Yuchen Liu - Volume upper bounds for K\'ahler-Einstein Q-Fano varieties

May 9, 2016 - Princeton University
This talk was part of the Princeton-Tokyo Algebraic Geometry Conference

A complex projective variety is said to be Q-Fano if it has klt singularities and the anti-canonical divisor is Q-Cartier and ample. Starting from dimension 2, the anti-canonical volume of a Q-Fano variety can be arbitrarily large, such as volumes of weighted projective spaces. Recently, Fujita showed that if an n-dimensional Q-Fano variety admits a K\"ahler-Einstein metric, then its volume is at most (n+1)^n. In this talk, I will discuss a refinement of Fujita's volume upper bounds involving invariants of the local singularities. If time permits, I will also talk about an equivalent relation between K-semistability and de Fernex-Ein-Musta\c{t}\u{a} type inequalities. Part of this work is joint with Chi Li.