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The volume of Kahler-Einstein varieties and convex bodies

Journal article
Authors Bo Berndtsson
Robert Berman
Published in Journal für die Reine und Angewandte Mathematik
Issue 723
Pages 127–152
ISSN 0075-4102
Publication year 2017
Published at Department of Mathematical Sciences
Pages 127–152
Language en
Subject categories Mathematics


We show that the complex projective space Pn has 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 Cn of independent interest. The paper supersedes our previous preprint [5] concerning the case of toric Fano manifolds.

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