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arXiv · 1204.1308

The volume of Kähler-Einstein Fano varieties and convex bodies

Robert J. Berman·Bo Berndtsson
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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.

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Robert J. Berman, Bo Berndtsson. 2012-04-05. The volume of Kähler-Einstein Fano varieties and convex bodies. https://arxiv.org/abs/1204.1308

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