Reading group "Sasaki geometry and valuations"
Sasaki manifolds are a class of Riemannian manifolds that
naturally arise as links of complex affine algebraic varieties with an
isolated singularity. They contain as a special case Kähler geometry on
a polarized projective manifold, through the affine cone over it.
In recent years, there has been a flurry of activity in the study of
“canonical” (e.g. Einstein) metrics in Sasaki geometry. Extending the
Yau-Tian-Donaldson picture for polarized manifolds, the goal is to
relate the existence of a canonical metric on the link of an isolated
singularity to an algebro-geometric condition of stability. For
Sasaki-Einstein metrics, this program was indeed completed a couple
of years ago by Collins and Szekelyhidi.
On the algebro-geometric side of the story, exciting new developments
involving the “non-Archimedean link” (i.e. the Berkovich space of
valuations centered at the singularity) and the Minimal Model Program
have appeared in a series of works by Chi Li, Yuchen Liu and Chenyang
Xu.
The goal of this reading group is to study some of these works, starting with a gentle introduction of the objects involved.
Practical information
Meetings will take place at the Centre de
Mathématiques Laurent Schwartz (Ecole Polytechnique), usually in the
conference room of the CMLS, and sometimes in that of the CPHT nearby.
The program so far is as follows:
Monday, February 5 (CMLS)
11.00-12.00: General introduction (Sebastien Boucksom)
14.00-15.00: The non-Archimedean link of an isolated singularity (Charles Favre)
Wednesday, February 14 (CMLS)
10.30-12.00: Introduction to Sasaki geometry (Eleonora Di Nezza)
14.00-15.30: K-stability and Sasaki geometry (Ruadhai Dervan)
Thursday, February 22 (conference room of the CPHT, very close to the usual one)
10.30-12.00: Log discrepancy and normalized volume I (Charles Favre)
14.00-15.30: Log discrepancy and normalized volume II (Sébastien Boucksom)
Monday, March 19 (CMLS)
10.30-12.00: Blum's existence of minimizing valuation I (Matteo Ruggiero)
13.30-15.00: Blum's existence of minimizing valuation II (Thibaut Delcroix)
15.30-16-30: Giulio Codogni: Positivity of the Chow-Mumford line bundle
for families of K-stable klt Fano varieties (seminar talk)
Thursday, April 5 (CMLS)
10.30-12.00: K-stability and valuations, after T.Fujita (Ruadhai Dervan)
14.00-15.30: K-stability and volume minimization, after C.Li (Sébastien Boucksom).
Wednesday, May 2 (CMLS)
10.30-12.00: Stable degeneration conjecture I (Chenyang Xu)
14.00-15.30: Stable degeneration conjecture II (Chi Li)
References
H.Blum. Existence of valuations with smallest normalized volume.
T.Collins, G.Szekelyhidi. K-semistability for irregular Sasakian manifolds.
T.Collins, G.Szekelyhidi. Sasaki-Einstein metrics and K-stability.
C.Li. K-semistability is equivariant volume minimization.
C.Li. Minimizing normalized volumes of valuations.
C.Li, Y.Liu. Kähler-Einstein metrics and volume minimization.
C.Li, C.Xu. Stability of Valuations and Kollár Components.
C.Li, C.Xu. Stability of Valuations: Higher Rational Rank.
D.Martelli, J.Sparks, S.T.Yau. Sasaki–Einstein Manifolds and Volume Minimisation.
C.Xu. Interaction between singularity theory and the Minimal Model Program.