Robert Berman: The volume of Kähler-Einstein Fano varieties

In this talk I will discuss a recent joint work with Bo Berndtsson, showing  that the complex projective space has maximal degree (volume) among all n−dimensional Kähler-Einstein Fano manifolds admitting a holomorphic vector field with simple zeroes. When specialized to toric varieties this confirms a conjecture of Nill-Paffenholz, which in turn, when translated to the realm of convex geometry, confirms Ehrhart’s volume conjecture for duals of lattice polytopes.  The proof uses Moser-Trudinger type inequalities for the complex Monge-Ampere operator.