- 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. |
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