The volume of Kähler-Einstein Fano varieties and convex bodies

Abstract

We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a 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 C^n of independent interest. The paper supersedes our previous preprint concerning the case of toric Fano manifolds.

arXiv comment28 pages

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Cited by in corpus (5)

• Optimal bounds for the volumes of Kähler-Einstein Fano manifolds
• On K-stability and the volume functions of $\mathbb{Q}$-Fano varieties
• A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone
• On extensions of Minkowski's theorem on successive minima
• A volume stability theorem on toric manifolds with positive Ricci curvature