Regularization of closed positive currents and Intersection Theory

By J. Demailly. 2007

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Citations in the corpus

Kähler currents and null loci (2013, methodology): “Thanks to a result of Demailly-Păun [21], a compact complex manifold is in class C if and only if it admits a Kähler current.”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, background): “One can also observe that the existence of such a metric ω forces the cohomology class { ω } to be K¨ahler by Demailly’s regularization theorem [ Dem82, Dem92 ].”
Growth of balls of holomorphic sections and energy at equilibrium (2008, methodology): “16 When L is ample one can show using Demaily’s regularisation theorem [34] that the smooth weights φj in both parts of Proposition 2.”
The pseudo-effective cone of a non-Kählerian surface and applications (2006, methodology): “Following the standard terminology used in the algebraic and K¨ahlerian case [ De ], we define”
A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem (2011, background): “An extensive treatment of these matters can be found in [11].”
Kähler–Einstein metrics along the smooth continuity method (2015, unknown): “First, if we let η = ω′ + √ −1ds ∧ ds̄, then by the approximation theorem of Demailly [23] (see also Blocki-Kolodziej [13]) there exists a decreasing sequence of smooth metrics ρs,ǫ ց p∗τs such that √ −1∂∂s,W ′(ρs,ǫ) ≥ −Cη.”
A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry (2013, background): “An extensive treatment of these matters can be found in [12].”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, unknown): “Thanks to Demailly’s regularization theorem [24, 26], we have a Kähler current T = α + √ −1∂∂ψ with analytic singularities along the Zariski closed set EnK(α) ⊂ M, such that T ≥ δω weakly onM, for some δ > 0.”
The metric geometry of singularity types (2019, methodology): “of all θ-psh functions on Xwhich are not identically −∞. The class {θ} is big if there exists ψ∈ PSH(X,θ) satisfying θ+ i∂∂ψ¯ ≥ εωfor some ε>0. By the fundamental approximation theorem of Demailly [Dem92], if {θ} is big there are plenty of θ-psh functions. Given u,v∈ PSH(X,θ), we say that • uis m”
Extension of plurisubharmonic functions with growth control (2010, result): “ofsmoothfunctions ϕj∈ PSH(V,ω) whichdecrease pointwise onV sothatlim ϕj= ϕonX. When Xis smooth this regularization result is well known to hold even when the cohomology class of ωis not integral (see [D3], [BK]). Corollary C allows to show that the singular Ka¨hler-Einstein currents constructed in [”
Regularity of Plurisubharmonic Upper Envelopes in Big Cohomology Classes (2009, methodology): “ that the inequality ϕalg can 6 ϕ can is an equality. The proof is an immediate consequence of the following statement based in turn on the Ohsawa-Takegoshi theorem and the approximation technique of [Dem92]. (5.19) Proposition. With L= G+ ∆, ω, α= ΘK X+L,h K X+L , γas above and KX+Lassumed to be bi”