Optimal bounds for the volumes of Kähler-Einstein Fano manifolds

Abstract

We show that any $n$-dimensional Fano manifold $X$ admitting Kähler-Einstein metrics satisfies that the anti-canonical volume is less than or equal to the value $(n+1{)}^n$. Moreover, the equality holds if and only if $X$ is isomorphic to the $n$-dimensional projective space.

25 pages

References in corpus (8)

• A generalization of Ross-Thomas' slope theory
• K-stability and Kähler-Einstein metrics
• K-stability of constant scalar curvature polarization
• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• A stronger concept of K-stability
• The volume of Kähler-Einstein Fano varieties and convex bodies
• The projective space has maximal volume among all toric Kähler-Einstein manifolds
• Examples of K-unstable Fano manifolds with the Picard number one

Cited by in corpus (17)

• Stability of Valuations and Kollár Components
• Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces
• On K-stability and the volume functions of $\mathbb{Q}$-Fano varieties
• K-stability of Fano varieties: an algebro-geometric approach
• A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties
• Uniqueness of K-polystable degenerations of Fano varieties
• Kähler-Einstein metrics and volume minimization
• Minimizing normalized volumes of valuations
• Calabi-Yau manifolds with isolated conical singularities
• Stability of Valuations: Higher Rational Rank
• Explicit Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds
• Generalized Kähler Einstein metrics and uniform stability for toric Fano manifolds
• A Note on Equivariant K-stability
• Interaction Between Singularity Theory and the Minimal Model Program
• On the semi-continuity problem of normalized volumes of singularities
• On exceptional sets in Manin's Conjecture
• On the spectral rigidity of Einstein-type Kähler manifolds