On the projective geometry of rational homogeneous

arXiv:math.AG/9810140v222Dec2000ONTHEPROJECTIVEGEOMETRYOFRATIONALHOMOGENEOUSVARIETIESJ.M.LANDSBERGANDLAURENTMANIVEL1.IntrodutionThisistherstpaperinaseriesestablishingnewrelationsbetweentherepresentationtheoryofomplexsimpleLiegroupsandthealgebraianddierentialgeometryoftheirhomogeneousvarieties.Inthispaperwedeterminethevarietiesoflinearspaesonrationalhomogeneousvarieties,provideexpliitgeometrimodelsforthesespaes,andestablishbasifatsabouttheloaldierentialgeometryofrationalhomogeneousvarieties.LetGbeaomplexsimpleLiegroup,Pamaximalparabolisubgroup.ThespaeoflinesonG=Pinitsminimalhomogeneousembeddingwasdeterminedin[4℄intermsofLieinidenesystems.ThereisadihotomybetweentheasesforwhihthesimplerootassoiatedtoPisshortornot:fornon-shortroots,thespaeoflinesonG=PisG-homogeneousandanbedesribedusingideasofTits;forshortroots,itisnotG-homogeneous.Wepresentarenementoftheirresult,dueto[1℄(foraparabolisubgroupPwhihdoesnotneedtobemaximal),thateahonnetedomponentofthespaeoflinesonsistsofexatlytwoG-orbits.TheproofrequiresthestudyoflinesthroughagivenpointofG=P.Forsimpliity,assumePismaximalinthisparagraph.Wenotethatthespaeoftangentdiretionstolinesthroughapointalwaysontains(andinfatequalsexeptintheaseofnon-shortroots)ahomogeneousspaeY1=H=QPT1,whereHisthesemi-simplepartofPandT1isaP-submoduleofthetangentspaewhihisaominusuleH-module.Motivatedbythisobservation,westudytheloalgeometryofsuhminusulevarietiesindetail.Inpartiular,weprovethattheirseondfundamentalformsompletelydeterminetheirhigherfundamentalforms;thebaseloiofthehigherfundamentalformsaresimplytheseantvarietiesofthebaselousoftheseondfundamentalforom.Weexploitthesuprisingonnetionbetweenseantvarieties,prolongationsandnilpotentorbitstruturesin[14,15℄.Weexplainhowtodeterminethehigherdimensionallinearspaesassoiatedtonon-shortrootsusingTitsmethods.Forshortroots,weprovideexpliitdesriptionsofthespaeswestudy,espeiallyintheexeptionalaseswhereweuseCayley’sotonions.Inallases,eahonnetedomponentofthevarietyoflinearspaesonaG=Pisquasi-homogeneous;morepreisely,itistheunionofanitenumberofG-orbits.Weusetheorderingoftherootsasin[2℄.2.Underthemirosope2.1.Fundamentalformsofprojetivevarieties.LetXnPn+abeaprojetivevariety,andletx2Xbeasmoothpoint.Takeloaloordinates(x1;:::;xn;xn+1;:::;xn+a)adaptedtoxandwriteXloallyasagraph(1;n;n+a;n+a):x=qxx+rxxx+:::Date:June2000.KEYWORDS:FLAGVARIETY,FUNDAMENTALFORMS,LINEARSPACES,OCTONION.AMSCLASSIFICATION:14M15,20G05.12J.M.LANDSBERGANDLAURENTMANIVELThegeometriinformationintheseries(thatis,informationindependentofhoieofadaptedoordinates)anbeenodedinaseriesoftensors,thesimplestofwhihistheprojetiveseondfundamentalformFF2X;x=qdxÆdxx2S2TxXNxX.Ifxisageneralpoint,FF2X;xevenontainsinformationabouttheglobalgeometryofX,see[9℄,[12℄.ItisusefultoonsiderthesystemofquadrisjFF2X;xj:=P(FF2X;x(NxX))PS2TxX,andBasejFF2X;xjPTxX,theirommonzerolous.Moregenerally,thek-thprojetivefundamentalformofXatx,FFk=FFkX;xisamapFFk:Nk!SkTxXwhereTxXdenotestheholomorphitangentspae,andNk=Nk;xXisthek-thnormalspae.ThenotationissuhthatTxX=N1.TodeneFFkandNk,oneanusethesamedenitionsasonedoesfortheEulideanfundamentalforms,eitherinoordinatesorasthederivativesofsuessiveGaussmappings(see[12℄).Foreahsmoothpointx2X,theosulatingspaes^Tx(k)X=(lkNl)?VdetermineaagofV,0^x^TxX^Tx(2)X:::^Tx(f)=V:Moregenerally,givenamapping:Y!PV,onedenesitsfundamentalformsFFkinthesamemanner.FF2;xquotientedbykerxisisomorphitotheseondfundamentalformoftheimage,FF2(Y);(x).See[13℄fordetails.Inwhatfollows,wewillslightlyabusenotationbyignoringtwists,whihwillnotmatteraswestudyfundamentalformsonlyatsomexedbasepoint.WewilluseNktodenotebothNkandNkO(1)anduseatwistofFFk.WeletjFFkjxPSkTxXdenotetheimageofFFkxandBasejFFkjPTxXdenoteitsbaselous.LetVbeavetorspae,letASdVbealinearsubspae,andletA(l):=(ASlV)\Sd+lV;thel-thprolongationofA.LetJa(A):=fvyPjv2V;P2AgSd1V,theJaobianidealofA.NotethatA(1)=fP2Sd+1VjJa(P)Ag.Abasifataboutfundamentalforms,duetoCartan([3℄,p377)(andredisoveredin[9℄),isthatifx2Xisageneralpoint,thentheprolongationpropertyholdsatx:jFFkX;xjjFFk1X;xj(1):Ageometrionsequeneisasfollows.Denethek-thseantvarietyk(Y)ofaprojetivevarietyYPNtobethelosureoftheunionofthelinearspaesspannedbykpointsofY.Proposition2.1.LetXnPn+abeavarietyandx2Xageneralpoint.ThenBasejFFkX;xjk1(BasejFF2X;xj):Proposition2.1isaonsequeneofthefollowinglemma:Lemma2.2.LetAS2VbeasystemofquadrisandletB(A)Vdenotetheoneoveritsbaselous.Thenk(B(A))B(A(k1)).Moreover,ifB(A)islinearlynondegenerate,thenfork2,Ik(k(B(A))=0andIk+1(k(B(A))=A(k),whereId(Z)SdVistheomponentoftheidealofZPVindegreed.Proof.LetP2A(1)andletx;y2B(A).ConsiderP(x+ty;x+ty;x+ty)=P(x;x;x)+3tP(x;x;y)+3t2P(x;y;y)+t3P(y;y;y)=Px(x;x)+3t2Px(y;y)+3tPy(x;x)+t3Py(y;y)=0;wherePx2S2VdenotesthederivativeofPwithrespettox.(SayingP2A(1)isequivalenttodemandingallderivativesofPlieinA.)Thegeneralizationislear.ONTHEPROJECTIVEGEOMETRYOFRATIONALHOMOGENEOUSVARIETIES13Remark2.3.IfXisminusuletheninfattheidealofk(BasejFF2X;xj)isgeneratedindegreek+1.Thisisnotalw