The complex shade of a real space and its applicat

arXiv:math/0104060v1[math.GT]5Apr2001THECOMPLEXSHADEOFAREALSPACEANDITSAPPLICATIONSTOBIASEKHOLMAbstract.Anaturaloriented(2k+2)-chaininCP2k+1withboundarytwiceRP2k+1,itscomplexshade,isconstructed.Viaintersectionnumberswiththeshade,anewinvariant,theshadenumber,ofk-dimensionalsubvarietieswithnormalvectorfieldsalongtheirrealpart,isintroduced.Foraneven-dimensionalrealvariety,theshadenumberandtheEulernumberofthecom-plementofthenormalvectorfieldintherealnormalbundleofitsrealpartagree.Foranodd-dimensionalorientablerealvariety,alinearcombinationoftheshadenumberandthewrappingnumber(self-linkingnumber)ofitsrealpartisindependentofthenormalvectorfieldandequalstheencomplexedwritheasdefinedbyViro[8].Shadenumbersofvarietieswithoutrealpointsandencomplexedwrithesofodd-dimensionalrealvarietiesare,inasense,Vassilievinvariantsofdegree1.Complexshadesofodd-dimensionalspheresareconstructed.Shadenum-bersofrealsubvarietiesinsphereshavepropertiesanalogoustothoseoftheirprojectivecounterparts.1.IntroductionThemanifoldRPn,n0,isorientableifandonlyifnisodd.Withitsstandardorientation,RP2k+1⊂CP2k+1ishomologoustozero,sinceH2k+1(CP2k+1;Z)=0.(Noticethedifferencebetweenevenandodddimensions:RP2krepresentsagen-eratorofH2k(CP2k;Z2).Ourmainconcernaresubvarietiesin“doubledimen-sion+1”andsoweconsideronlytheodd-dimensionalcase.)ThecomplexshadeofRP2k+1consistsofthecomplexificationsofallreallinesthroughapointpinRP2k+1.Itisanoriented(2k+2)-chainΓpinCP2k+1withboundarytwiceRP2k+1,seeDefinition3.1.Geometricallydistinctshadesarein1-1correspondencewithpointsinRP2k+1.Theyallrepresentthesamehomologyclass,theshadeclass[Γ]∈H2k(CP2k+1,RP2k+1).Thishomologyclass[Γ]hasthreecharacterizingprop-erties:itisinvariantundercomplexconjugation,ithasboundarytwiceRP2k+1,anditsintersectionnumberwiththeclassrepresentedbyacomplexk-dimensionallinearspacewithoutrealpointshasabsolutevalueequalto1,seeProposition3.7.Theshadeclasswillbeusedtomeasurecertainlinkingphenomena.RecallthatifMisanm-dimensionalorientedmanifold,andBandCaredisjointorientedcyclesofdimensionskandm−k−1,respectively,whichareweaklyzero-homologous(i.e.representtorsionclassesinhomology)thenthelinkingnumber1991MathematicsSubjectClassification.14P25.Keywordsandphrases.Algebraicvariety;Complexification;Encomplexedwrithe;Isotopy;Linkingnumber;Realalgebraicknot;Realalgebraicvariety;Rigidisotopy;Shade;Vassilievinvariant;Wrappingnumber.Duringthepreparationofthismanuscripttheauthorreceivedapost-doctoralgrantfromStiftelsenf¨orinternationaliseringavforskningochh¨ogreutbildning.12TOBIASEKHOLMlk(B,C)isdefinedas1ntimestheintersectionnumberofanoriented(k+1)-chainAwithboundaryn·BandCinM.Anoriented2k-cycleCinthecomplementofRP2k+1inCP2k+1neednotbeweaklyzero-homologousandthelinkingnumberofRP2k+1andCisingeneralnotdefined.Infact,therearemanypossiblechoicesofacounterpartoflinkingnum-berinthiscase:theintersectionnumberofanyoriented2k-chainwithboundaryRP2k+1andCwoulddo.Theshadeclassprovidesachoice:wemeasurethelink-ingofCandRP2k+1inCP2k+1as12timestheintersectionnumberoftheshadeclassandC.Thisconstructionhasstraightforwardapplicationstok-dimensionalcomplexprojectivevarietieswithoutrealpointsandgiverisetowhatwillbecalledshadenumbers,see§2.A.Mostk-dimensionalcomplexvarietiesinprojective(2k+1)-space(varietiesinopendensesubsetoftheChow-variety)arewithoutrealpoints.Butthedefinitionofshadenumbercanbeextendedtoarbitraryk-dimensionalprojectivesubvari-etiesprovidedtheyhavebeenequippedwithadditionalstructure.(TheadditionalstructureisacertainkindofvectorfieldalongtherealpartinRP2k+1,onwhichtheshadenumberwilldepend.Anyk-dimensionalvarietyadmitssuchadditionalstructureandinspecialcasesitappearsnaturally,see§7.A)Themostinterestingapplicationsofshadenumbersariseinthestudyoftheinterplaybetweentherealandthecomplexgeometry/topologyofgenericrealpro-jectivevarieties(see§2.B):thereareconnectionsbetweenshadenumbersandtopologicalinvariantsoftherealpartequippedwiththeabovementionedvectorfield.Thenatureoftheseconnectionsdependsontheparityofthedimensionofthevarietyandthedifferencesbetweenevenandodddimensionsarestriking,see§2.Dand§2.E.Sincespheresfrequentlyoccursasrealalgebraicambientspaces,wedefinecomplexshadesofspheresaswell,see§2.C.1.A.Applicationstorealalgebraiclinks.ThewritheofagenericprojectionofaknotinR3toaplane(aknotdiagram)isthesignedsumofdoublepointsofitsimage.ThisnotionhasastraightforwardgeneralizationtoplaneprojectionsofknotsinRP3.Ingeneralthewrithechangesastheprojectionvaries.ItwasobservedbyViro[7]thatthewritheofaknotmaybeenhancedifitisrealalgebraic(arealalgebraicknotinprojective3-spaceisasmooth1-dimensionalprojectivevarietywithconnectedrealpart).In[8],Virointroducedtheencomplexedwrithewhichshowsthattheclassificationofrealalgebraicknotsuptorigidisotopy(asmoothisotopywhichisalsoacontinuousfamilyofrealalgebraiccurves,see[4]and[6])ismorerefinedthanthecorrespondingclassificationuptosmoothisotopy.Viro’sdefinitionisdiagrammatic:theencomplexedwritheofarealalgebraicknotVisexpressedasthesignedsumofrealdoublepointsinagenericrealprojec-tionofVtoaplane.Notethatthepreimageofadoublepointinsuchap