A variational approach to complex Monge-Ampère equations

By Robert J. Berman, S. Boucksom, Vincent Guedj, Ahmed Zeriahi. Publications mathématiques (Bures-sur-Yvette), 2009

284 citations recorded by the source corpus.

We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7a4db1065bd431c20bbcebf8fa0df7ee5d28af3b

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Citations in the corpus

Geodesic rays and stability in the cscK problem (2020, methodology): “Using the property of mixed Monge-Ampère operators and the non-Archimedean estimates developed in [19], it is easy to adapt the proof in the Archimedean case (see [8, 35]) to prove the following useful result.”
Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, unknown): “[4], [9], [5], [10]) 3.”
Smooth and singular Kahler-Einstein metrics (2014, background): “The weak convergence when τ ≤ 1 in general was first established in [22] (and the strong convergence then can be deduced from arguments of [214, 141]) using a new pluripotential estimate from [23, 18] that can be stated as follows: Lemma 6.”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, methodology): “Moreover, the current ω will be shown to be of ﬁnite energy, in the sense of [ GZ07, BBGZ13 ] and as discussed in the last section of the present paper this allows one to deﬁne a canonical (singular) Weil-Peterson metric on the compact moduli space in terms of Deligne pairings.”
A variational approach to the Yau–Tian–Donaldson conjecture (2015, background): “k a Kahler form in this class. The equation Ric(! u) = s! u+ + with u2E1 corresponds to a Monge-Ampere equation of the form MA(u) = e2(su ˆ)!n 0 with ddc and ˆsmooth, and hence admits a solution [BBGZ13]. It follows that s (X;L) 0 =)s (X;L) = (X;L), which proves the theorem in that case. Assume now”
Plurisubharmonic envelopes and supersolutions (2017, methodology): “er the equations (1.3) MA(ϕ) = eλϕµ, where λ∈ Ris a constant. When λ= 0 a normalization condition µ(X) = Vol(θ) should be imposed. We will use the following result obtained by a variational method in [BBGZ13]: Theorem 1.12. Assume that λ > 0. Then there exists a unique ϕ ∈ E1(X,θ) solving (1.3). ”
Geometric pluripotential theory on Kähler manifolds (2019, methodology): “In Chapter 2, we develop some background in finite energy pluripotential theory, necessary for later developments, closely following the original treatises of Guedj–Zeriahi and collaborators [71, 72, 11, 12], that were inspired by work of Cegrell [33] in the local case.”
Kahler-Einstein metrics with edge singularities (2011, unknown): “10 ([11], [9]).”