lectures on complex geometry-clabi-yau manifolds and toric geometry[bouchard]

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1、Lecturesoncomplexgeometry,Calabi–YaumanifoldsandtoricgeometryVincentBouchard♣MathematicalInstituteUniversityofOxford24-29St.Giles’,OxfordOX13LB,UKAbstractThesearelecturenotessupplementingafour-hourintroductorycourseoncom-plexgeometry,Calabi–Yaumanifoldsandtoricgeometry,gi

2、venattheFirstModaveSummerSchoolinMathematicalPhysics.Wefirstdefinebasicconceptsofcomplexgeometry,leadingtoananalysisofthedifferentdefinitionsofCalabi–Yaumanifolds.Thelastsectionprovidesashortintroductiontotoricgeometry,aimedatconstruct-ingCalabi–Yaumanifoldsintwodifferentwaysi

3、ntoricgeometry;ashypersurfacesintoricvarietiesandaslocaltoricCalabi–Yauthreefolds.June2005♣bouchard@maths.ox.ac.ukCONTENTS2Contents1Complexgeometry51.1Complexmanifolds...............................51.2Tensorsoncomplexmanifolds.........................81.3Exteriorformsonc

4、omplexmanifolds.....................91.4Cohomology...................................101.5Chernclasses..................................121.5.1Cherncharacter.............................141.6Holomorphicvectorbundles..........................151.7Divisorsandlinebundles....

5、........................182K¨ahlergeometry182.1K¨ahlermanifolds................................192.2FormsonK¨ahlermanifolds...........................202.3Cohomology...................................232.4Holonomy....................................233Calabi–Yaugeometry25

6、3.1Calabi–Yaumanifolds..............................253.2Cohomology...................................273.3Examples....................................283.3.1ChernclassofCPm...........................293.3.2Calabi–Yauconditionforcompleteintersectionmanifolds......293.3.3T

7、hequinticinCP4...........................313.3.4TheTian-Yaumanifold.........................343.4‘Local’Calabi–Yaumanifolds.........................384Toricgeometry384.1Homogeneouscoordinates...........................394.1.1Toricdivisors..............................42CO

8、NTENTS34.2ToricCalabi–Yauthreefolds..........................424.3Toricdiagramsandsymplecticquot