The volume of Kahler-Einstein varieties and convex bodies

Journal article, 2017

We show that the complex projective space ℙnhas maximal degree (volume) among all n-dimensional Kähler-Einstein Fano manifolds admitting a non-trivial holomorphic ℂ∗-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart's volume conjecture for a large class of rational polytopes, including duals of lattice polytopes. The case of spherical varieties/multiplicity free symplectic manifolds is also discussed. The proof uses Moser-Trudinger type inequalities for Stein domains and also leads to criticality results for mean field type equations in ℂnof independent interest. The paper supersedes our previous preprint [5] concerning the case of toric Fano manifolds.