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Algebraic Geometry Seminar - Yuji Odaka (RIMS/Imperial College)

posted Apr 3, 2012, 1:40 AM by Jihye Jung   [ updated Apr 3, 2012, 1:40 AM ]
 - Title: On moduli and stability of Q-Fano varieties
- Speaker : Yuji Odaka (RIMS/Imperial College)
- Date & Time : April 5(Thu) , 2012 14:00~
- Place : Math Science building room 404
- Abstract
   : Originally, the concept of (GIT-) stability of varieties is introduced to construct their MODULI space and succeeded in curve case. Having been known that ``good” degenerations of canonical models (which form projective moduli) are NOT stable from surface case (Shepherd-Barron, Kollar, Alexeev..), stud y of GIT-stability has not been focused for a few decades. One observation of the speaker is that those good degenerations actually do satisfy K-STABILITY: a version of original stability, expectedly equivalent to the existence of ``canonical” Kahler metric, due to differential geometers Tian, Donaldson.

Thus, it now becomes quite natural to expect that varieties satisfying such stability form quasi-projective moduli spaces (let us call it K-MODULI!), which indeed has natural differential geometric explanation with moment map. In the way, it also turned out that discrepancy and MMP-gadget are well-suited for the study of the stabilities.

In this talk, we focus on most classical but non-trivial case, Q-Fano varieties, and consider the problem that “concretely when are polarized varieties K-stable?”, which is also studied by Hwang-Kim-Lee-Park before the speaker's works. I explain two approaches and some concrete partial results. Newer approach is via Tian’s conjecture (recently settled by Li-Xu) that we only need to see normal degenerations, to apply ``separatedness of moduli”, which is actually also one of our tentative goal toward K-moduli. Both approaches reveal interesting relations with log-canonical thresholds, biratonal superrigidity.