The volume of Kahler-Einstein varieties and convex bodies
Artikel i vetenskaplig tidskrift, 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.


Bo Berndtsson

Chalmers, Matematiska vetenskaper, Matematik

Göteborgs universitet

Robert Berman

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Journal für die Reine und Angewandte Mathematik

0075-4102 (ISSN)

Vol. 2017 127-152




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