A Crash Course on compact complex surfaces.pdf

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1、ACrashCourseonCompactComplexSurfacesKennethChuchu@math.utexas.eduDepartmentofMathematicsUniversityofTexasatAustinApril24,2007ACrashCourseonCompactComplexSurfaces–p.1/24Abstract“Analyticinvariants”ofcomplexmanifoldsthatthegeneralizationsofthegenusofcurves,andtheirbirationallyi

2、nvariantnature.Blow-upofasurfaceatapoint.Birationalclassificationofcomplexsurfacesviaminimalmodels.Enriques-Kodairaclassification.Canonicalmodels.Calabi-YaumanifoldsandK3surfaces.FanomanifoldsanddelPezzosurfaces.ACrashCourseonCompactComplexSurfaces–p.2/24ExamplesofCompactComple

3、xSurfaces1.P2,P1×P1(∼=smoothquadricsurfaceinP3),smoothhypersurfacesinP3,two-dimensionalsubmanifoldsofPn,CartesianproductsoftwocompactRiemannsurfaces.2.fakeprojectiveplanes:=compactcomplexsurfaceswithb1=0,b2=1notisomorphictoP2.Suchasurfaceisprojectivealgebraicanditisthequotien

4、toftheopenunitballinC2byadiscretesubgroupofPU(2,1).Thefirstexample(Mumfordsurface)wasconstructedMumfordusingp-adictecnhniques.Recently,allpossible(17knownfiniteclassesplusfourpossiblecandidatesandnomore)fakeprojectiveplaneshavebeenenumeratedbyGopalPrasadandSai-KeeYeung.Seeabstr

5、actforcolloquiumonMarch26,2007.3.Ruledsurface:=P1-bundleoveracompactRiemannsurface.Canbeshown:Allruledsurfacesareprojectivizationsofrank-twovectorbundlesovercompactRiemannsurfaces.Hirzebruchsurfaces:P(OP1⊕OP1(−n)),n=0,1,2,...4.Ellipticsurface:=totalspaceofaholomorphicfibration

6、overacompactRiemannsurfacewithgenericfiberbeingasmoothellipticcurve.5.2-dimensionalcomplextori:C2/Λ,whereΛ∼=Z4isadiscretelatticeinC2.6.Hopfsurface:=compactcomplexsurfacewithuniversalcoverC2−{0}.Forexample,`´C2−{0}/Z,wheretheactionofZonC2isgeneratedbyC2−→C2:z7→2z.(TheHopfsurfac

7、eiscompactandnon-Kähler.)ACrashCourseonCompactComplexSurfaces–p.3/24IntheBeginning...GoddessSaidLettherebe...CURVES.Iamnotjoking;askthestringtheorists.ACrashCourseonCompactComplexSurfaces–p.4/24Classificationofsmoothcompactcomplexcurvesbygenusanalytic/topologicalgenusg(C)=h0(C

8、,Ω1)=h0(C,K)CCdegreeofcanonicalbundledeg(KC)=2g(C)−2g(C)=h0(K)Cecurv