspecialcaseinanAppendix(whichislogicallyindependentoftherestofthepaper,exceptforbasicfactsanddefinitionsgivenin§1).Webelievethatthisspecialcaseshedslightonthegeneralcase.Inessence,whatmakestheproofeasierinthiscaseisthefactthatwehaveaveryphysicalandconcreterealiza