CONNECTEDNESS OF CLASSES OF FIELDS AND ZERO CYCLES

CONNECTEDNESSOFCLASSESOFFIELDSANDZEROCYCLESONPROJECTIVEHOMOGENEOUSVARIETIESVLADIMIRCHERNOUSOVANDALEXANDERMERKURJEV1.IntroductionLetXbeaproperschemeof¯nitetypeovera¯eldF.Azero-cycleonXistheformalsumPni[xi]whereni2Zandxiareclosed(zero-dimensional)pointsofthevarietyX.Thefactorgroupofthegroupofzero-cyclesmodulorationalequivalenceiscalledChowgroupofdimensionzeroandisdenotedbyCH0(X).Theassignmentx7!deg(x)extendstothedegreehomomorphismdeg:CH0(X)!Z:Theimageofdegcoincideswithn(X)Zwheren(X)isthegreatestcommondivisorofthedegreesdeg(x)=[F(x):F]overallclosedpointsx2X.WedenotethekernelofdegbyCH0(X).ThemainpurposeofthepaperistopresentacharacteristicfreeuniformmethodofcomputingthegroupCH0(X)forprojectivehomogeneousvari-etiesofsemisimplealgebraicgroups.Themethodisbasedontheideaofparametrizationof¯eldsoverwhichXhasapoint.WeillustratethemethodbyprovingthatinmanycasesthegroupCH0(X)istrivialandgiveexamplesofvarietieswhenthisgroupisnottrivial.Themainresultsofthepapercanbesummarizedasfollows.LetXbeaschemeoverF.WedenotebyA(X)theclassofall¯eldex-tensionsL=FsuchthatX(L)6=;.Wesaythattwo¯eldsL0;L12A(X)ofthesamedegreenoverFaresimplyX-equivalentiftheyaremembersofacontinuousfamilyof¯eldsLt2A(X),t2A1,ofdegreenoverF(forprecisede¯nitionseeSection6).WesaythatLandL0areX-equivalentiftheycanbeconnectedbyachainof¯eldsL0=L;L1;:::;Lr=L0suchthatLiandLi+1aresimplyX-equivalentfori=0;:::;r¡1.FurthermorewesaythattheclassA(X)isconnectedifeverytwo¯eldsinA(XE)ofdegreen(XE)overEareXE-equivalentoveranyspecialextensionE=F(seeSection6).Our¯rstresult(Theorem6.5)assertsthatifXisanarbitraryproperschemeoverFsuchthattheclassA(X)isconnectedandCH0(XL)=0forany¯eldL2A(X),thenCH0(X)=0.NotethattheconditionCH0(XL)=0alwaysholdsforprojectivehomogeneousvarietiesX.ThustheconnectednessoftheclassA(X)forsuchXimpliesCH0(X)=0.2000MathematicsSubjectClassi¯cation.Primary20G15,14M15,14M17.The¯rstauthorwassupportedinpartbyCanadaResearchChairsProgramandNSERCCanadaGrantG121210944.ThesecondauthorwassupportedinpartbyNSFGrant#0355166.12V.CHERNOUSOVANDA.MERKURJEVWeprovetheconnectednessofA(X)forvariousclassesofprojectivehomo-geneousvarieties.Theseinclude:Severi-Brauervarieties,certaingeneralizedSeveri-Brauervarieties,quadrics,involutionvarieties,projectivehomogeneousvarietiesrelatedtogroupsofexceptionaltypes3;6D4;G2;F4;1;2E6;E7withtriv-ialTitsalgebras.AsanapplicationwegetthatCH0(X)=0,i.e.theChowgroupCH0(X)isin¯nitecyclicforallabovementionedvarieties.Weborrowfrom[11]theideaofusingsymplecticinvolutionsinthecaseofgeneralizedSeveri-Brauervarietiesandinvolutionvarieties.Someofourresultswereknownbeforeundercertainrestrictionsoncharac-teristicoftheground¯eldF.TrivialityofCH0(X)inthecaseoftheSeveri-BrauervarietyX=SB(A)wasprovenbyPaninin[15]ifcharFdoesnotdivideind(A).Quadricsover¯eldsofcharacteristic6=2wereconsideredbySwan[27]andKarpenko[8].ThecasesofcertaingeneralizedSeveri-BrauervarietiesandinvolutionvarietiesweretreatedbyKrashenin[11]undertheassumptioncharF=0.Involutionvarietiesofalgebrasofindexatmost2wereconsideredin[13]bythesecondauthorunderassumptioncharF6=2.Petrov,SemenovandZainoulline[19],independently,haverecentlyshownthatCH0(X)=0forprojectivehomogeneousvarietiesXrelatedtogroupsoftypesG2;F2;1E6over¯eldsofcharacteristic0withtrivialTitsalgebras.ThenotionofX-equivalenceusedinthepaperasthemaintechnicaltoolisformulatedintermsof¯eldextensionsanddiscretevaluations,sothatweavoidsymmetricpowerconstructionsusedin[11]todescribeclosedpoints.FlexibilityofthenotionofX-equivalenceallowsusnottoimposeanychar-acteristicrestrictionsonF(exceptforthetrialitarianD4andE6;E7whereweassumethatcharacteristicis6=2;3).Anotheradvantageofourmethodisitstransparencyandshortness.Evenforthosevarietieswheretheresultwasalreadyknownourproofsaresimpler.Finallyweremarkthatmostlikelyourresultsonalgebraicgroupsofex-ceptionaltypesareclosetooptimal.ItlookshopelesstoweakenrestrictionsonTitsalgebrasandprovethatCH0(X)isin¯nitecyclicforlargerclassesofprojectivehomogeneousvarietiesofexceptionalgroups.Attheendofthepaperwegivetwoexamplesofprojectivehomogeneousva-rietiesXwithCH0(X)6=0relatedtoalgebraicgroupsoftypesA1+A1+A1andB3withnontrivialTitsalgebras.Notethatour¯rstresultisminimalpossiblesinceCH0(X)=0forallprojectivehomogeneousvarietiesXofdimensionatmost2(seeProposition4.5).2.Preliminaryfactsonalgebraicgroups2.1.Parabolicsubgroups.LetGbeasimplesimplyconnectedalgebraicgroupovera¯eldF.FixamaximaltorusT½GoverFandabasis¢oftherootsystem§=§(G;T)ofGwithrespecttoT.RecallthatforeachsubsetS½¢onecanassociatetheparabolicsubgroupPSinG,whosesemisimplepartisgeneratedbythecorrespondingrootsubgroupsU§®ofGforallroots®2S.Itisde¯nedoveraseparableclosureFsepofFandiscalledthestandardZEROCYCLES3parabolicsubgroupoftypeS.EveryparabolicsubgroupPinGoverFsepisconjugatetoauniquestandardparabolicsubgroupPS.WesaythatPisoftypeS.LetXSbethevarietyofallparabolicsubgroupsofGoftypeS.IfSisstablewithrespecttotheso-called¤-actionoftheGaloisgroupGal(Fsep=F)(see[28]),thevarietyXSisde¯nedoverF.IthasarationalpointifandonlyifGcontainsanF-de¯nedparabolicsubgroupoftypeS.IfXS(F)6=;,the