Regularity of Plurisubharmonic Upper Envelopes in Big Cohomology Classes

By Robert J. Berman, J. Demailly. 2009

117 citations recorded by the source corpus.

The goal of this work is to prove the regularity of certain quasiplurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of Hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes are locally bounded outside an analytic set of singularities. It is furthermore shown that a parametrized version of this result yields a priori inequalities for the solution of the Dirichlet problem for a degenerate Monge–Ampere operator; applications to geodesics in the space of Kahler metrics are discussed. A similar technique provides a logarithmic modulus of continuity for Tsuji’s “supercanonical” metrics, that generalize a well-known construction of Narasimhan and Simha.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7c03deefc882501737980a08b4d8f7d94ee7fff5

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Complex Monge-Ampere equations and totally real submanifolds (2009, background): “see also Mabuchi [51] and Semmes [57]. There has been interesting work in this direction, e.g. by Chen [16], Chen and Tian [17], Phong and Sturm [54], [55], [56], Blocki [10], and Berman and Demailly [7]. The HCMA equation (ψ≡ 0 in (1.1)), which is well deﬁned on general complex manifolds, also aris”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, unknown): “ Lemmas 4.1 and 4.3 give uniform C1,1 bounds for fε,βon compact subsets away from EnK(α), it follows easily that uis C1,1 loc(M\EnK(α)), as desired. Lastly we prove (1.4), following the same lines as [2, 6, 44]. Away from EnK(α) we can deﬁne the Monge-Amp`ere operator (α+ √ −1∂∂u)n (either in the se”
Kähler currents and null loci (2013, background): “Ample and non-Kähler loci were introduced by Boucksom [5], and they have been much studied recently in connection with the regularity for complex Monge–Ampère equations in big cohomology classes [2,7,20].”
On the singularity type of full mass currents in big cohomology classes (2016, methodology): “It follows from Demailly’s approximation theorem that Vθ is continuous in the ample locus Amp(θ).”
Geometric pluripotential theory on Kähler manifolds (2019, methodology): “7 was first proved by Berman–Demailly using methods from pluripotential theory[15].”
Plurisubharmonic envelopes and supersolutions (2017, background): “By the fundamental approximation theorem of Demailly [Dem92] any quasi-plurisubharmonic function u can be approximated from above by a sequence (uj) of (α+ εjω)-psh functions with analytic singularities, where α is a closed smooth (1, 1)-form such that α + ddcu ≥ 0.”
Growth of balls of holomorphic sections and energy at equilibrium (2008, background): “This result has now been extended to the case of an arbitrary big cohomology class in H 1,1(X,R) in [11].”