Monge–Ampère equations in big cohomology classes

By S. Boucksom, Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi. 2008

430 citations recorded by the source corpus.

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L1+ε-density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/43beb3a66844ead3b46b9d05b1143ff2c998d82a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Nakai-Moishezon criterions for complex Hessian equations (2020, methodology): “We expect that the solution ω should be a Kähler current with vanishing Lelong number similar to degenerate complex Monge-Ampere equations studied in [2].”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, methodology): “Actually, [ BEGZ10 , Proposition 2.11] gives another characterization of this last space: a function u ∈ PSH ( X, ω ) belongs to E 1 ( X, ω ) if and only if u ∈ E ( X, ω ) and X ( u − V θ )MA( u ) + ∞ (and for any u ∈ E ( X, ω ), the explicit integral formula for E ( u ) above is still valid).”
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, methodology): “Following the terminology in [15] in the Kähler geometry setting we will denote this subclass of P(Rn) by EP (R n) (compare Remark 2.”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, background): “Since u has minimal singularities, it follows from [10] that”
Relations between the Kähler cone and the balanced cone of a Kähler manifold (2012, methodology): “Replacing Yau’s theorem by Boucksom-Eyssidieux-Guedj-Zeriahi’s theorem [6] on complex Monge-Ampère equations in big cohomology classes, we can generalize the above proposition on the map b to the map b.”
The metric geometry of singularity types (2019, background): “same singularity type as Vθ, i.e., [u] = [Vθ]. Given θ1,...,θn smooth closed (1,1)-forms and ϕj ∈ PSH(X,θj), j= 1,...n, following BedfordTaylor [BT76, BT82] in the local setting, it has been shown in [BEGZ10] that the sequence of positive measures 1 T j{ϕj>V θj − k}θ 1 max(ϕ1 ,V−k)∧...∧θ n n θn− ”