The volume of singular Kahler-Einstein Fano varieties

We show that the anti-canonical volume of an n-dimensional Kahler-Einstein Q-Fano variety is bounded from above by certain invariants of the local singularities, namely lct(n). mult for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kahler-Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex-Ein-Mustata type inequality holds on its affine cone.