# Kahler-Einstein metrics and volume minimization

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Chi Li, Purdue University
Fine Hall 314

Futaki invariant is an obstruction to the existence of Kahler-Einstein metrics on Fano manifolds. Martelli-Sparks-Yau showed that the Futaki invariant is the derivative of a normalized volume functional on the space of Reeb vector fields of associated affine cones and derived the volume minimization principle for more general Sasaki-Einstein metrics. I will show that this volume minimization principle can be extended to work on a much bigger space of centered real valuations. This gives an equivalent characterization of K-semistability (which is equivalent to almost Kahler-Einstein”) and has an interesting algebra-geometric consequence. If time permits, I will also discuss the generalization to the case of Sasaki-Einstein metrics and some relation to metric tangent cones on singular Kahler-Einstein varieties. This talk is partly based on joint works with Yuchen Liu and Chenyang Xu.