The volume of Kähler–Einstein Fano varieties and convex bodies

Abstract

Abstract We show that the complex projective space ℙ n ${\mathbb{P}^{n}}$ has maximal degree (volume) among all n-dimensional Kähler–Einstein Fano manifolds admitting a non-trivial holomorphic ℂ * ${\mathbb{C}^{*}}$ -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 ℂ n ${\mathbb{C}^{n}}$ of independent interest. The paper supersedes our previous preprint [5] concerning the case of toric Fano manifolds.