A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem

Abstract

For $\varphi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $${\int }_X{e}^{-\varphi }.$$ We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $X$. As consequences we get a simplified proof of the Bando-Mabuchi uniqueness theorem for Kähler - Einstein metrics and a generalization of this theorem to 'twisted' Kähler-Einstein metrics.

arXiv comment22 pages, revised and expanded

References in corpus (6)

• A variational approach to complex Monge-Ampere equations
• Analytic torsion, vortices and positive Ricci curvature
• A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics
• Curvature of vector bundles associated to holomorphic fibrations
• Regularity of plurisubharmonic upper envelopes in big cohomology classes
• Strict and non strict positivity of direct image bundles

Cited by in corpus (29)

• K-stability and Kähler-Einstein metrics
• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• Ricci flow on surfaces with conic singularities
• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
• Conical Kahler-Einstein metric revisited
• Kahler-Einstein metrics and stability
• The conical Kähler-Ricci flow on Fano manifolds
• Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's "hypothèse fondamentale"
• Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds
• Connecting toric manifolds by conical Kahler-Einstein metrics
• Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
• Smooth and singular Kahler-Einstein metrics
• A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics
• On the existence of conic Kahler-Einstein metrics
• Stability of Valuations: Higher Rational Rank
• The Continuity Method to Deform Cone Angle
• Geodesic Rays and Kähler-Ricci Trajectories on Fano Manifolds
• On the limit of spectral measures associated to a test configuration
• Properness of log $F$-functionals
• Curvatures of direct image sheaves of vector bundles and applications I
• The conical Kähler-Ricci flow with weak initial data on Fano manifold
• On the Spectrum of weighted Laplacian operator and its application to uniqueness of Kähler Einstein metrics
• On the Geodesics in the space of Kähler metrics with prescribed singularities
• On the uniqueness of conical Kähler-Einstein metrics
• A Bando-Mabuchi Uniqueness Theorem
• Isometry group of Sasaki-Einstein metric
• Stability of the conical Kähler-Ricci flows on Fano manifolds
• The space of Kähler potentials on an asymptotically locally euclidean Kähler manifold