Fano manifolds

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1、FanomanifoldsMarcoAndreattaDipartimentodiMatematicaTrentoFanomanifolds–p.1/22GeneralframeLetXbea(complex)compactmanifoldofdimensionnandTXitstangentbundle.Define−KX=detTX.Fanomanifolds–p.2/22GeneralframeLetXbea(complex)compactmanifoldofdimensionnandTXitstangentbund

2、le.Define−KX=detTX.XissaidtobeaFanomanifoldif−KXisample.Fanomanifolds–p.2/22GeneralframeLetXbea(complex)compactmanifoldofdimensionnandTXitstangentbundle.Define−KX=detTX.XissaidtobeaFanomanifoldif−KXisample.Example.TXisample(iffXisPn).Fanomanifolds–p.2/22Generalfram

3、eLetXbea(complex)compactmanifoldofdimensionnandTXitstangentbundle.Define−KX=detTX.XissaidtobeaFanomanifoldif−KXisample.Example.TXisample(iffXisPn).Conjecture.Xisuniruled(equivalentlyTXisnotgenericallyseminegative)iffXisbirationaltoafibrationsofFanovarieties.Fanoman

4、ifolds–p.2/22GeneralframeLetXbea(complex)compactmanifoldofdimensionnandTXitstangentbundle.Define−KX=detTX.XissaidtobeaFanomanifoldif−KXisample.Example.TXisample(iffXisPn).Conjecture.Xisuniruled(equivalentlyTXisnotgenericallyseminegative)iffXisbirationaltoafibration

5、sofFanovarieties.TheifpartwasprovedbyKollár-Miyaoka-Mori,theonlyifpartfollowsfromtheMinimalModelconjecture.Fanomanifolds–p.2/22GeneralframeLetXbea(complex)compactmanifoldofdimensionnandTXitstangentbundle.Define−KX=detTX.XissaidtobeaFanomanifoldif−KXisample.Example

6、.TXisample(iffXisPn).Conjecture.Xisuniruled(equivalentlyTXisnotgenericallyseminegative)iffXisbirationaltoafibrationsofFanovarieties.TheifpartwasprovedbyKollár-Miyaoka-Mori,theonlyifpartfollowsfromtheMinimalModelconjecture.FanomanifoldsarethebuildingblocksoftheMMPa

7、ndtheyareuniruled,i.e.coveredbyrationalcurves.Fanomanifolds–p.2/22NumericalinvariantsLetXbeaFanomanifold.Wedefinetheindex:rX=max{m∈N

8、−KX=mLforsomedivisorL},Fanomanifolds–p.3/22NumericalinvariantsLetXbeaFanomanifold.Wedefinetheindex:rX=max{m∈N

9、−KX=mLforsomedivisorL}

10、,andthepseudoindex:iX=min{m∈N

11、−KX·C=m,C⊂Xrationalcurve}.Fanomanifolds–p.3/22NumericalinvariantsLetXbeaFanomanifold.Wedefinetheindex:rX=max{m∈N

12、−KX=mLforsomedivi