On the abundance for threefolds in positive characteristic
On the abundance for threefolds in positive characteristic
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Zheng Xu , Chinese Academy of Sciences
OverÌýtheÌýpastÌýdecade,ÌýtheÌýMinimalÌýModelÌýProgramÌý(MMP)ÌýhasÌýbeenÌýlargelyÌýestablishedÌýforÌýthreefoldsÌýoverÌýperfectÌýfieldsÌýofÌýcharacteristic $>3$.ÌýAÌýcentralÌýconjectureÌýremainingÌýisÌýtheÌýabundanceÌýconjecture,ÌýwhichÌýpredictsÌýthatÌýif $(X,B)$ isÌýaÌýprojectiveÌýlogÌýcanonicalÌýthreefoldÌýpairÌýoverÌýanÌýalgebraicallyÌýclosedÌýfieldÌýofÌýcharacteristic $>3$ and $K_{X}+B$ isÌýnef,Ìýthen $K_{X}+B$ isÌýsemiample.ÌýInÌýthisÌýtalk,ÌýIÌýwillÌýdiscussÌýsomeÌýrecentÌýresultsÌýtowardsÌýthisÌýconjecture.