idealbw1

JOURNALOFTHEAMERICANMATHEMATICALSOCIETYVolume15,Number3,Pages531–572S0894-0347(02)00396-XArticleelectronicallypublishedonApril5,2002TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPSDANABRAMOVICH,KALLEKARU,KENJIMATSUKI,ANDJAROSLAWWLODARCZYKContents0.
Introduction5311.
Preliminaries5372.
Birationalcobordisms5453.
Torication5524.
Aproofoftheweakfactorizationtheorem5605.
Generalizations5636.
Problemsrelatedtoweakfactorization567Acknowledgements569References5690.
IntroductionWeworkoveranalgebraicallyclosedeldKofcharacteristic0.
WedenotethemultiplicativegroupofKbyK.
0.
1.
Statementofthemainresult.
Thepurposeofthispaperistogiveaproofforthefollowingweakfactorizationconjectureofbirationalmaps.
Wenotethatanotherproofofthistheoremwasgivenbythefourthauthorin[82].
Seesection0.
13forabriefcomparisonofthetwoapproaches.
Theorem0.
1.
1(WeakFactorization).
Letφ:X1X2beabirationalmapbetweencompletenonsingularalgebraicvarietiesX1andX2overanalgebraicallyclosedeldKofcharacteristiczero,andletUX1beanopensetwhereφisanisomorphism.
ThenφcanbefactoredintoasequenceofblowingsupandblowingsReceivedbytheeditorsMarch14,2000and,inrevisedform,June1,2000.
2000MathematicsSubjectClassication.
Primary14E05.
TherstauthorwaspartiallysupportedbyNSFgrantDMS-9700520andbyanAlfredP.
Sloanresearchfellowship.
Inaddition,hewouldliketothanktheInstitutdesHautesEtudesScientiques,CentreEmileBorel(UMS839,CNRS/UPMC),andMaxPlanckInstitutf¨urMath-ematikforafruitfulvisitingperiod.
ThesecondauthorwaspartiallysupportedbyNSFgrantDMS-9700520.
ThethirdauthorhasreceivednonancialsupportfromNSForNSAduringthecourseofthiswork.
ThefourthauthorwassupportedinpartbyPolishKBNgrant2P03A00516andNSFgrantDMS-0100598.
c2002AmericanMathematicalSociety531532D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKdownwithnonsingularirreduciblecentersdisjointfromU,namely,thereexistsasequenceofbirationalmapsbetweencompletenonsingularalgebraicvarietiesX1=V01V12···i1Vi1iVii+1···l1Vl1lVl=X2where(1)φ=ll1···21,(2)iareisomorphismsonU,and(3)eitheri:Vi1Vior1i:ViVi1isamorphismobtainedbyblowingupanonsingularirreduciblecenterdisjointfromU.
Furthermore,thereisanindexi0suchthatforalli≤i0themapViX1isaprojectivemorphism,andforalli≥i0themapViX2isaprojectivemorphism.
Inparticular,ifX1andX2areprojective,thenalltheViareprojective.
0.
2.
Strongfactorization.
Ifweinsistintheassertionabovethat111i0andi0+1,lbemorphismsforsomei0,weobtainthefollowingstrongfactor-izationconjecture.
Conjecture0.
2.
1(StrongFactorization).
LetthesituationbeasinTheorem0.
1.
1.
ThenthereexistsadiagramYψ1ψ2X1φX2wherethemorphismsψ1andψ2arecompositesofblowingsupofnonsingularcen-tersdisjointfromU.
Seesection6.
1forfurtherdiscussion.
0.
3.
Generalizationsofthemaintheorem.
Weconsiderthefollowingcate-gories,inwhichwedenotethemorphismsby"brokenarrows":(1)theobjectsarecompletenonsingularalgebraicspacesoveranarbitraryeldLofcharacteristic0,andbrokenarrowsXYdenotebirationalL-maps,and(2)theobjectsarecompactcomplexmanifolds,andbrokenarrowsXYdenotebimeromorphicmaps.
Giventwobrokenarrowsφ:XYandφ:XYwedeneanabsoluteisomorphismg:φ→φasfollows:InthecaseXandYarealgebraicspacesoverL,andX,YareoverL,thengconsistsofanisomorphismσ:SpecL→SpecL,togetherwithapairofbiregularσ-isomorphismsgX:X→XandgY:Y→Y,suchthatφgX=gYφ.
Intheanalyticcase,gsimplyconsistsofapairofbiregularisomorphismsgX:X→XandgY:Y→Y,suchthatφgX=gYφ.
Theorem0.
3.
1.
Letφ:X1X2beasincase(1)or(2)above.
LetUX1beanopensetwhereφisanisomorphism.
Thenφcanbefactored,functoriallywithrespecttoabsoluteisomorphisms,intoasequenceofblowingsupandblowingsdownwithnonsingularcentersdisjointfromU.
Namely,toanysuchφweassociateadiagraminthecorrespondingcategoryX1=V01V12···i1Vi1iVii+1···l1Vl1lVl=X2TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS533where(1)φ=ll1···21,(2)iareisomorphismsonU,and(3)eitheri:Vi1Vior1i:ViVi1isamorphismobtainedbyblowingupanonsingularcenterdisjointfromU.
(4)Functoriality:ifg:φ→φisanabsoluteisomorphism,carryingUtoU,andi:Vi1Viisthefactorizationofφ,thentheresultingrationalmapsgi:ViVigiveabsoluteisomorphisms.
(5)Moreover,thereisanindexi0suchthatforalli≤i0themapViX1isaprojectivemorphism,andforalli≥i0themapViX2isaprojectivemorphism.
(6)LetEiVibetheexceptionaldivisorofVi→X1(respectively,Vi→X2)incasei≤i0(respectively,i≥i0).
ThentheabovecentersofblowingupinVihavenormalcrossingswithEi.
If,moreover,X1U(respectively,X2U)isanormalcrossingsdivisor,thenthecentersofblowinguphavenormalcrossingswiththeinverseimagesofthisdivisor.
Remarks.
(1)Notethat,inordertoachievefunctoriality,wecannotrequirethecentersofblowinguptobeirreducible.
(2)Functorialityimplies,asimmediatecorollaries,theexistenceoffactorizationoveranyeldofcharacteristic0,aswellasfactorization,equivariantundertheactionofagroupG,ofaG-equivariantbirationalmap.
Ifoneassumestheaxiomofchoice,thenastandardargumentshowsthatequivarianceimpliesfunctoriality.
Inourproofswedonotusetheaxiomofchoice,withtheexceptionsof(1)existenceofanalgebraicclosure,and(2)section5.
6,whereshowingfunctorialitywithouttheassumptionoftheaxiomofchoicewouldrequirerevisingsomeoftheargumentsof[56].
Wehopethattheinterestedreaderwillbeabletoreworkourargumentswithouttheassumptionoftheaxiomofchoiceifthisbecomesdesirable.
(3)Thesametheoremholdstrueforvarietiesoralgebraicspacesofdimensiondoveraperfecteldofcharacteristicp>0assumingthatcanonicalembed-dedresolutionofsingularitiesholdstrueforvarietiesoralgebraicspacesofdimensiond+1incharacteristicp.
Theproofforvarietiesgoesthroughwordforwordasinthispaper,whileforthealgebraicspacecaseoneneedstorecastsomeofourstepsfromtheZariskitopologytotheetaletopology(see[38],[53]).
(4)Whilethistheoremclearlyimpliesthemaintheoremasaspecialcase,weprefertocarryouttheproofofthemaintheoremthroughoutthetext,andtoindicatethechangesoneneedstoperformforprovingTheorem0.
3.
1insection5.
0.
4.
Applyingthetheorem.
Supposeoneisgivenabiregularinvariantofnon-singularprojectivevarietiesandoneisinterestedinthebehaviorofthisinvariantunderbirationaltransformations.
Traditionally,onewould(1)studythebehavioroftheinvariantunderblowingsupwithnonsingularcenters,(2)formaconjec-tureaccordingtothisstudy,andnally(3)attempttoprovetheconjectureusingadditionalideas.
Sometimessuchadditionalideasturnouttobefairlysimple(e.
g.
birationalinvarianceofspacesofdierentialforms).
Sometimestheyuseknownbutdeepresults(e.
g.
HodgetheoryforshowingthebirationalinvarianceofHi(X,OX)in534D.
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WLODARCZYKcharacteristic0;abelianvarietiesforthebirationalinvarianceofH1(X,OX)ingeneral;orDeligne'sworkontheWeilconjecturesfortheresultsof[47]).
Sometimestheyleadtothedevelopmentofbeautifulnewtheories(e.
g.
MotivicintegrationfortheinvarianceofHodgenumbersofbirationalCalabi-Yauvarieties,[45],[7],[8],[22],[50];seealso[10]whereourtheoremisapplied).
Ourtheoremimpliesthat,incharacteristic0,step(3)aboveisnolongernec-essary:oncesuchaconjectureiscompatiblewithblowingsupwithnonsingularcenters,itholdsforanybirationalmap.
Atthetimeoftherevisionofthispaperweknowoftwoannouncedapplicationsforwhichnoalternativemethodsofproofareknown:(a)constructionofellipticgeneraofsingularvarietiesbyL.
BorisovandA.
Libgober[11],and(b)showingthatthealgebraiccobordismringofaeldistheLazardring,byM.
LevineandF.
Morel([48],Theor`eme1.
1,[49]).
Whenwesetouttowritethispaper,weattemptedtogiveastatementdetailedenoughandgeneralenoughtoapplyinallapplicationswehadimagined.
Assoonasthepaperwascirculated,itbecameclearthatthereareapplicationsnotcoveredbyTheorem0.
3.
1,eventhoughthemethodsapply.
Inthepreprint[27]ofH.
GilletandCh.
Soule,theauthorsusethebehavioroflocalizedToddclassesunderproperbirationalmapsofschemeswhichareprojectiveoveradiscretevaluationringofresiduecharacteristic0.
Intheirprooftheyrelyondeep(andyetunpublishedincompleteform)resultsofJ.
Franke[25];alternatively,theycouldhaveusedweakfactorizationforsuchmaps.
Whileprovingthiscasemaybeastraightfor-wardexerciseusingourmethods,thiswouldstillleaveaplethoraofotherpossibleapplications(moregeneralbaseschemes,realanalyticgeometry,p-adicanalyticgeometry,tonameafew).
Onecouldimagineastatementofageneral"weakfactorization–type"resultrelyingonaminimalsetofaxiomsneededtocarryoutourlineofproofofweakfactorization.
Wedecidedtospareourselvesandthereadersuchformalisminthispaper.
0.
5.
Earlyoriginsoftheproblem.
ThehistoryofthefactorizationproblemofbirationalmapscouldbetracedbacktotheItalianschoolofalgebraicgeome-ters,whoalreadyknewthattheoperationofblowinguppointsonsurfacesisafundamentalsourceofrichnessforsurfacegeometry:theimportanceofthestrongfactorizationtheoremindimension2(see[83])cannotbeoverestimatedintheanalysisofthebirationalgeometryofalgebraicsurfaces.
WecanonlyguessthatZariski,possiblyevenmembersoftheItalianschool,contemplatedtheprobleminhigherdimensionearlyon,butrefrainedfromstatingitbeforeresultsonresolutionofsingularitieswereavailable.
ThequestionofstrongfactorizationwasexplicitlystatedbyHironakaas"Question(F)"in[30],Chapter0,§6,andthequestionofweakfactorizationwasraisedin[61].
Theproblemremainedlargelyopeninhigherdimensionsdespitetheeortsandinterestingresultsofmany(seee.
g.
Crauder[15],Kulikov[46],Moishezon[55],Schaps[72],Teicher[76]).
ManyoftheseweresummarizedbyPinkham[64],wheretheweakfactorizationconjectureisexplicitlystated.
0.
6.
Thetoriccase.
Fortoricbirationalmaps,theequivariantversionsoftheweakandstrongfactorizationconjectureswereposedin[61]andcametobeknownasOda'sweakandstrongconjectures.
Whilethetoricversioncanbeviewedasaspecialcaseofthegeneralfactorizationconjectures,manyoftheexamplesdemon-stratingthedicultiesinhigherdimensionsareinfacttoric(seeHironaka[29],TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS535Sally[70],Shannon[73]).
ThusOda'sconjecturepresentedasubstantialchallengeandcombinatorialdiculty.
Indimension3,Danilov'sproofofOda'sweakconjec-ture[21]waslatersupplementedbyEwald[24].
Oda'sweakconjecturewassolvedinarbitrarydimensionbyJ.
Wlodarczykin[80],andanotherproofwasgivenbyR.
Morelliin[56](seealso[57],and[4],wheretheresultisgeneralizedtothetoroidalsituation).
AnimportantcombinatorialnotionwhichMorelliintroducedintothisstudyisthatofacobordismbetweenfans.
Thealgebro-geometricreal-izationofMorelli'scombinatorialcobordismisthenotionofabirationalcobordismintroducedin[81].
Ourproofofthemaintheoremreliesontoricweakfactorization.
Thisremainsasoneofthemostdiculttheoremsleadingtoourresult.
In[56],R.
MorellialsoproposedaproofofOda'sstrongconjecture.
Agapinthisproof,whichwasnotnoticedin[4],wasrecentlydiscoveredbyK.
Karu.
Asfarasweknow,Oda'sstrongconjecturestandsunprovenatpresentevenindimension3.
0.
7.
Alocalversion.
Thereisalocalversionofthefactorizationconjecture,for-mulatedandprovedindimension2byAbhyankar([1],Theorem3).
Christensen[13]posedtheproblemingeneralandsolveditforsomespecialcasesindimension3.
HerethevarietiesX1andX2arereplacedbyappropriatebirationallocalringsdominatedbyaxedvaluation,andblowingsuparereplacedbymonoidaltrans-formssubordinatetothevaluation.
Theweakformofthislocalconjecture,aswellasthestrongversioninthethreefoldcase,wasrecentlysolvedbyS.
D.
Cutkoskyinaseriesofpapers[16,17].
CutkoskyalsoshowsthatthestrongversionoftheconjecturefollowsfromOda'sstrongfactorizationconjecturefortoricmorphisms.
Inasense,Cutkosky'sresultsaysthattheonlylocalobstructionstosolvingtheglobalstrongfactorizationconjecturelieinthetoriccase.
0.
8.
Birationalcobordisms.
Ourmethodisbaseduponthetheoryofbirationalcobordisms[81].
Asmentionedabove,thistheorywasinspiredbythecombinatorialnotionofpolyhedralcobordismsofR.
Morelli[56],whichwasusedinhisproofofweakfactorizationfortoricbirationalmaps.
Givenabirationalmapφ:X1X2,abirationalcobordismBφ(X1,X2)isavarietyofdimensiondim(X1)+1withanactionofthemultiplicativegroupK.
ItisanalogoustotheusualcobordismB(M1,M2)betweendierentiablemanifoldsM1andM2givenbyaMorsefunctionf(andinfactintheK¨ahlercasethemomentummapofCisaMorsefunction,makingtheanalogymoredirect).
InthedierentialsettingonecanconstructanactionoftheadditiverealgroupR,wherethe"time"t∈Ractsasadieomorphisminducedbyintegratingthevectoreldgrad(f);hencethemultiplicativegroup(R>0,*)=exp(R,+)actsaswell.
Thecriticalpointsoffarepreciselythexedpointsoftheactionofthemultiplicativegroup,andthehomotopytypeofbersoffchangeswhenwepassthroughthesecriticalpoints(see[54]).
Analogously,inthealgebraicsetting"passingthrough"thexedpointsoftheK-actioninducesabirationaltransformation.
Lookingattheactiononthetangentspaceateachxedpoint,weobtainalocallytoricdescriptionofthetransformation.
Thisalreadygivesthemainresultof[81]:afactorizationofφintocertainlocallytoricbirationaltransformationsamongvarietieswithlocallytoricstructures.
Moreprecisely,itisshownin[81]thattheintermediatevarietieshaveabelianquotientsingularities,andthelocallytoricbirationaltransformationscanbefactoredintermsofweightedblowingsup.
Suchbirationaltransformations536D.
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WLODARCZYKcanalsobeinterpretedusingtheworkofBrion-Procesi,Thaddeus,Dolgachev-Huandothers(see[12,77,78,23]),whichdescribesthechangeofGeometricInvariantTheoryquotientsassociatedtoachangeoflinearization.
Weusesuchmethodsinsection2.
5inshowingthattheintermediatevarietiesareprojectiveoverX1orX2.
AvariantofourconstructionusingGeometricInvariantTheory,intermsofThaddeus's"MasterSpace",isgivenbyHuandKeelin[34].
0.
9.
Locallytoricversustoroidalstructures.
Consideringthefactthatweakfactorizationhasbeenprovenfortoroidalbirationalmaps([80],[56],[4]),onemightna¨velythinkthatalocallytoricfactorization,asindicatedinthepreviouspara-graph,wouldalreadyprovideaproofforTheorem0.
1.
1.
However,inthelocallytoricstructureobtainedfromacobordism,theembeddedtorichosenmayvaryfrompointtopoint,whileatoroidalstructure(seeDenition1.
5.
1)requirestheembeddedtoritobeinducedfromonexedopenset.
Thusthereisstillagapbetweenthenotionoflocallytoricbirationaltransformationsandthatoftoroidalbirationalmaps.
Developingamethodforbridgingoverthisgapisthemaincontributionofthispaper.
0.
10.
Torication.
Inordertobridgeoverthisgap,wefollowideasintroducedbyAbramovichanddeJongin[2],andblowupsuitableopensubsets,calledquasi-elementarycobordisms,ofthebirationalcobordismBφ(X1,X2)alongtoricideals.
ThisoperationinducesatoroidalstructureinaneighborhoodofeachconnectedcomponentFofthexedpointset,onwhichtheactionofKisatoroidalaction(wesaythattheblowinguptoriestheactionofK).
Nowthebirationaltransfor-mation"passingthroughF"istoroidal.
Weusecanonicalresolutionofsingularitiestodesingularizetheresultingvarieties,bringingourselvestoasituationwherewecanapplythefactorizationtheoremfortoroidalbirationalmaps.
ThiscompletestheproofofTheorem0.
1.
1.
0.
11.
Relationwiththeminimalmodelprogram.
ItisworthwhiletonotetherelationofthefactorizationproblemtothedevelopmentofMori'sprogram.
Hironaka[28]usedtheconeofeectivecurvestostudythepropertiesofbirationalmorphisms.
ThisdirectionwasfurtherdevelopedandgivenadecisiveimpactbyMori[58],whointroducedthenotionofextremalraysandsystematicallyuseditinanattempttoconstructminimalmodelsinhigherdimension,calledtheminimalmodelprogram.
Danilov[21]introducedthenotionofcanonicalandterminalsin-gularitiesinconjunctionwiththetoricfactorizationproblem.
ThiswasdevelopedbyReidintoageneraltheoryofthesesingularities[66,67],whichappearinanessentialwayintheminimalmodelprogram.
Theminimalmodelprogramissofarprovenuptodimension3([59],seealso[39,40,41,44,74]),andfortoricvarietiesinarbitrarydimension(see[68]).
Inthestepsoftheminimalmodelprogramoneisonlyallowedtocontractadivisorintoavarietywithterminalsingularities,ortoperformaip,modifyingsomecodimension≥2loci.
Thisallowsafactorizationofagivenbirationalmorphismintosuch"elementaryoperations".
Analgorithmtofactorbirationalmapsamonguniruledvarieties,knownasSarkisov'sprogram,hasbeendevelopedandcarriedoutindimension3(see[71,69,14],andsee[52]forthetoriccaseinarbitrarydimension).
Still,wedonotknowofawaytosolvetheclassicalfactorizationproblemusingsuchafactorization.
TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS5370.
12.
Relationwiththetoroidalizationproblem.
In[3],Theorem2.
1,itisproventhatgivenamorphismofprojectivevarietiesX→B,therearemodicationsmX:X→XandmB:B→B,withaliftingX→Bwhichhasatoroidalstructure.
Thetoroidalizationproblem(see[3],[4],[43])isthatofobtainingsuchmXandmBwhicharecompositesofblowingsupwithnonsingularcenters(maybeevenwithcenterssupportedonlyoverthelocuswhereX→Bisnottoroidal).
Theproofin[3]reliesontheworkofdeJong[36]andmethodsof[2].
Theauthorsofthepresentpaperhavetriedtousethesemethodstoapproachthefactorizationconjectures,sofarwithoutsuccess;onenotionwedouseinthispaperisthetoricidealof[2].
Itwouldbeinterestingifonecouldturnthisapproachonitsheadandprovearesultontoroidalizationusingfactorization.
Moreonthisinsection6.
2.
0.
13.
Relationwiththeproofin[82].
Anotherproofoftheweakfactorizationtheoremwasgivenindependentlybythefourthauthorin[82].
Themaindierencebetweenthetwoapproachesisthefollowing:inthecurrentpaperweareusingobjectssuchastoricidealsdenedlocallyoneachquasi-elementarypieceofacobordism.
Theblowingupofatoricidealgivesthequasi-elementarycobordismatoroidalstructure.
Thesetoroidalmodicationsarethenpiecedtogetherusingcanonicalresolutionofsingularities.
Incontrast,in[82]oneworksglobally:anewcombinatorialtheoryofstratiedtoroidalvarietiesandappropriatemorphismsbetweenthemisdeveloped,whichallowsonetoapplyMorelli'sπ-desingularizationalgorithmdirectlytotheentirebirationalcobordism.
Thisstratiedtoroidalvarietystructureonthecobordismissomewhereinbetweenournotionsoflocallytoricandtoroidalstructures.
0.
14.
Outlineofthepaper.
Insection1wediscusslocallytoricandtoroidalstructures.
WealsouseeliminationofindeterminaciesofarationalmaptoreducetheproofofTheorem0.
1.
1tothecasewhereφisaprojectivebirationalmorphism.
Supposenowwehaveaprojectivebirationalmorphismφ:X1→X2.
Insection2weapplythetheoryofbirationalcobordismstoobtainaslightlyrenedversionoffactorizationintolocallytoricbirationalmaps,rstprovenin[81].
OurcobordismBisrelativelyprojectiveoverX2,andusingageometricinvarianttheoryanalysis,inspiredbyThaddeus'swork,weshowthattheintermediatevarietiescanbechosentobeprojectiveoverX2.
Insection3weutilizeafactorizationofthecobordismBintoquasi-elementarypiecesBai,andforeachpiececonstructanidealsheafI(Denition3.
1.
4)whoseblowinguptoriestheactionofKonBai(Proposition3.
2.
5).
Inotherwords,KactstoroidallyonthevarietyobtainedbyblowingupBaialongI.
Insection4weprovetheweakfactorizationtheorembyputtingtogetherthetoroidalbirationalmapsobtainedfromthetoricationofthequasi-elementarycobordisms(Proposition4.
2.
1),andapplyingtoroidalweakfactorization.
Themaintoolinthisstepiscanonicalresolutionofsingularities.
Insection5weproveTheorem0.
3.
1.
Wethendiscusssomeproblemsrelatedtostrongfactorizationinsection6.
1.
Preliminaries1.
1.
Quotients.
Weusethefollowingdenitionsforquotients.
Supposeareduc-tivegroupGactsonanalgebraicvarietyX.
WedenotebyX/Gthespaceoforbits,538D.
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WLODARCZYKandbyX//Gthespaceofequivalenceclassesoforbits,wheretheequivalencere-lationisgeneratedbytheconditionthattwoorbitsareequivalentiftheirclosuresintersect;suchaspaceisendowedwithaschemestructurewhichsatisestheusualuniversalproperty,ifsuchastructureexists.
Insuchacase,thespaceX//GiscalledacategoricalquotientandthespaceX/Giscalledageometricquotient.
AspecialcasewhereX//Gexistsasaschemeisthefollowing:supposethereisananeG-invariantmorphismπ:X→Y.
ThenwehaveX//G=SpecY(πOX)G.
WhenthisconditionholdswesaythattheactionofGonXisrelativelyane.
Aparticularcaseofthisoccursingeometricinvarianttheory(discussedinsection2.
5),wheretheactionofGontheopensetofpointswhicharesemistablewithrespecttoaxedlinearizationisrelativelyane.
1.
2.
Canonicalresolutionofsingularitiesandcanonicalprincipalization.
Inthefollowing(especiallyLemma1.
3.
1,section4.
2,section5),wewillusecanon-icalversionsofHironaka'stheoremsonresolutionofsingularitiesandprincipaliza-tionofanideal,provedin[9,79].
1.
2.
1.
Canonicalresolution.
FollowingHironaka,byacanonicalembeddedresolu-tionofsingularitiesW→WwemeanadesingularizationprocedureuniquelyassociatingtoWacompositeofblowingsupwithnonsingularcenters,satisfyinganumberofconditions.
Inparticular:(1)"Embedded"meansthefollowing:assumethesequenceofblowingsupisappliedwhenWUisaclosedembeddingwithUnonsingular.
DenotebyEitheexceptionaldivisoratsomestageoftheblowingup.
Then(a)Eiisanormalcrossingsdivisor,andhasnormalcrossingswiththecenterofblowingup,and(b)atthelaststageWhasnormalcrossingswithEi.
(2)"Canonical"means"functorialwithrespecttosmoothmorphismsandeldextensions",namely,ifθ:V→Wiseitherasmoothmorphismoraeldextension,thentheformationoftheidealsblownupcommuteswithpullingbackbyθ;henceθcanbeliftedtoasmoothmorphismθ:V→W.
Inparticular:(a)ifθ:W→Wisanautomorphism(ofschemes,notnecessarilyoverK),thenitcanbeliftedtoanautomorphismW→W,and(b)thecanonicalresolutionbehaveswellwithrespecttoetalemorphisms:ifV→Wisetale,wegetanetalemorphismofcanonicalresolutionsV→W.
AnimportantconsequenceoftheseconditionsisthatallthecentersofblowinguplieoverthesingularlocusofW.
WenotethattheresolutionprocessesintheworkofBierstoneandMilmanandofVillamayorcommutewitharbitraryformallysmoothmorphisms(inparticularsmoothmorphisms,eldextensions,andformalcompletions),thoughthetreatmentinanyofthepublishedworksdoesnotseemtostatethatexplicitly.
1.
2.
2.
Compatibilitywithanormalcrossingsdivisor.
IfWUisembeddedinanonsingularvariety,andDUisanormalcrossingsdivisor,thenavariantoftheresolutionprocedureallowsonetochoosethecentersofblowinguptohavenormalcrossingswithDi+Ei,whereDiistheinverseimageofD.
Thisfollowssincetheresolutionsetup,asin[9],allowsincludingsuchadivisorin"year0".
1.
2.
3.
Principalization.
Bycanonicalprincipalizationofanidealsheafinanonsin-gularvarietywemean"thecanonicalembeddedresolutionofsingularitiesofthesubschemedenedbytheidealsheafmakingitadivisorwithnormalcrossings";TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS539i.
e.
,acompositeofblowingsupwithnonsingularcenterssuchthatthetotaltrans-formoftheidealisadivisorwithsimplenormalcrossings.
Canonicalembeddedresolutionofsingularitiesofanarbitrarysubscheme,notnecessarilyreducedorirre-ducible,isdiscussedinsection11of[9],andthisimpliescanonicalprincipalization,asonesimplyneedstoblowupWatthelaststep.
1.
2.
4.
Eliminationofindeterminacies.
Nowletφ:W1W2beabirationalmapandUW1anopensetonwhichφrestrictstoamorphism.
Byeliminationofindeterminaciesofφwemeanamorphisme:W1→W1,obtainedbyasequenceofblowingsupwithnonsingularcentersdisjointfromU,suchthatthebirationalmapφeisamorphism.
Eliminationofindeterminaciescanbereducedtoprincipalizationofanidealsheaf:ifoneisgivenanidealsheafIonW1withblowingupW1=BlI(W1)suchthatthebirationalmapW1→W2isamorphism,andifW1→W1istheresultofprincipalizationofI,thenthebirationalmapW1→W1isamorphism,thereforethesameistrueforW1→W2.
IfthesupportoftheidealIisdisjointfromtheopensetUwhereφisanmorphism,thenthecentersofblowingupgivingW1→W1aredisjointfromU.
ProvingthatsuchanidealIexists(say,inthenonprojectivecase),andinasucientlynaturalmannerforprovingfunctoriality(evenifWiareprojective),isnontrivial.
WemakeuseofHironaka'sversionofChow'slemma,asfollows.
Wemayassumethatφ1isamorphism;otherwisewereplaceW2bytheclosureofthegraphofφ.
NowweuseChow'slemma,provenbyHironakaingeneralin[31],Corollary2,p.
504,asaconsequenceofhisatteningprocedure:thereexistsanidealsheafIonW1suchthattheblowingupofW1alongIfactorsthroughW2.
HencethecanonicalprincipalizationofIalsofactorsthroughW2.
AlthoughitisnotexplicitlystatedbyHironaka,theidealIistheunitidealinthecomplementoftheopensetU:theblowingupofIconsistsofasequenceofpermissibleblowingsup([31],Denition4.
4.
3,p.
537),eachofwhichissupportedinthecomplementofU.
AnotherimportantfactisthattheidealIisinvariant,namely,itisfunctorialunderabsoluteisomorphisms:ifφ:W1W2isanotherproperbirationalmap,withcorrespondingidealI,andθi:Wi→Wiareisomor-phismssuchthatφθ1=θ2φ,thenθ1I=I.
ThisfollowssimplybecauseatnopointinHironaka'satteningprocedureisthereaneedforanychoice.
ItmustbepointedoutthatHironaka'satteningprocedure,andthereforethechoiceoftheidealI,doesnotcommutewithsmoothmorphismsingeneral—infactHironakagivesanexamplewhereitdoesnotcommutewithlocalization.
Thesameresultsholdforanalyticandalgebraicspaces.
WhileHironakastateshisresultonlyintheanalyticsetting,theargumentsholdinthealgebraicsettingaswell.
See[65]foranearliertreatmentofthecaseofvarieties.
WeemphasizeagainthatChow'slemmaintheanalyticsetting,anditsdelicatepropertiesinboththealgebraicandanalyticsettings,relyonHironaka'sdicultatteningtheorem(see[31],orthealgebraiccounterpart[65]).
1.
3.
Reductiontoprojectivemorphisms.
Westartwithabirationalmapφ:X1X2betweencompletenonsingularalgebraicvarietiesX1andX2denedoverKandrestrictingtoanisomorphismonanopensetU.
540D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKLemma1.
3.
1(Hironaka).
ThereisacommutativediagramX1φ→X2g1↓↓g2X1φX2suchthatg1andg2arecompositesofblowingsupwithnonsingularcentersdisjointfromU,andφisaprojectivebirationalmorphism.
Proof.
ByHironaka'stheoremoneliminationofindeterminacies(see1.
2.
4above),thereisamorphismg2:X2→X2whichisacompositeofblowingsupwithnonsingularcentersdisjointfromU,suchthatthebirationalmaph:=φ1g2:X2→X1isamorphism:X2h↓g2X1φX2Bythesametheorem,thereisamorphismg1:X1→X1whichisacompositeofblowingsupwithnonsingularcentersdisjointfromU,suchthatφ:=h1g1:X1→X2isamorphism.
Sincethecompositehφ=g1isprojective,itfollowsthatφisprojective.
ThuswemayreplaceX1X2byX1→X2andassumefromnowonthatφisaprojectivemorphism.
Notethat,bythepropertiesofcanonicalprincipalizationandHironaka'sat-tening,theformationofφ:X1→X2isfunctorialunderabsoluteisomorphisms,andtheblowingsuphavenormalcrossingswiththeappropriatedivisors.
ThiswillbeusedintheproofofTheorem0.
3.
1(seesection5).
1.
4.
Toricvarieties.
LetN=ZnbealatticeandσNRastrictlyconvexrationalpolyhedralcone.
WedenotetheduallatticebyMandthedualconebyσ∨MR.
TheanetoricvarietyX=X(N,σ)isdenedasX=SpecK[M∩σ∨].
Form∈M∩σ∨wedenoteitsimageinthesemigroupalgebraK[M∩σ∨]byzm.
Moregenerally,thetoricvarietycorrespondingtoafanΣinNRisdenotedbyX(N,Σ);see[26],[62].
IfX1=X(N,Σ1)andX2=X(N,Σ2)aretwotoricvarieties,theembeddingsofthetorusT=SpecK[M]inbothofthemdeneatoric(i.
e.
,T-equivariant)birationalmapX1X2.
SupposeKactseectivelyonananetoricvarietyX=X(N,σ)asaone-parametersubgroupofthetorusT,correspondingtoaprimitivelatticepointa∈N.
Ift∈Kandm∈M,theactiononthemonomialzmisgivenbyt(zm)=t(a,m)·zm,where(·,·)isthenaturalpairingonN*M.
TheK-invariantmonomialscorre-spondtothelatticepointsM∩a⊥,henceX//K=SpecK[M∩σ∨∩a⊥].
Ifa/∈±σ,thenσ∨∩a⊥isafull-dimensionalconeina⊥,anditfollowsthatX//Kisagainananetoricvariety,denedbythelatticeπ(N)andconeπ(σ),whereTORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS541π:NR→NR/R·aistheprojection.
Thisquotientisageometricquotientpreciselywhenπ:σ→π(σ)isabijection.
1.
5.
Locallytoricandtoroidalstructures.
Thereissomeconfusionintheliteraturebetweenthenotionoftoroidalembeddingsandtoroidalmorphisms([42],[3])andthatoftoroidalvarieties(see[20]),whichweprefertocalllocallytoricvarieties.
Acrucialissueinthispaperisthedistinctionbetweenthetwonotions.
Denition1.
5.
1.
(1)AvarietyWislocallytoricifforeveryclosedpointp∈WthereexistsanopenneighborhoodVpWofpandanetalemorphismηp:Vp→XptoatoricvarietyXp.
Suchamorphismηpiscalledatoricchartatp.
(2)AnopenembeddingUWisatoroidalembeddingifforeveryclosedpointp∈Wthereexistsatoricchartηp:Vp→XpatpsuchthatU∩Vp=η1p(T),whereTXpisthetorus.
Wecallsuchchartstoroidal.
SometimesweomittheopensetUfromthenotationandsimplysaythatavarietyistoroidal.
(3)Wesaythatalocallytoric(respectively,toroidal)chartonavarietyiscom-patiblewithadivisorDWifη1p(T)∩D=,i.
e.
,DcorrespondstoatoricdivisoronXp.
AtoroidalembeddingUXcanequivalentlybespeciedbythepair(X,DX),whereDXisthereducedWeildivisorsupportedonXU.
WewillsometimesinterchangebetweenUXand(X,DX)fordenotingatoroidalstructureonX.
AdivisorDiscompatiblewiththetoroidalstructure(X,DX)ifitissupportedinDX.
Forexample,theanelineA1isclearlylocallytoric,A1{0}A1isatoroidalembedding,andA1A1isadierenttoroidalembedding,whereachartatthepoint0canbeobtainedbytranslationfromthepoint1.
Toroidalembeddingscanbenaturallymadeintoacategory:Denition1.
5.
2.
LetUiWi(i=1,2)betoroidalembeddings.
Aproperbirationalmorphismf:W1→W2issaidtobetoroidalif,foreveryclosedpointq∈W2andanyp∈f1q,thereisadiagramofbersquaresXp←VpW1φ↓↓↓fXq←VqW2whereηp:Vp→Xpisatoroidalchartatp,ηq:Vq→Xqisatoroidalchartatq,andφ:Xp→Xqisatoricmorphism.
Remarks.
(1)Atoroidalembeddingasdenedaboveisatoroidalembeddingwithoutself-intersectionaccordingtothedenitionin[42],andabirationaltoroidalmorphismsatisestheconditionofallowabilityin[42].
(2)Toatoroidalembedding(UWW)onecanassociateapolyhedralcomplexW,suchthatproperbirationaltoroidalmorphismstoW,uptoisomor-phisms,areinone-to-onecorrespondencewithcertainsubdivisionsofthecomplex(see[42]).
ItfollowsfromthisthatthecompositionoftwoproperbirationaltoroidalmorphismsW1→W2andW2→W3isagaintoroidal:therstmorphismcorrespondstoasubdivisionofW2,thesecondonetoa542D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKsubdivisionofW3,hencetheircompositionistheuniquetoroidalmorphismcorrespondingtothesubdivisionW1ofW3.
(3)Someofthemanyissuessurroundingthesedenitionsweavoideddiscussinghereareaddressedinthethirdauthor'slecturenotes[53].
Wenowturntobirationalmaps:Denition1.
5.
3([30],[35]).
Letψ:W1W2bearationalmapdenedonadenseopensubsetU.
DenotebyΓψtheclosureofthegraphofψUinW1*W2.
WesaythatψisproperiftheprojectionsΓψ→W1andΓψ→W2arebothproper.
Denition1.
5.
4.
LetUiWibetoroidalembeddings.
Aproperbirationalmapψ:W1W2issaidtobetoroidalifthereexistsatoroidalembeddingUZZandacommutativediagramZW1ψW2whereZ→Wi(i=1,2)areproperbirationaltoroidalmorphisms.
Inparticular,aproperbirationaltoroidalmapinducesanisomorphismbetweentheopensetsU1andU2.
Remarks.
(1)Itfollowsfromthecorrespondencebetweenproperbirationalto-roidalmorphismsandsubdivisionsofpolyhedralcomplexesthatthecom-positionoftoroidalbirationalmapsgivenbyW1←Z1→W2andW2←Z2→W3isagaintoroidal.
Indeed,ifZ1→W2andZ2→W2correspondtotwosubdivisionsofW2,thenacommonrenementofthetwosubdivisionscorrespondstoatoroidalembeddingZsuchthatZ→Z1andZ→Z2aretoroidalmorphisms.
Forexample,thecoarsestrenementcorrespondstotakingforZthenormalizationoftheclosureofthegraphofthebirationalmapZ1Z2.
ThecompositemapsZ→Wiarealltoroidalbirationalmorphisms.
(2)ItcanbeshownthatamorphismbetweentoroidalembeddingswhichisatoroidalbirationalmapinthesenseofDenition1.
5.
4isatoroidalmorphisminthesenseofDenition1.
5.
2.
Inotherwords,Denitions1.
5.
2and1.
5.
4arecompatible.
Forlocallytoricvarieties,therearenosatisfactoryanaloguesofthedenitionsoftoroidalmorphismsandbirationalmaps.
Onecandenea"locallytoricmorphism"tobeonewhichistoriconsuitabletoriccharts,butthisnotionisneitherstableundercompositionnoramenabletocombinatorialmanipulations.
Anextensiveandquitedelicatetheoryinvolvingstraticationsoflocallytoricvarietiesisdevelopedin[82]inordertoresolvethisissue.
Hereweuseadierentremedy.
Wedenearestrictiveclassofbirationaltransformationsbetweenlocallytoricandtoroidalvarieties,inwhichallchartsare"uniform"overacommonbaseY.
Thesearestillnotstableundercomposition,buttheirlocalcombinatorialnaturesucesforourgoals.
Thesearetheonlytransformationswewillneedintheconsiderationsofthecurrentpaper.
TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS543Denition1.
5.
5.
(1)Atightlylocallytoricbirationaltransformationisaproperbirationalmapψ:W1W2togetherwithadiagramofbirationalmapsW1ψW2YbetweenlocallytoricvarietiesW1andW2satisfyingthefollowingcondition:Foreveryclosedpointq∈Ythereexistatoricchartηq:Vq→Xqatq,andadiagramofberedsquaresW1→Y←W2∪∪∪V1→Vq←V2↓↓↓X1→Xq←X2suchthat(a)Vi→XiaretoricchartsforWi,i=1,2,and(b)Xi→Xqaretoricmorphisms(2)Analogously,letUiWibetoroidalembeddings.
Atightlytoroidalbi-rationaltransformationbetweenthemisatightlylocallytoricbirationaltransformationψ:W1W2wherethetoricchartsabovecanbechosentobetoroidal.
Remark.
Whiletightlylocallytoricbirationaltransformationsareessentialinourarguments,tightlytoroidaltransformationsarenot:theargumentusedbeforetoshowthatacompositionoftoroidalbirationalmapsistoroidalshowsthatatightlytoroidalbirationaltransformationgivesatoroidalbirationalmap.
Thisistheonlypropertyofsuchtransformationswewilluse.
1.
6.
Weakfactorizationfortoroidalbirationalmaps.
Theweakfactoriza-tiontheoremforproperbirationaltoricmapscanbeextendedtothecaseofproperbirationaltoroidalmaps.
Thisisprovedin[4]fortoroidalmorphisms,usingthecorrespondencebetweenbirationaltoroidalmorphismsandsubdivisionsofpolyhe-dralcomplexes.
ThegeneralcaseofatoroidalbirationalmapW1←Z→W2canbededucedfromthis,asfollows.
BytoroidalresolutionofsingularitieswemayassumeZisnonsingular.
WeapplytoroidalweakfactorizationtothemorphismsZ→Wi,togetasequenceoftoroidalbirationalmapsW1=V1V2···Vl1Vl=ZVl+1···Vk1Vk=W2consistingoftoroidalblowingsupanddownwithnonsingularcenters.
Westatethisresultforlaterreference:Theorem1.
6.
1.
LetU1W1andU2W2benonsingulartoroidalembeddings.
Letψ:W1W2beapropertoroidalbirationalmap.
Thenφcanbefactoredintoasequenceoftoroidalbirationalmapsconsistingoftoroidalblowingsupanddownofnonsingularcentersinnonsingulartoroidalembeddings.
Thisdoesnotimmediatelyimplythatonecanchooseafactorizationsatisfyingaprojectivitystatementasinthemaintheorem,orinafunctorialmanner.
Wewillshowthesefactsinsections2.
7and5,respectively.
Itshouldbementionedthatiftoricstrongfactorizationistrue,thenthetoroidalcasefollows.
544D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYK1.
7.
Locallytoricandtoroidalactions.
Denition1.
7.
1(see[60],p.
198).
LetVandXbevarietieswithrelativelyaneK-actions,andletη:V→XbeaK-equivariantetalemorphism.
Thenηissaidtobestronglyetaleif(i)thequotientmapV//K→X//Kisetale,and(ii)thenaturalmapV→X*X//KV//Kisanisomorphism.
Denition1.
7.
2.
(1)LetWbealocallytoricvarietywithaK-action,suchthatW//Kexists.
Wesaythattheactionislocallytoricifforanyclosedpointp∈Wwehaveatoricchartηp:Vp→Xpatpandaone-parametersubgroupKTpofthetorusinXp,satisfyingVp=π1πVp,whereπ:W→W//Kistheprojection;ηpisK-equivariantandstronglyetale.
(2)IfUWisatoroidalembedding,wesaythatKactstoroidallyonWifthechartsabovecanbechosentoroidal.
Thedenitionaboveisequivalenttotheexistenceofthefollowingdiagramofbersquares:Xp←VpW↓↓↓fX//K←Vp//KW//Kwherethehorizontalmapsprovidetoric(resp.
toroidal)chartsinWandW//K.
Itfollowsthatthequotientofalocallytoricvarietybyalocallytoricactionisagainlocallytoric;thesameholdsinthetoroidalcase.
Remark.
Ifwedonotinsistonthechartsbeingstronglyetale,thenthemor-phismofquotientsmayfailtobeetale.
Consider,forinstance,thespaceX=SpecK[x,x1,y]withtheactiont(x,y)=(t2x,t1y).
ThequotientisX/K=SpecK[xy2].
ThereisanequivariantetalecoverV=SpecK[u,u1,y]withtheactiont(u,y)=(tu,t1y),wherethemapisdenedbyx=u2.
ThequotientisV/K=SpecK[uy],whichisabranchedcoverofX/K,sincexy2=(uy)2.
ThefollowinglemmashowsthatlocallytoricK-actionsareubiquitous.
Wenotethatitcanbeprovenwithfewerassumptions;see[81],[53].
Lemma1.
7.
3.
LetWbeanonsingularvarietywitharelativelyaneK-action,thatis,theschemeW//KexistsandthemorphismW→W//Kisananemorphism.
ThentheactionofKonWislocallytoric.
Proof.
TakingananeopensetinW//K,wemayassumethatWisane.
WeembedWequivariantlyintoaprojectivespaceandtakeitscompletion(see,e.
g.
,[75]).
Afterapplyingequivariantresolutionofsingularitiestothiscompletion(seesection1.
2)wemayalsoassumethatWisanonsingularprojectivevarietywithaK-action,andWWisananeinvariantopensubset.
Letp∈Wbeaclosedpoint.
SinceWiscomplete,theorbitofphasalimitpointq=limt→0t(p)inW.
NowqisxedbyK,henceKactsonthecotangentspacemq/m2qatq.
SinceKisreductive,wecanliftasetofeigenvectorsofthisTORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS545actiontosemi-invariantlocalparametersx1,xnatq.
TheselocalparametersdeneaK-equivariantetalemorphismηq:Vq→XqfromananeKinvariantopenneighborhoodVqofqtothetangentspaceXq=Spec(Symmq/m2q)atq.
Thelatterhasastructureofatoricvariety,wherethetorusisthecomplementofthezerosetofxi.
SeparatingtheparametersxiintoK-invariantsandnoninvariants,wegetafactorizationXq=X0q*X1q,wheretheactionofKonX1qistrivialandtheactiononX0qhas0asitsuniquexedpoint.
ThuswegetaproductdecompositionXq//K=X0q//K*X1q.
ByLuna'sFundamentalLemma([51],Lemme3),thereexistaneK-invariantneighborhoodsVqofqandXqof0,suchthattherestrictionηq:Vq→Xqisstronglyetale.
Considerrstthecaseq∈W,inwhichcasewemayreplacepbyq.
DenoteZ=XKq∩Xq.
ThenZXKqX1qisaneopen,and,usingthedirectproductdecompositionabove,X0q*ZXqisaneopen.
DenoteXq=Xq∩X0q*Z.
ThisisaneopeninXq,anditiseasytoseethatXq//K→Xq//Kisanopenembedding:anorbitinXqisclosedifandonlyifitisclosedinXq.
WritingVq=ηq1Xq,itfollowsthatVq→Xqisastronglyetaletoricchart.
Inthecaseq/∈W,replaceVqbyVq.
Nowηqisinjectiveonanyorbit,andthereforeitisinjectiveontheorbitofp.
LetXpXqbetheaneopentoricsubvarietyinwhichthetorusorbitofηq(p)isclosed,andletVp=η1qXp∩W.
Nowconsidertherestrictionη:Vp→Xp,wheretheK-orbitsofpandη(p)areclosed.
ByLuna'sFundamentalLemmathereexistaneopenK-invariantneighborhoodsVpVpandXpXpofηp(p)suchthattherestrictionη:Vp→Xpisastronglyetalemorphism.
SinceXp/Kisageometricquotient,wehaveanopenembeddingXp/KXp/KandwehaveastronglyetaletoricchartVp→Xp.
Itremainstoshowthatthechartscanbechosensaturatedwithrespecttotheprojectionπ:W→W//K.
Iftheorbitofphasalimitpointq=limt→0t·porq=limt→∞t·pinW,whichisnecessarilyuniqueasπisane,thenanequivarianttoricchartatqalsocoversp.
Sowemayreplacepbyqandassumethattheorbitofpisclosed.
Nowπ(WVp)isclosedanddoesnotcontainπ(p),sowecanchooseananeneighborhoodYinitscomplement,andreplaceVpbyπ1Y.
2.
Birationalcobordisms2.
1.
Denitions.
Denition2.
1.
1([81]).
Letφ:X1X2beabirationalmapbetweentwoalgebraicvarietiesX1andX2overK,isomorphiconanopensetU.
AnormalalgebraicvarietyBiscalledabirationalcobordismforφanddenotedbyBφ(X1,X2)ifitsatisesthefollowingconditions:(1)ThemultiplicativegroupKactseectivelyonB=Bφ(X1,X2).
(2)ThesetsB:={x∈B:limt→0t(x)doesnotexistinB}andB+:={x∈B:limt→∞t(x)doesnotexistinB}arenonemptyZariskiopensubsetsofB.
(3)ThereareisomorphismsB/K→X1andB+/K→X2.
546D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYK(4)Consideringtherationalmapψ:BB+inducedbytheinclusions(B∩B+)Band(B∩B+)B+,thefollowingdiagramcommutes:BψB+↓↓X1φX2WesaythatBrespectstheopensetUifUiscontainedintheimageof(B∩B+)/K.
Denition2.
1.
2([81]).
LetB=Bφ(X1,X2)beabirationalcobordism,andletFBKbeasubsetofthexed-pointset.
WedeneF+={x∈B|limt→0t(x)∈F},F={x∈B|limt→∞t(x)∈F},F±=F+∪F.
Denition2.
1.
3([81]).
LetB=Bφ(X1,X2)beabirationalcobordism.
Wede-nearelationamongconnectedcomponentsofBKasfollows:letF1,F2BKbetwoconnectedcomponents,andsetF1F2ifthereisapointx/∈BKsuchthatlimt→0t(x)∈F1andlimt→∞t(x)∈F2.
Denition2.
1.
4.
AbirationalcobordismB=Bφ(X1,X2)issaidtobequasi-elementaryifanytwoconnectedcomponentsF1,F2BKareincomparablewithrespectto.
Notethatthisconditionprohibits,inparticular,theexistenceofa"loop",namelyaconnectedcomponentFandapointy/∈Fsuchthatbothlimt→0t(x)∈Fandlimt→∞t(x)∈F.
Denition2.
1.
5([81]).
Aquasi-elementarycobordismBissaidtobeelementaryifthexedpointsetBKisconnected.
Denition2.
1.
6(cf.
[56],[81]).
WesaythatabirationalcobordismB=Bφ(X1,X2)iscollapsibleiftherelationisastrictpre-order,namely,thereisnocyclicchainofxedpointcomponentsF1F2FmF1.
2.
2.
Themainexample.
Wenowrecallafundamentalexampleofanelementarybirationalcobordisminthetoricsetting,discussedin[81]:Example2.
2.
1.
LetB=An=SpecK[z1,zn]andlett∈Kactbyt(z1,zi,zn)=(tα1z1,tαizi,tαnzn).
WeassumeKactseectively,namelygcd(α1,αn)=1.
WeregardAnasatoricvarietydenedbyalatticeN=Znandanonsingularconeσ∈NRgeneratedbythestandardbasisσ=v1,vn.
Thedualconeσ∨isgeneratedbythedualbasisv1vn,andweidentifyzvi=zi.
TheK-actionthencorrespondstoaone-parametersubgroupa=(α1,αn)∈N.
TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS547Weassumethata/∈±σ.
WehavetheobviousdescriptionofthesetsB+andB:B={(z1,zn);zi=0forsomeiwithαi=(vi,a)0}.
Wedenetheupperboundaryandlowerboundaryfansofσtobeσ={x∈σ;x+·a∈σforall>0},+σ={x∈σ;x+·(a)∈σforall>0}.
ThenweobtainthedescriptionofB+andBasthetoricvarietiescorrespondingtothefans+σandσinNR.
Letπ:NR→NR/R·abetheprojection.
ThenB//Kisagainananetoricvarietydenedbythelatticeπ(N)andconeπ(σ).
Similarly,onecancheckthatthegeometricquotientsB/KandB+/Karetoricvarietiesdenedbyfansπ(+σ)andπ(σ).
Sincebothπ(+σ)andπ(σ)aresubdivisionsofπ(σ),wegetadiagramofbirationaltoricmapsB/KB+/KB//KItiseasytosee(see,e.
g.
,[81])thatthevarietiesB±/Khaveonlyabelianquotientsingularities.
Moreover,themapφcanbefactoredasaweightedblowingupfollowedbyaweightedblowingdown.
Moregenerally,onecanprovethatifΣisasubdivisionofaconvexpolyhedralconeinNRwithlowerboundaryΣandupperboundary+Σrelativetoanelementa∈N±Σ,thenthetoricvarietycorrespondingtoΣ,withtheK-actiongivenbytheone-parametersubgroupa∈N,isabirationalcobordismbetweenthetwotoricvarietiescorrespondingtoπ(Σ)andπ(+Σ)asfansinNR/R·a.
Forthedetails,wereferthereaderto[56],[81]and[4].
2.
3.
Constructionofacobordism.
Itwasshownin[81]thatbirationalcobor-dismsexistforalargeclassofbirationalmapsX1X2.
Herewedealwithaveryspecialcase.
Theorem2.
3.
1.
Letφ:X1→X2beaprojectivebirationalmorphismbetweencompletenonsingularalgebraicvarieties,whichisanisomorphismonanopensetU.
ThenthereisacompletenonsingularalgebraicvarietyBwithaneectiveK-action,satisfyingthefollowingproperties:(1)Thereexistclosedembeddingsι1:X1→BKandι2:X2→BKwithdisjointimages.
(2)TheopensubvarietyB=B(ι1(X1)∪ι2(X2))isabirationalcobordismbetweenX1andX2respectingtheopensetU.
(3)ThereisacoherentsheafEonX2,withaK-action,andaclosedK-equivariantembeddingBP(E):=ProjX2SymE.
Proof.
LetJOX2beanidealsheafsuchthatφ:X1→X2istheblowingupmorphismofX2alongJandJU=OU.
LetI0betheidealofthepoint0∈P1.
ConsiderW0=X2*P1andletp:W0→X2andq:W0→P1betheprojections.
LetI=(p1J+q1I0)OW0.
LetWbetheblowingupofW0alongI.
(PaoloAluhaspointedoutthatthisWisusedwhenconstructingthedeformationtothenormalconeofJ.
)548D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKWeclaimthatX1andX2lieinthenonsingularlocusofW.
ForX2=X2*{∞}X2*A1Wthisisclear.
SinceX1isnonsingular,embeddedinWasthestricttransformofX2*{0}X2*P1,toprovethatX1liesinthenonsingularlocusitsucestoprovethatX1isaCartierdivisorinW.
Welookatlocalcoordinates.
LetA=Γ(V,OV)forsomeaneopensubsetVX2,andlety1,ymbeasetofgeneratorsofJonV.
ThenontheaneopensubsetV*A1X2*P1withcoordinateringA[x],theidealIisgeneratedbyy1,ym,x.
Thechartsoftheblowingupcontainingthestricttransformof{x=0}areoftheformSpecAy1yiymyi,xyi=SpecAy1yiymyi*SpecKxyi,whereKactsonthesecondfactor.
Thestricttransformof{x=0}isdenedbyxyi,henceitisCartier.
LetB→Wbeacanonicalresolutionofsingularities.
Thenconditions(1)and(2)areclearlysatised.
Forcondition(3),notethatB→X2*P1,beingacompositionofblowingsupofinvariantideals,admitsanequivariantamplelinebundle.
TwistingbythepullbackofOP1(n)weobtainanequivariantlinebundlewhichisampleforB→X2.
ReplacingthisbyasucientlyhighpowerandpushingforwardwegetE.
Wereferthereaderto[81]formoredetails.
WecallavarietyBasinthetheoremacompactied,relativelyprojectivecobor-dism.
2.
4.
Collapsibilityandprojectivity.
LetB=Bφ(X1,X2)beabirationalcobor-dism.
WeseekacriterionforcollapsibilityofB.
LetCbethesetofconnectedcomponentsofBφ(X1,X2)K,andletχ:C→Zbeafunction.
WesaythatχisstrictlyincreasingifFFχ(F)ai}).
ThefollowingisanimmediateextensionofProposition1of[81].
Proposition2.
4.
3.
(1)Baiisaquasi-elementarycobordism.
(2)Fori=1,m1wehave(Bai)+=(Bai+1).
TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS549ThefollowingisananalogueofLemma1of[81]inthecaseofthecobordismswehaveconstructed.
Proposition2.
4.
4.
LetEbeacoherentsheafonX2withaK-action,andletBP(E)beacompactied,relativelyprojectivecobordismembeddedK-equi-variantly.
ThenthereexistsastrictlyincreasingfunctionχforthecobordismB=B(X1∪X2).
Inparticular,thecobordismiscollapsible.
Proof.
SinceKactstriviallyonX2,andsinceKisreductive,thereexistsadirectsumdecompositionE=b∈ZEbwhereEbisthesubsheafonwhichtheactionofKisgivenbythecharactert→tb.
Denotebyb0,bkthecharacterswhichgureinthisrepresentation.
NotethattherearedisjointembeddingsP(Ebj)P(E).
Letp∈BbeaxedpointlyingintheberP(Eq)overq∈X2.
Wechooseabasis(xb0,1,xb0,d0xbk,1,xbk,dk)ofEqwherexbj,ν∈Ebjandusethefollowinglemma:Lemma2.
4.
5.
Supposep∈P(Eq)Kisaxedpointwithhomogeneouscoordinates(pb0,1,pb0,d0pbk,1,pbk,dk).
Thenthereisajpsuchthatpbj,ν=0wheneverj=jp.
Inparticular,p∈P(Ebjp)P(E).
IfFBKisaconnectedcomponentofthexedpointset,thenitfollowsfromthelemmathatFP(Ebj)forsomej.
Wedeneχ(F)=bj.
Tocheckthatχisstrictlyincreasing,considerapointp∈Bsuchthatlimt→0t(p)∈F1andlimt→∞t(p)∈F2forsomexedpointcomponentsF1andF2.
Letthecoordinatesofpintheberoverq∈X2be(pb0,1,pb0,d0pbk,1,pbk,dk).
Nowlimt→0t(p)∈P(Ebmin),limt→∞t(p)∈P(Ebmax),wherebmin=min{bj:pbj,ν=0forsomeν},bmax=max{bj:pbj,ν=0forsomeν}.
Thus,ifpisnotxedbyK,thenχ(F1)=bminai+1(thisisatechnicalconditionwhichcomesinhandyinwhatfollows).
Denotebyρ0(t)theactionoft∈KonE.
Foranyr∈Zconsiderthe"twisted"actionρr(t)=tr·ρ0(t).
NotethattheinducedactiononP(E)doesnotdependonthe"twist"r.
ConsideringthedecompositionE=Ebj,weseethatρr(t)actsonEbjbymultiplicationbytbjr.
Wecanapplygeometricinvarianttheoryinitsrelativeform(see,e.
g.
,[63],[33])totheactionρr(t)ofK.
Recallthatapointp∈P(E)issaidtobesemistablewithrespecttoρr,writtenp∈(P(E),ρr)ss,ifthereisapositiveintegernandaρr-invariantlocalsections∈(Symn(E))ρr,suchthats(p)=0.
ThemainresultofgeometricinvarianttheoryimpliesthatProjX2∞n≥0(Symn(E))ρr=(P(E),ρr)ss//K;moreover,thequotientmap(P(E),ρr)ss→(P(E),ρr)ss//Kisane.
Wecandene(B,ρr)ssanalogously,andweautomaticallyhave(B,ρr)ss=B∩(P(E),ρr)ss.
Thenumericalcriterionofsemistability(see[60])immediatelyimpliesthefol-lowing:Lemma2.
5.
1.
For0≤i≤mwehave(1)(B,ρai)ss=Bai.
(2)(B,ρai+1)ss=(Bai)+.
(3)(B,ρai1)ss=(Bai).
Inotherwords,thetriangleofbirationalmaps(Bai)/Ki(Bai)+/KBai//KisinducedbyachangeoflinearizationoftheactionofK.
Inparticularweobtain:Proposition2.
5.
2.
Themorphisms(Bai)+/K→X2,(Bai)/K→X2andBai//K→X2areprojective.
2.
6.
Themainresultof[81].
LetBbeacollapsiblenonsingularbirationalcobor-dism.
ThenwecanwriteBasaunionofquasi-elementarycobordismsB=iBai,with(Bai)+=(Bai+1).
ByLemma1.
7.
3eachBaihasalocallytoricstructuresuchthattheactionofKislocallytoric.
Lemma2.
6.
1.
LetBaibeaquasi-elementarycobordism,witharelativelyanelocallytoricK-action.
ThenBai//K,(Bai)/K,(Bai)+/KarelocallytoricTORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS551varietiesandwehaveadiagramoflocallytoricmaps(Bai)/Ki(Bai)+/KBai//Kwhereiisatightlylocallytoricbirationaltransformation.
IncaseBaiisnonsingular,thediagramabovecanbedescribedintoricchartsbythemainexampleinsection2.
2.
IftheactionofKonBaiistoroidal,thenallthesevarietiesandmapsarealsotoroidal,andiisatoroidalbirationalmap.
Proof.
Letηp:Vp→XpbeastronglyetaleK-equivarianttoricchartinBaigivingalocallytoricstructuretotheactionofK.
Then(Vp)=(Bai)∩Vpandthemorphism(Vp)→(Xp)isagainstronglyetale,providinglocallytoricstructuresonthevariety(Bai)/Kandthemorphism(Bai)/K→Bai//K.
Similarlyfor(Bai)+.
NowweassumeBBisopeninacompactied,relativelyprojectivecobordism.
WhenwecomposethebirationaltransformationsobtainedfromeachBaiwegetaslightrenementofthemainresultof[81].
Theorem2.
6.
2.
Letφ:X1X2beabirationalmapbetweencompletenonsin-gularalgebraicvarietiesX1andX2overanalgebraicallyclosedeldKofchar-acteristiczero,andletUX1beanopensetwhereφisanisomorphism.
ThenthereexistsasequenceofbirationalmapsbetweencompletelocallytoricalgebraicvarietiesX1=W01W12···i1Wi1iWii+1···m1Wm1mWm=X2where(1)φ=mm121,(2)iareisomorphismsonU,and(3)foreachi,thebirationaltransformationiistightlylocallytoricandetalelocallyequivalenttoamapdescribedin2.
2.
InparticularWihaveniteabelianquotientsingularities,andicanbeobtainedasaweightedblowingupfollowedbyaweightedblowingdown.
Furthermore,thereisanindexi0suchthatforalli≤i0themapWiX1isaprojectivemorphism,andforalli≥i0themapWiX2isaprojectivemorphism.
Inparticular,ifX1andX2areprojective,thenalltheWiareprojective.
Remark.
Fortheprojectivityclaim(2),wetakethersti0termsinthefactorizationtocomefromHironaka'seliminationofindeterminaciesinLemma1.
3.
1,whichisprojectiveoverX1,whereasthelasttermscomefromB,whichisprojectiveoverX2,andthegeometricinvarianttheoryconsiderationsasinProposition2.
5.
2.
2.
7.
Projectivityoftoroidalweakfactorization.
ThefollowingisarenementofTheorem1.
6.
1,inwhichaprojectivitystatementisadded:Theorem2.
7.
1.
LetU1W1andU2W2benonsingulartoroidalembeddings.
Letψ:W1W2beapropertoroidalbirationalmap.
Thenφcanbefactoredintoasequenceoftoroidalbirationalmapsconsistingoftoroidalblowingsupanddownofnonsingularcenters,namely:W1=V01V12···i1Vi1iVii+1···l1Vl1lVl=W2552D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKwhere(1)φ=ll121,(2)iareisomorphismsonU,theembeddingsUViaretoroidal,andiaretoroidalbirationalmaps,and(3)eitheri:Vi1Vior1i:ViVi1isatoroidalmorphismobtainedbyblowingupanonsingularirreducibletoroidalcenter.
Furthermore,thereisanindexi0suchthatforalli≤i0themapViW1isaprojectivemorphism,andforalli≥i0themapViW2isaprojectivemorphism.
Inparticular,ifW1andW2areprojective,thenalltheViareprojective.
Proof.
Asin[4],Lemma8.
7,wereducetothecasewherethepolyhedralcomplexofW2isembeddableasaquasi-projectivetoricfan2inaspaceNR.
IndeedthatlemmagivesanembeddingpreservingtheQ-structureforthebarycentricsubdivisionofanysimplicialcomplex,andsince2isnonsingularthisembeddingpreservesintegralstructuresaswell.
Afurthersubdivisionensuresthatthefanisquasi-projective.
(WenotethatthisembeddingisintroducedforthesolepurposeofapplyingMorelli'sπ-desingularizationlemmadirectly,ratherthanobservingthattheproofworkswordforwordinthetoroidalcase.
)Asin1.
3.
1wemayassumeW1W2isaprojectivemorphism.
Thusthecom-plex1ofW1isaprojectivesubdivisionof2.
OurconstructionofacompactiedrelativelyprojectivecobordismBforthemorphismφyieldsatoroidalembeddingBwhosecomplexBisaquasi-projectivepolyhedralcobordismlyingin(NZ)Rsuchthatπ(+B)=2andπ(B)=1,whereπistheprojectionontoNR.
Moreover,thetoroidalmorphismB→W2givesapolyhedralmorphismB→2inducedbytheprojectionπ.
Morelli'sπ-desingularizationlemmagivesaprojec-tivesubdivisionB→B,isomorphicontheupperandlowerboundaries±B,suchthatBisπ-nonsingular.
WestillhaveapolyhedralmorphismB→2.
ThecomplexBcorrespondstoatoroidalbirationalcobordismBbetweenW1andW2.
SinceBisπ-nonsingular,anyelementarypieceBFBcorrespondstoatoroidalblowingupfollowedbyatoroidalblowingdownbetweennonsingulartoroidalembeddings,withnonsingularcenters.
Itfollowsthatthesameholdsforeveryquasi-elementarypieceofB(herethecentersmaybereducible,butblowingupareduciblecenteristhecompositionofblowingsupofitsconnectedcompo-nentsoneatatime).
AsinTheorem2.
6.
2above,thesetoroidalembeddingscanbechosentobeprojectiveoverW2.
3.
TorificationWewishtoreplacethelocallytoricfactorizationofTheorem2.
6.
2byatoroidalfactorization.
ThisamountstoreplacingBwithalocallytoricK-actionbysomeBwithatoroidalK-action.
Wecallsuchaproceduretorication.
Thebasicidea,whichgoesbackatleasttoHironaka,isthatifoneblowsupanideal,theexceptionaldivisorsprovidetheresultingvarietywithusefulextrastructure.
Theidealweconstruct,calledatoricideal,iscloselyrelatedtothetoricidealof[2].
3.
1.
Constructionofatoricideal.
LetBbeanormalvarietywitharelativelyaneK-action.
Wedenotebyπ:B→B//Kthequotientmorphism,whichbyassumptionisane.
Considerthequasi-coherentsheafofalgebrasA=πOBtogetherwithaK-actiononit.
ForanintegerαwedenotebyAαAthesubsheafofsemi-invariantTORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS553sectionsf∈AofK-characterα:t(f)=tαf.
Denition3.
1.
1.
TheK-equivariantidealsheafIBαonB,generatedbyAα,iscalledtheα-toricidealsheafoftheactionofK.
WesometimesomitthesuperscriptBandwriteIαifthereisnoriskofconfusion.
LetBbealocallytoric,quasi-elementarycobordismwitharelativelyane,locallytoricK-action;B=Baiforsomeiaccordingtoourpreviousnotation.
Wecontinuetodenotebyπ:B→B//Kthequotientmorphism.
RecallthatbyDenition1.
7.
2ofalocallytoricaction,thebirationalcobordismBiscoveredwithlocallytoricchartsoftheformXpηp←VpB↓↓↓πX//K←Vp//KB//KwithbothsquaresCartesianandthehorizontalmapsetale.
Forachartasabove,letIXpαbetheα-toricidealsheafonXp.
Lemma3.
1.
2.
WehaveIBα|Vp=η1pIXpα.
Moreover,theidealsheafIXpαisgeneratedbymonomialsofK-characterα.
Proof.
Assumethatf∈Aαisregularatp∈B.
ReplacingVpbyasmalleropensetifnecessary,wemayassumethatfisregularonVp.
Wehavef∈O(Vp)=O(Xp)O(Xp//K)O(Vp//K).
NowKactstriviallyonsectionsofOVp//K,hencefliesintheidealgeneratedbypullbacksofsectionsofOXpofK-characterα.
Thesecondstatementisclear.
NotethatthezerofunctionliesineveryAc,anditisconceivablethatsomeIcisthezeroideal.
Thisdoesnothappenforacobordism:Lemma3.
1.
3.
Foranyc∈Z,theidealIBcisnonzero.
Proof.
ByLemma3.
1.
2itsucestoprovethisfortheidealsIXpconthetoricchartsXp.
LetX=X(N,σ)beananetoricvariety,withaneectiveK-actiononXgivenbyaprimitivelatticepointa∈N.
SinceX∩X+isanonemptyopensubset,itfollowsthata/∈±σ.
Thisimpliesthatthesetofpointsm∈M∩σ∨suchthat(m,a)=cisnonempty.
Thusthesetofnonzerof∈OXpofK-characterαisnonempty.
HencetheidealIXcisnonzero.
Assumefurtherthatthelocallytoricquasi-elementarycobordismBisnonsin-gular,coveredwithanitenumberoflocallytoricchartsasabove.
Foreachchartηp:Vp→Xpwechoosemonomialcoordinatesz1,zngeneratingOXp.
LetC={c1,c}beanitesetofintegerscontainingthecharactersofK-actiononthecoordinatesziforallcharts.
LetI=IB=IBc1···IBcbetheproductoftheci-toricideals,andletBtor→BbethenormalizedblowingupofBalongI.
SinceIisK-equivariant,theactionofKliftstoBtor.
DenotebyDBtorthetotaltransformofthezerosetofI,andUBtor=BtorD.
554D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKDenition3.
1.
4.
WecallIBatoricidealandBtor→Batoricblowingup.
ItfollowsthatBtor,beingthenormalizedblowingupoftheproductIc1···Ic,satisesauniversalproperty:itistheminimalnormalmodicationofBsuchthattheinverseimageofIciisprincipalforalli.
ThisimpliesthatBtoriscanonicallyisomorphictothenormalizationofthevarietyobtainedfromBbyrstblowingupIc1,thentheinverseimageofIc2,andsoon.
3.
2.
Thetorifyingpropertyofthetoricideal.
TojustifytheterminologyofDenition3.
1.
4wearegoingtoshowthatUBtorBtorisatoroidalembeddingonwhichKactstoroidally.
ItclearlysucestoprovethisforthetoricvarietiesXtorpobtainedbyblowingupthelocallytoricchartsXpalongmonomialidealsIXp.
Wearethusledtothefollowingproblem:givenatoricvarietyXwithaK-actionandadivisorDXT,whenaretheembeddingXDXandtheK-actiononittoroidalInthissituationwenditusefultokeepinmindthepair(X,D)insteadoftheopenembeddingXDX.
DenotebyDXthereducedWeildivisorXT,andwriteDX=D∪D.
Following[3],section3,wesaythat(X,D)isobtainedbyremovingthedivisorDfromthetoroidalstructure(X,DX).
Thereforethequestionabovecanberephrasedasfollows:whichreducedWeildivisorsDDXcanberemovedfromthetoroidalstructuresothattheresultingpairistoroidal,withtoroidalK-actionExample3.
2.
1.
ConsidertheanelineX=SpecK[x],atoricvarietywiththestandardK-actionx→tx,t∈K,andtoricdivisorD=DX={x=0}.
Thepair(X,D)istoroidalandtheactionofKistoroidal.
Thepair(X,)isalsotoroidal,buttheactiononthispairisnottoroidal.
Example3.
2.
2.
ConsidertheaneplaneX=SpecK[x,y],atoricvarietywithtoricdivisorDX={xy=0}.
ConsidertheK-action(x,y)→(tx,y),t∈K.
IfwedenoteD={x=0},D={y=0},thepair(X,D)istoroidalandtheactionofKistoroidal.
ThusthedivisorDcanberemovedfromthestandardtoroidalstructure(X,DX)keepingtheactiontoroidal.
Westartwithsomecombinatorics.
LetX=X(N,σ)beananetoricvariety.
Ifρσ∨isafaceofσ∨,wesaythatρsplitsofromσ∨withcomplementτifwehaveσ∨=τ*ρ,M=Mτ*Mρ,whereτσ∨isasubcone,notnecessarilyaface,andMτ(resp.
Mρ)isthesublatticeofMgeneratedbyM∩τ(resp.
M∩ρ).
Lemma3.
2.
3.
Letρ1,ρkbeasubsetofthecodimension1facesofσ∨,andletw1,wk∈M∩σ∨.
Thefollowingareequivalent:(1)Foreachi=1,kthefaceρisplitsofromσ∨withcomplementwi:σ∨=wi*ρi,M=Zwi*Mρi.
TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS555(2)w1,wkarelinearlyindependent,generatingaunimodularsublatticeofM,andthefaceρ1∩.
.
.
∩ρksplitsofromσ∨withcomplementw1,wk:σ∨=w1,wk*ρ1ρk,M=Zw1Zwk*Mρ1∩.
.
.
∩ρk.
Proof.
Theimplication(1)(2)istrivial.
Theconversefollowsbyinductiononk.
Onewritesσ∨=w1*ρ1andshowsthatforalli=2,kthefaceρ1∩ρisplitsofromρ1withcomplementwi.
Thegeometriccontentofthelemmaisthefollowing:Lemma3.
2.
4.
LetXbeananetoricvarietywithD1,Dk,E1,Eltheir-reducibletoricdivisorsofX.
AssumethatD1,DkareCartierandletzw1zwkbeasetofmonomialsdeningthesedivisors.
Thefollowingareequivalent:(1)Foreachi=1,kwecanwriteXasaproductoftoricvarietiesX=SpecK[zwi]*Xi=A1*Xi,Di={0}*Xi.
(2)WecanwriteXasaproductoftoricvarietiesX=SpecK[zw1zwk]*X=Ak*X,Di=Di*X,whereDiaretheirreducibletoricdivisorsinAkdenedbyzwi.
Iftheseconditionsaresatised,then(X,iEi)isatoroidalpair,i.
e.
,(XiEi)Xisatoroidalembedding.
If,moreover,KactsonXasasubgroupofthetorusandzwiareinvariantforalli=1,k,thenKactstoroidallyonthisembedding.
Proof.
Theequivalenceofthetwoconditionsfollowsfromthepreviouslemma.
(NotethatbecausezwidenedistinctdivisorsDi,thecomplementaryfacesρiofσ∨aredistinct.
)ForthelaststatementsitsucestocoverX=Ak*XwithchartsoftheformGkm*X.
Notethat(X,iEi)inthelemmaisthetoroidalpairobtainedfromXbyremovingthedivisorsD1,Dkfromthetoroidalstructure(X,(iEi)∪(jDj)).
Wearenowreadytoprovethemainresultofthissection.
RecallthatBisanonsingularquasi-elementarybirationalcobordism,withrelativelyaneK-action,andp:Btor→Bisthetorifyingblowupconstructedinsection3.
1.
Proposition3.
2.
5.
(1)ThevarietyBtorisaquasi-elementarycobordism,with(Btor)+=Btor*BB+and(Btor)=Btor*BB.
(2)TheembeddingUBtorBtoristoroidalandKactstoroidallyonthisem-bedding.
Proof.
LetusrstseethattheactionofKonBtorisrelativelyane,whichalsoimpliesthatBtorisquasi-elementary:otherwisetheclosureofaK-orbitisacompleterationalcurve,whichcannotbecontainedintheberofananemorphism.
556D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKTheidealIisdenedasIc1···Icforsomenitesetofcharacters{c1,c}.
EachIciistheidealgeneratedbyAci,thesubsheafofπOBoffunctionswithK-characterci.
ThereforeIisgeneratedbyJ:=Ac1···Ac,whichisacoherentsheafonB//K.
WethushaveasurjectivesheafhomomorphismJOB//KOB→I,inducingaclosedembeddingProjBk≥0IkB*B//KProjB//Kk≥0Jk.
SincethenormalizationmorphismisniteandthequotientmorphismB→B//Kisane,itfollowsthatBtor→ProjB//Kk≥0Jkisananeinvariantmor-phism,showingthattheactionisrelativelyane.
Wenotethat(Btor)KistheinverseimageofBK.
ForthisitsucestoshowthattheberofBtor→Boveraxedpointconsistsofxedpoints.
ThisfollowssincethecoordinateringofananechartinaK-invariantberofthemorphismBtor→Bisgeneratedbyfractionsf=f1/f2wherefiaregeneratorsoftheidealI,henceKactstriviallyonf.
CombiningthiswiththefactthatBtor→Bisproper,wegetthatx∈(Btor)+ifandonlyifitsimageisinB+,andsimilarlyfor(Btor).
Thisprovestherstpartoftheproposition.
Forthesamereason,ifanopensetVBissaturated(i.
e.
,V=π1π(V)),thenthesameholdsforitsinverseimageVtorBtor.
ToprovethatUBtorBtoristoroidalandKactstoroidallyonthisembedding,weconsidertoricchartsηp:Vp→XpinBgivingtheactionofKonBalocallytoricstructure.
ForsimplicitywewriteV=Vp,X=Xp.
ByLemma3.
1.
2theidealI=IBrestrictedtoVistheinverseimageoftheidealIX=IXc1···IXcinX.
ItfollowsthatthenormalizationXtoroftheblowingupofIXinXprovidesatoricchartηtor:Vtor→XtorforBtorsuchthattheactionofKonBtorisagainlocallytoric.
LetDtorXtorbethesupportofthedivisordenedbythetotaltransformofIX.
ThenUBtor∩Vtor=ηtor1XtorDtor,andwearereducedtoprovingthat(XtorDtor)XtorisatoroidalembeddingonwhichKactstoroidally.
Inotherwords,wehavetoshowthattheirreducibletoricdivisorsDXtorthatdonotlieinDtorcanberemovedfromthestandardtoroidalstructuregivenbythetoricstructure,keepingtheK-actiontoroidal.
ByLemma3.
2.
4wecanremovethemoneatatime.
WriteX=X(N,σ)whereσ=v1,vm,σ∨=v1,vm,±vm+1,vn,andletKactonzi=zvibycharacterci.
TheonlyirreducibletoricdivisorsinXtorthatdonotlieinthetotaltransformofIXareamongthestricttransformsofthedivisors{zi=0}X.
Considerthedivisor{z1=0}.
TheidealIXc1containsz1.
IfIXc1isprincipal,thenthestricttransformof{z1=0}isacomponentofDtorandthereisnothingtoprove.
AssumethatthisisnotthecaseandchoosemonomialgeneratorsforIXc1correspondingtolatticepointsv1,m1,mlinM∩σ∨.
Wemayassumethatzmiarenotdivisiblebyz1.
Tostudythestricttransformof{z1=0}inXtorwerstblowupIXc1,thentherestoftheIXci,andthennormalize.
LetYbeananechartoftheblowingupofXalongIXc1(whichisnotnecessarilynormal),obtainedbyinvertingoneofthegeneratorsofIXc1,andletDbethestricttransformof{z1=0}inY.
ThenDisnonemptyifandonlyifYisthechartofTORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS557theblowingupwhereweinvertoneofthezmi,sayzm1.
WehaveY=SpecKz1zm1,zm2zm1zmlzm1,z2,zm,z±1m+1,z±1n=SpecKz1zm1*SpecKzm2zm1zmlzm1,z2,zm,z±1m+1,z±1n=A1*Y,wherethesecondequalityfollowssincez1doesnotdividezmi.
HerethestricttransformDof{z1=0}isdenedbyz1/zm1,onwhichKactstrivially.
ItremainstobeshownthatifweblowuptheidealsIXcifori=1pulledbacktoYandnormalize,thisproductstructureispreserved.
WedenetheidealsIYcionYgeneratedbyallmonomialsonwhichKactsbycharacterci.
ThelemmabelowshowsthatIYciisequaltotheinverseimageofIXci.
HencewemayblowupIYciinsteadoftheinverseimageofIXci.
SinceKactstriviallyonz1/zm1,theidealsIYciaregeneratedbymonomialsinthesecondtermoftheproduct.
Thus,blowingupIYcipreservestheproduct,andsodoesnormalization.
Lemma3.
2.
6.
ForananetoricvarietyXwithanactionofKasaone-parametersubgroupofthetorus,letIXαbetheidealgeneratedbyallmonomialsonwhichKactsbycharacterα.
Ifφ:Y→XisachartoftheblowingupofIXα,thenIYβ=IXβOYforallβ.
Proof.
ClearlyIXβOYIYβ.
Fortheconverse,letthemonomialgeneratorsofthecoordinateringofYbez1/zm1,zm2/zm1zml/zm1,z1,zm,z±1m+1,z±1nforsomegeneratorszmiofIα.
ThusaregularmonomialonYcanbewrittenasaproductzm=(z1zm1)b1(zm2zm1)b2···(zmlzm1)bl·zd11···zdnnforsomeintegersbi,dj,wherebi,dj≥0fori=1,l,j=1,m.
IfzmhappenstobeageneratorofIYβ,i.
e.
,Kactsonzmbycharacterβ,thenalsoKactsonzm=zd11···zdnnbycharacterβ,andzmisinIXβOY.
Corollary3.
2.
7.
TheembeddingsUBtor±/KBtor±/Karetoroidalembeddings,andthebirationaltransformationBtor/KBtor+/Kistoroidal.
Proof.
ThisisimmediatefromthepropositionandLemma2.
6.
1.
Infact,asthefollowinglemma,inconjunctionwith3.
2.
6,shows,themapBtor/KBtor+/Kisanisomorphismiftheset{c1,c}inthedenitionofthetoricidealI=IBc1···IBcischosenlargeenough.
Sincewedonotneedthisresult,weonlygiveasketchoftheproof.
Lemma3.
2.
8.
LetB=X(N,σ)=SpecK[z1,zm,z±1m+1,z±1n]beanon-singularanetoricvariety,andassumethatKactsonzibycharacterci.
Letα∈Zbedivisiblebyallci,andletIαandIαbetheidealsgeneratedbyallmono-mialsofK-characterαandα,respectively.
IfBisthenormalizationofthe558D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKblowingupofIα·Iα,thenthebirationalmapB/KB+/Kisanisomorphism.
Thesameholdsforanytoricidealcorrespondingtoasetofcharacterscontainingαandα.
Sketchoftheproof.
Letσ=v1,vm,andletπ:NR→NR/R·abetheprojectionfroma.
Ifπmapsσisomorphicallytoπ(σ),thenBandB+areisomorphicalready.
Otherwise,thereexistuniqueraysr++σandrσsuchthatthestarsubdivisionofπ(+σ)atπ(r+)isequaltothestarsubdivisionofπ(σ)atπ(r).
NowthenormalizedblowingsupofIαandIαturnouttocorrespondtostarsubdivisionsofσatr+andr.
TheresultingsubdivisionΣclearlysatisesπ(Σ)=π(+Σ).
InourargumentsinthenextsectionwewilluseamoredetaileddescriptionofthecoordinateringofsomeanetoricchartsofBtor.
IfKactsonthevariableziviatci,andiftheidealIciisnotprincipal,thenthestricttransformofthedivisorDi={zi=0}isremovedfromthetoroidalstructurein(Btor,Dtor),i.
e.
,itisnotcontainedinDtor.
Assumeτisaconeinthesubdivisionassociatedtothenormalizationoftheblowingupofatoricideal,anddenotetheraysinτcorrespondingtothedivisorsDiwhichareremovedfromthetoroidalstructurebyvi.
Afterrenumbering,wemayassumethatthesearev1,vk.
Wehaveseenabovethatforeachi=1,kthecorrespondinganetoricvarietyYdecomposesasY=SpecK[zi/zmi]*Yi.
HerethestricttransformofDiisthezerolocusofzi/zmi.
Sincevj∈τ,wehavethat(vimi,vj)≥0fori,j=1,k.
Sincemiispositiveonτ,wehave(mi,vj)=0,i,j=1,k,whichmeansthatzjdoesnotdividezmifori,j=1,k.
WealsohavethatzidoesnotappearinanymonomialintheringofYifori=1,k.
ApplyingLemma3.
2.
4withwi=vimi,weobtainthefollowing:Corollary3.
2.
9.
LetB=X(N,σ)=SpecK[z1,zm,z±1m+1,z±1n]beanon-singularanetoricvariety,andassumethatKactsonzibycharacterci.
LetYBtorbeananetoricchartcorrespondingtoaconeτ,andassumethattheraysinτcorrespondingtodivisorswhichareremovedfromthetoroidalstructurearev1,vk.
Thenthereexistmi∈σ∨andatoricvarietyY,suchthat(1)(mi,vj)=0fori,j=1,k,(2)zi/zmiareK-invariant,(3)zidoesnotappearinanymonomialintheringofY,and(4)Y=SpecKz1zm1zkzmk*Y.
Example3.
2.
10.
ConsiderB=A3=SpecK[z1,z2,z3],wheret∈Kactsast·(z1,z2,z3)=(t2z1,t3z2,t1z3).
TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS559WehavethefollowinggeneratorsofthetoricidealsIα:I2=(z1,z2z3),I3=(z2,z21z3),I1=(z3).
ToillustrateLemma3.
2.
8wealsoconsiderI6=(z31,z22,z21z2z3).
TheidealI6=(z63)isunnecessaryhere,beingprincipal.
LetI=I2I3I6I1.
IfweregardB=X(N,σ)asthetoricvarietycorrespondingtotheconeσ=v1,v2,v3NR,thenBtorisdescribedbythefancoveredbythefollowingfourmaximalcones:σ1=v1,v1+v3,v1+v2,σ2=v1+v2,v1+v3,2v1+3v2,v3,σ3=2v1+3v2,v3,2v2+v1,v1+v2,v2+v3,σ4=2v2+v1,v2+v3,v2.
IfwedonotincludethefactorI6inI,thentheconesσ2andσ3arecombinedtoonenonsimplicialcone.
IncludingI6hastheeectthatBtor/KBtor+/Kbecomesanisomorphism.
v1v3v2v1+v3v2+v3v1+v22v1+3v2v1+2v2I2I6I3Thetorifyingpropertycanbeillustratedontheanetoricvarietycorrespondingtoσ1.
Thedualconeσ∨1hastheproductdescriptionσ∨1=v1(v2+v3),v2,v3=v1(v2+v3)*v2,v3.
560D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKThus,evenifweremovethedivisor{z1/z2z3=0}fromtheoriginaltoricstructureofX(N,σ1)=Speck[z1/z2z3,z2,z3],westillhavethetoroidalembeddingstructureX(N,σ1)({z2=0}∪{z3=0})X(N,σ1).
Asz1/z2z3isinvariant,theactionofKistoroidal.
Forexample,at0∈X(N,σ1)wehaveatoricchartK*K2→K*K2=X(N,σ1),(x1,x2,x3)→(x11,x2,x3).
Globally,thedivisorscorrespondingtothenewraysDv1+v2,Dv1+v3,D2v1+3v2,Dv1+2v2,Dv2+v3togetherwithDv3comingfromI1,areobtainedthroughtheblowingupofthetoricideals.
ConsideringUBtor=Btor(Dv1+v2∪Dv1+v3∪D2v1+3v2∪Dv1+2v2∪Dv2+v3∪Dv3)weobtainatoroidalstructureUBtorBtorwithatoroidalK-action.
4.
Aproofoftheweakfactorizationtheorem4.
1.
Thesituation.
InTheorem2.
6.
2wehaveconstructedafactorizationofthegivenbirationalmapφintotightlylocallytoricbirationaltransformationsX1=W1W1+=W2W2+.
.
.
WmWm+=X2,Ba1//KBa2//KBam//KwhereWi±=(Bai)±/K(hereWiisWi1inthenotationofTheorem2.
6.
2,andWi+isWi).
SinceBisnonsingular,wecanapplytheresultsofsection3.
ForachoiceofatoricidealI=Ic1···IconBai,denotebyBtorai→Baithecorrespondingtoricblowingup.
WriteWtori±=Btorai±/K,andUtori±=UBtorai±/K.
WehaveanaturaldiagramofbirationalmapsWtoriWtori+↓fi↓fi+WiBtorai//KWi+↓Bai//KByCorollary3.
2.
7theembeddingsUtori±Wtori±aretoroidal,andthebirationaltransformationtori:WtoriWtori+istoroidal.
WesaythattheidealI=Ic1···Icisbalancedifcj=0.
ItfollowsfromLemma3.
1.
3thatwecanalwaysenlargetheset{c1,c}togetabalancedtoricidealI.
Asinsection3wedenotebyπ:Bai→Bai//Kthequotientmorphism.
Lemma4.
1.
1.
SupposethetoricidealIisbalanced.
Thenthemorphismfi±isthenormalizedblowingupoftheidealsheafIi±denedasthepullbacktoWi±ofπI∩OBai//K.
TORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS561Proof.
ByLemma3.
1.
2,theidealIisgeneratedbyK-invariantsections,andwecanidentifyIastheinverseimageofanidealsheafinBai//Kgeneratedbythesamesections—whichwecantaketobeπI∩OBai//K.
LetIi±bethepullbackofthisidealsheafto(Bai)±/Kviathemap(Bai)±/K→Bai//K.
Thenfi±isthenormalizedblowingupofIi±becausetakingthequotientbyKcommuteswithblowingupthesheafI.
Fromnowonweassumethatthetoricidealsarechosentobebalanced.
Theproofofthemaintheoremcanbecarriedoutwithoutthisassumption,butitwouldmakethepresentationmorecomplicated.
NotethatifthevarietiesWi±werenonsingularandthemorphismsfi±werecompositesofblowingsupofnonsingularcenters,wewouldgettheweakfactor-izationbyapplyingTheorem1.
6.
1toeachtori.
Thisisnotthecaseingeneral.
InthissectionwereplaceWi±bynonsingularvarietiesandfi±bycompositesofblowingsupwithnonsingularcenters.
4.
2.
Liftingtoroidalstructures.
LetWresi±→Wi±bethecanonicalresolutionofsingularities.
Notethat,sinceWi+=W(i+1),wehaveWresi+=Wres(i+1).
DenoteIresi±=Ii±OWresi±.
LetWcani±→Wresi±bethecanonicalprincipalizationoftheidealIresi±,andlethi±:Wcani±→Wtori±betheinducedmorphism.
Wcanihi→WtoriWtori+hi+←Wcani+↓↓fi↓fi+↓Wresi→WiBtorai//KWi+←Wresi+↓Bai//KDenoteUcani±=h1i±Utori±.
Thecrucialpointnowistoshow:Proposition4.
2.
1.
TheembeddingUcani±Wcani±isatoroidalembedding,andthemorphismWcani±→Wtori±istoroidal.
Proof.
Forsimplicityofnotationwedropthesubscriptsiandai,aswetreateachquasi-elementarypieceseparately.
WemayassumethatallthevarietiesB,W±,Wtor±,Wres±,Wcan±andthemorphismsbetweenthemaretoric.
Indeed,ifVp→Xpisatoricchartatsomepointp∈W±,obtainedfromatoricchartinB,wegetatoricchartforWtor±byblowingupatoricidealinXp,whichisatoricidealsinceitisgeneratedbymonomials.
Similarly,resolutionofsingularitiesandprincipalizationoverthetoricvarietyXpprovidetoricchartsforWres±andWcan±.
Thecanonicityofresolutionandprincipalizationimpliesthatthemapsaretoric(i.
e.
,torusequivariant).
Considernowthediagramoftoricmorphismsbetweentoricvarietiesandthecorrespondingdiagramoffans:Wcan±→Wtor±Σcan±→Σtor±↓↓W±Σ±LetXτWtor±beananeopentoricsubvarietycorrespondingtoaconeτ∈Σtor±,andwriteXτ=Ak*Xτ,562D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKwherethetoricdivisorsD1,DkpulledbackfromAkaretheonesremovedinordertodenethetoroidalstructureonWtor±.
LetXcanτbetheinverseimageofXτinWcan±.
WeneedtoshowthatwehaveadecompositionXcanτ=Ak*Xcanτ,suchthattheresultingmapAk*Xcanτ→Ak*Xτisaproduct,withtherstfactorbeingtheidentitymap.
WriteXτ=Bτ/K,whereBτBtor±istheaneopentoricsubvarietylyingoverXτ.
ByCorollary3.
2.
9,thecoordinateringsofBτandXτcanbewrittenasAXτ=K[z1zm1zkzmk]AXτ,ABτ=K[z1zm1zkzmk]ABτ,whereXτ=Bτ/K,andwherezmjaremonomialsonwhichKactswiththesamecharacterasonzj,suchthatzizmjfori,j=1,k.
Lemma4.
2.
2.
Foreachy=(y1,yk)∈KkconsidertheautomorphismθyofB=SpecK[z1,zm,z±1m+1,z±1n]denedbyθy(zi)=zi+yi·zmi,i≤k,θy(zi)=zi,i>k.
Then:(1)θydenesanactionoftheadditivegroupKkonB.
(2)TheactionofθycommuteswiththegivenK-action.
(3)TheidealsIcareinvariantunderthisaction.
(4)TheactionleavesB±invariant,anddescendstoW±.
(5)TheactionliftstoBtor.
(6)ThisactiononBtorleavestheopensetBτinvariant.
(7)TheinducedactiononBτdescendstoaxed-point-freeactionofKkonXτ.
(8)TheresultingactiononXτisgivenbyθy(zi/zmi)=zi/zmi+yi;θy(f)=fforf∈AXτ.
Proof.
Sincezimjfori,j=1,k,wehavethattheθycommutewitheachother,andθyθy=θy+ythusdeningaKk-action.
SinceKactsonziandmithroughthesamecharacter,itcommuteswithθy.
ForthesamereasontheidealsIcareinvariant:zα=zαiihasK-charactercifandonlyifθy(zα)does,thereforeθyIc=Ic.
SinceB=BV(cBW1BW2.
Thisisclearlyfunctorialinφ.
NowZ→BWiaretoroidalbirationalmorphisms,correspondingtosubdivisionsZ→Bi.
LetHG1bethesubgroupstabilizingthesubdivisionZ→B1.
FixarepresentativeintheisomorphismclassofZ→B1,and,usingtheaxiomofchoice,xanisomorphismofanyelementoftheisomorphismclasswiththisrepresentative.
NotethattheabsoluteautomorphismgroupofZ→W1mapstoH.
WeclaimthatinordertoconstructafunctorialfactorizationofZ→W1itsuf-cestoconstructanH-equivariantcombinatorialfactorizationofourrepresentativeoftheisomorphismclass,whichbyabuseofnotationwecallZ→B1.
Indeed,suchacombinatorialH-equivariantfactorizationcorrespondstoasequenceofH-equivariantsubdivisionsΣi→B1suchthateitherΣi→Σi+1oritsinverseisa566D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKnonsingularstarsubdivisiononeachcone,suchthatZ→B1factorsthroughanisomorphismZ→Σ0,andsuchthatΣm=B1.
Pullingbackbythechosenisomorphism,wegetanequivariantcombinatorialfactorizationforeveryelementintheisomorphismclass,whichttogethertogiveafunctorialcombinatorialfac-torization.
Accordingtotheconstructionof[42],thisfunctoriallycorrespondstoasequenceofmodicationsVi→BW1whichttogetherasafunctorialfactorizationofZ→BW1.
NowZ/H→B1/Hisasubdivisionofnonsingularpolyhedralcomplexes,andthetoroidalweakfactorizationtheoremsaysthatitadmitsacombinatorialfactorization,asasequencecomposedofnonsingularstarsubdivisionsandinversenonsingularstarsubdivisions.
LiftingthesesubdivisionstoZ→B1,wegetanH-equivariantfactorization,whichinturncorrespondstoafunctorialtoroidalfactorizationofZBW1.
WenowapplythesameproceduretoZ→BW2.
Thisgivesthedesiredfunctorialtoroidalfactorizationofφ.
5.
7.
AnalytictoroidalC-actions.
ThenatureofC-actionsonanalyticspacesdiersignicantlyfromthecaseofvarieties.
However,thesituationisalmostthesameifonerestrictstorelativelyalgebraicactions.
Denition5.
7.
1.
LetX→SbeamorphismofanalyticspacesandLarelativelyamplelinebundleforX→S.
AnactionofConX,LoverSisrelativelyalgebraicifthereisanopencoveringS=Si,analgebraicactionofConaprojectivespacePNi,andaZariski-locally-closedC-equivariantembeddingSi*SXSi*PNi,suchthatforsomeintegerliwehavethatLliX*SSiisC-isomorphictothepullbackofOPNi(1).
ItiseasytoseethatifX→Sisaprojectivemorphism,Lalinebundle,witharelativelyalgebraicC-action,thenXProjSSymE,wherethesheafE=ki=1EiisacompletelyreducibleCsheaf.
Intheanalyticcategoryweuseembeddedchartsratherthanetaleones.
Ac-cordingly,wesaythataC-equivariantopensetVXisstronglyembeddedifforanyorbitOV,theclosureofOinXiscontainedinV.
ThisimpliesthatV//C→X//Cisanopenembedding.
WedeneananalyticlocallytoricC-actiononWusingstronglyembeddedtoricchartsηp:Vp→Xp(westillhavetherequirementthatVp=π1πVp,whereπ:W→W//Kistheprojection,whichmeansthatVpWisalsostronglyembedded).
Itisnotdiculttoshowthatastronglyembeddedtoricchartexistsforeachpointp∈B,theanalogueofLuna'sfundamentallemma.
Withthesemodications,Lemma1.
7.
3isproveninthesamemannerintheanalyticsetting.
Wealsonotethat,ifD=li=1DiWisasimplenormalcrossingsdivisor,thentoricchartscanbechosencompatiblewithD.
Indeed,weonlyneedtochoosesemi-invariantparametersx1,xnsothatxiisadeningequationforDi,fori=1,l.
5.
8.
Analyticbirationalcobordisms.
Analyticbirationalcobordismsarede-nedthesamewayasinthecaseofvarieties,withtheextraassumptionthattheC-actionisrelativelyalgebraic.
Givenaprojectivebirationalmorphismφ:X1→X2weconstructacompacti-ed,relativelyprojectivecobordismB→X2asinthealgebraicsituation,withthefollowingmodication:usingcanonicalresolutionofsingularitieswemakethein-verseimageofX2UinBintoasimplenormalcrossingsdivisor,crossingX1andTORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS567X2normally.
Notethattheseoperationsarefunctorialinabsoluteisomorphismsofφ.
Asindicatedbefore,thisconstructionendowsB→X2withafunctorialrela-tivelyamplelinebundle.
SincethisbundleisobtainedfromtheProjconstructionoftheblowingupofaninvariantideal,itcomeswithafunctorialC-actionaswell.
Theconsiderationsofcollapsibilityandgeometricinvarianttheoryworkasinthealgebraicsetting,leadingtoTheorem2.
6.
2.
Wenotethattheresultinglocallytoricfactorizationisfunctorial,andthetoricchartsonWicanbechosencompatiblewiththedivisorcomingfromX1UorX2U.
5.
9.
Functorialityoftoricationandcompatibilitywithdivisors.
Wenotethatthedenitionoftheα-toricidealsisclearlyfunctorial,anditiseasytomakeafunctorialchoiceofabalancedsetofcharactersintheconstructionofatoricideal(Denition3.
1.
4).
Theproofofitsexistenceworksasinthecaseofvarieties.
Thesameistrueforitstorifyingproperty.
Inordertomakethisconstructioncompatiblewithdivisors,wereplacethetotaltransformDofIbyaddingtheinverseimageofX2U.
ThisguaranteesthattheresultingtoroidalstructureonBtoriscompatiblewiththedivisorscomingfromX2U.
5.
10.
ConclusionoftheproofofTheorem0.
3.
1.
Canonicalresolutionofsin-gularitiesisfunctorial,thereforetheconstructionofWres±→W±isfunctorial.
WecannowreplaceWres±bythecanonicalprincipalizationoftheinverseimageofX2U,makingthelatterasimplenormalcrossingsdivisor.
SincetheidealIisfunctorial,theconstructionofWcan±→W±isfunctorial,andthelocallytoricstruc-tureimpliesthatthecentersofblowingupinWcan±→Wres±havenormalcrossingswiththeinverseimageofX2U.
Wecannowapplyfunctorialtoroidalfactor-izationtothetoroidalbirationalmapWcanWcan+.
Notethatthecentersofblowingup,beingtoroidal,automaticallyhavenormalcrossingswithWcan±Ucan±.
Thetheoremfollows.
6.
Problemsrelatedtoweakfactorization6.
1.
Strongfactorization.
Despiteourattempts,wehavenotbeenabletousethemethodsofthispapertoprovethestrongfactorizationconjecture,evenassum-ingthetoroidalcaseholdstrue.
Intheconstructionofthetoricidealin3.
1andtheanalysisofitsblowingupin3.
2and4.
2,theassumptionofthecobordismBaibeingquasi-elementaryisessential.
Itiseasytogiveexampleswheretheformationofthetoricidealdoesnotcommutewithtakinganeopensets,thereforewecannotgluetogethertheidealsdenedontheindividualquasi-elementarypiecesintooneidealovertheentirebirationalcobordismB.
Onecanextendeachoftheseidealsseparately,forinstancebytakingtheZariskiclosureofitszeroscheme,butthebehaviorofthisextension(aswellasotherswehaveconsidered)alongBBaiisproblematic.
Theweakfactorizationtheoremreducesthestrongfactorizationconjecturetothefollowingproblem:Problem6.
1.
1.
LetX1→X2Xnbeasequenceofblowingsupwithnonsingularcenters,withXnnonsingular,andsuchthatthecenterofblowingupofXi→Xi+1hasnormalcrossingswiththeexceptionaldivisorofXi+1→Xn.
568D.
ABRAMOVICH,K.
KARU,K.
MATSUKI,ANDJ.
WLODARCZYKLetY→Xnbeablowingupwithnonsingularcenter.
FindastrongfactorizationofthebirationalmapX1Y.
Webelievethatatleastthethreefoldcaseofthisproblemistractable.
6.
2.
Toroidalization.
Problem6.
2.
1(Toroidalization).
Letφ:X→Ybeasurjectivepropermor-phismbetweencompletenonsingularvarietiesoveranalgebraicallyclosedeldofcharacteristic0.
DothereexistsequencesofblowingsupwithnonsingularcentersνX:X→XandνY:Y→Ysothattheinducedmapφ:XYisatoroidalmorphismCansuchmapsbechoseninafunctorialmanner,andinsuchawaythattheypreserveanyopensetwhereφadmitsatoroidalstructureThiscanbeviewedasaproblemofndingaHironaka-typelogarithmicdesin-gularizationofamorphism.
Theresultof[3],Theorem2.
1,givesalogarithmicdesingularizationofamorphism,butnotusingblowingsupwithnonsingularcen-ters.
Asimilarconjecturewasproposedin[43].
Wenotethatthetoroidalizationcon-jectureconcernsnotonlybirationalmorphismsφbutalsogenericallynitemor-phismsormorphismswithdimX>dimY.
Thesolutiontotheaboveconjecturewouldreducethestrongfactorizationconjecturetothetoroidalcase,simplybycon-sideringthecaseofabirationalmorphismφandthenapplyingthetoroidalcasetoφ.
UntilrecentlytheauthorsknewofacompleteproofonlyifeitherdimX=2(seebelow),ordimY=1(whichfollowsimmediatelyfromresolutionofsingularities;see[42],II§3).
Recently,S.
D.
CutkoskyworkedoutahighlynontrivialsolutionofthecasedimX=3,dimY=2[18].
Theconjectureisfalseinpositivecharacteristicsduetowildramications.
See,e.
g.
,[19].
Onegeneralresultwhichwedoknowisthefollowing.
Theorem6.
2.
2.
Letφ:X→Ybeasurjectivemorphismbetweencompleteva-rietiesoveranalgebraicallyclosedeldofcharacteristic0.
ThenthereexistsamodicationνX:X→XandasequenceofblowingsupwithnonsingularcentersνY:Y→Ysothattheinducedmapφ:XYisatoroidalmorphism.
Proof.
In[3],Theorem2.
1,itisshownthatmodicationsνXandνYsuchthatφistoroidalexist,assumingXandYareprojectiveandthegenericberofφisgeometricallyintegral.
WecanreducetotheprojectivecaseusingChow'slemma.
Thecasewherethegenericberisnotgeometricallyintegralisresolvedinthesecondauthor'sthesis[37].
Sincethelatterisnotwidelyavailablewegiveasimilarargumenthere.
Theinductiveproofof[3],Theorem2.
1,reducestheproblemtothecasewhereφisgenericallynite.
ByHironaka'sattening(orbytakingaresolutionofthegraphofYHilbY(X)),wemayassumethatX→Yisnite.
Usingresolutionofsingularities,wemayassumeYisnonsingularandthebranchlocusisanormalcrossingsdivisor.
BynormalizingXwemayassumeXnormal.
DenotingthecomplementofthebranchlocusbyUYanditsinverseimageinXbyUX,Abhyankar'slemmasaysthatUXXisatoroidalembeddingandX→Yistoroidal,whichiswhatweneeded.
ItremainstobeshownthatνYcanbechosentobeasequenceofblowingsupwithnonsingularcenters.
LetY←Y→YbeaneliminationofindeterminaciesofYYandletY→YbethecanonicalprincipalizationofthepullbackoftheTORIFICATIONANDFACTORIZATIONOFBIRATIONALMAPS569idealofthetoroidaldivisorofY.
LetX→Y*YXbethenormalizationofthedominantcomponent.
ThenY→Yisasequenceofblowingsupwithnonsingularcenters.
Applying[3],Lemma6.
2,weseethatX→Yisstilltoroidal,whichiswhatweneeded.
Sinceeveryproperbirationalmorphismofnonsingularsurfacesfactorsasase-quenceofpointblowingsup,weget:Corollary6.
2.
3.
Thetoroidalizationconjectureholdsforagenericallynitemor-phismφ:X→Yofsurfaces.
Inthiscase,itisnotdiculttodeducethatthereexistsaminimaltoroidalization(sincethecongurationofintermediateblowingsupinX→XorY→Yformsatree).
ThisresulthasbeenproveninanalgorithmicmannerbyCutkoskyandPiltant[19].
Similarstatementscanbefoundin[6].
AcknowledgementsWeheartilythankE.
Bierstone,L.
Bonavero,S.
Iitaka,Y.
Kawamata,P.
Milman,Y.
Miyaoka,S.
Mori,N.
NakayamaB.
Siebert,andV.
Srinivasforhelpfulcomments.
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bu.
eduDepartmentofMathematics,HarvardUniversity,OneOxfordStreet,Cambridge,Massachusetts02139E-mailaddress:kkaru@math.
harvard.
eduDepartmentofMathematics,PurdueUniversity,1395MathematicalSciencesBuild-ing,WestLafayette,Indiana47907-1395E-mailaddress:kmatsuki@math.
purdue.
eduInstytutMatematykiUW,Banacha2,02-097Warszawa,PolandE-mailaddress:jwlodar@mimuw.
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pl