A variational approach to complex Monge-Ampere equations

Abstract

We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kaehler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kaehler metrics and Berndtsson's positivity of direct images we extend Ding-Tian's variational characterization and Bando-Mabuchi's uniqueness result to singular Kaehler-Einstein metrics. Finally using our variational characterization we prove the existence, uniqueness and convergence of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy in our sense.

54 pages, no figures

References in corpus (7)

• Invariance of Plurigenera
• The Space of Kähler metrics (II)
• Existence of minimal models for varieties of log general type
• Some numerical results in complex differential geometry
• A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring
• Regularity of plurisubharmonic upper envelopes in big cohomology classes
• Bergman kernels and equilibrium measures for line bundles over projective manifolds

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• Statistical mechanics of permanents, real-Monge-Ampere equations and optimal transport
• A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry
• On the limit of spectral measures associated to a test configuration
• Solution to a non-Archimedean Monge-Ampère equation
• Kähler-Einstein metrics with cone singularities on klt pairs
• On Dirichlet's principle and problem
• Energy functionals for Calabi-Yau metrics
• On the uniqueness of conical Kähler-Einstein metrics