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The volume of Kähler–Einstein Fano varieties and convex bodies

Robert J. Berman EMAIL logo and Bo Berndtsson


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


We are grateful to Benjamin Nill for helpful comments on the toric setting, Michel Brion for his help with spherical varieties, Gabor Székelyhidi for encouraging us to consider the relation to the invariant R(X) and Bo’az Klartag for allowing us to include here his beautiful reduction to Grunbaum’s inequality. Finally, thanks to the referees for their helpful comments.


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Received: 2013-5-21
Revised: 2014-3-20
Published Online: 2014-8-31
Published in Print: 2017-2-1

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