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  concerning the case of toric Fano manifolds.