Vector Bundle on Riemannian Surfaces

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1、VECTORBUNDLESONRIEMANNSURFACESSABINCAUTISContents1.DifferentiableManifolds22.ComplexManifolds32.1.RiemannSurfacesofGenusOne42.2.ConstructingRiemannSurfacesasCurvesinP262.3.ConstructingRiemannSurfacesasCovers92.4.ConstructingRiemannSurfacesbyGlueing103.TopologicalVecto

2、rBundles113.1.TheTangentandCotangentBundles133.2.Interlude:Categories,ComplexesandExactSequences143.3.MetricsonVectorBundles153.4.TheDegreeofaLineBundle163.5.TheDeterminantalLineBundle173.6.ClassificationofTopologicalVectorBundlesonRiemannSurfaces183.7.HolomorphicVect

3、orBundles193.8.SectionsofHolomorphicVectorBundles204.Sheaves214.1.CechCohomology234.2.LineBundlesandCechCohomology274.3.Riemann-RochandSerreDuality294.4.Vectorbundles,locallyfreesheavesanddivisors304.5.AproofofRiemann-Rochforcurves335.ClassifyingvectorbundlesonRieman

4、nsurfaces345.1.Grothendieck’sclassificationofvectorbundlesonP1345.2.Atiyah’sclassificationofvectorbundlesonellipticcurves35References3611.DifferentiableManifoldsTwotopologicalspacesXandYarehomeomorphicifthereexistcontinuousmapsf:X→Yandg:Y→Xsuchthatg◦f=idXandf◦g=idY.Thi

5、sisdenotedX∼=Y.Exercise1.ShowthatS1andtheopenunitinterval(0,1)arenothomeomorphic.Exercise2.Showthat(0,1)andthereallineRarehomeomorphic.Warning:toshowX∼=Yitisnotenoughtofindacontinuousmapf:X→Ywhichis1-1andontobecausetheinversemapf−1maynotbecontinuous.Forexample,thenatu

6、ralinclusionf:(0,1]→S1iscontinuousandbijectivebuttheinversef−1isnotcontinuousatf(1).AmanifoldofdimensionnisaHausdorfftopologicalspaceMsuchthateverypointp∈Mhasaneighbourhoodp∈U⊂MwhichishomeomorphictotheopenunitballinRn.Inotherwords,MlocallylookslikeRn.Example.Therealli

7、neR,theunitcircleS1andtheopeninterval(0,1)areone-dimensionalmanifolds.Thesemiopeninterval(0,1]isnotamanifoldsincethereisnoopenneighbourhoodof1homeomorphictoanopenintervalinR.Example.ThesphereS2,thetorusS1×S1,theopencylinder(0,1)×S1aswellasanyopensubsetofR2aretwo-dime

8、nsionalmanifolds.AnopencoverofatopologicalspaceXisacollectionofopensubspacesUαsuchthat∪αUα=X.AmanifoldMiscompactifeveryopencoverofM