Intrinsic capacities on compact Kähler manifolds

By Vincent Guedj, Ahmed Zeriahi. 2004

302 citations recorded by the source corpus.

We study fine properties of quasiplurisubharmonic functions on compact Kähler manifolds. We define and study several intrinsic capacities which characterize pluripolar sets and show that locally pluripolar sets are globally “quasi-pluripolar.”

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66401c2ea207c227304ac8bf8776f082fa5f8df6

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In Applied Mathematics

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Citations in the corpus

Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity (2017, unknown): “θ,φ) F>−∞. Since µ∈ MA, by (15) we have that F(uj) ≤ Iφ(uj) + C|Iφ(uj)|1/2 + C, ∀j. It thus follows that Iφ(uj) is uniformly bounded, hence supXujis also uniformly bounded. By compactness property [GZ05] a subsequence of (uj) converges to u∈ PSH(X,θ,φ), and supRXu= 0. Since Iφ is upper semicontin”
Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I (2011, background): “ respect to (E,Mτ,g) is deﬁned by Vg,E,τ(ζ) = sup{u(z) : u∈ PSH(Mτ),u|E ≤ 0,u|∂Mτ ≤ τ}. It is obvious that Vg,M,τ(ζ) ≥ √ ρ(ζ) and it is almost standard that Vg,M,τ(ζ) = √ ρ(ζ). See Proposition 4.1 of [GZ] or Corollary 9 of [BT2]. The set M= ( √ ρ)−1(0) is often called the center. As proved in [LS1],”
On regularization of plurisubharmonic functions on manifolds (2007, methodology): “They involve local approximation of plurisubharmonic functions by holomorphic functions from weighted Bergman spaces and use in particular the Hörmander and Ohsawa-Takegoshi L(2)-estimates for the ∂-operator (in [6] this method is applied globally for sections of a positive holomorphic line bundle).”
Plurisubharmonic envelopes and supersolutions (2017, background): “ the total mass of the Monge-Ampère measure of Vθ (see [Bou04, BEGZ10]). A set E ⊂ X is called pluripolar if locally it is contained in the −∞ locus of some plurisubharmonic function. It was shown in [GZ05] that Eis pluripolar if and only if E⊂ {φ= −∞} for some φ∈ PSH(X,ω). One can replace the Kähle”
The Monge-Ampère operator and geodesics in the space of Kähler potentials (2005, methodology): “Although there are now many powerful approximation theorems (see [8, 14 ] and references therein), we found it more convenient to extend the Bedford-Taylor uniqueness theorem to a situation adapted to the problem at hand, in the spirit of the earlier extension.”
Solution to a non-Archimedean Monge-Ampère equation (2011, methodology): “el functions. Finally, in §5 we prove the locality property. The results in §§3–5 and even some of the proofs parallel those in complex analysis (especially the ones on compact Ka¨hler manifolds, see [GZ05]). However, the non-Archimedean results ultimately originate in basic properties of the inters”
Bergman kernels and equilibrium measures for line bundles over projective manifolds (2007, methodology): “These inequalities are then combined with some L2-estimates and global pluripotential theory, the pluripotential part being based on the recent work [24] by Guedj-Zeriahi.”
Convergence of Calabi–Yau manifolds (2009, background): “ if there is a function f∈ Lp(dµ), p>1, such that ( √ −1∂∂u)n= fdµ, where dµis the standard Lebesgue measure, then Condition 2.3 is satisﬁed. In this case, we can choose h(y) = (1 +log(1+y))2n. In [32], the notion of relative capacity was generalized to global capacity on a compact Kahler manifol”
Equidistribution of zeros of random holomorphic sections (2013, unknown): “For a current T ∈ C k , we denote its action on a test form Φ by T, Φ . h i For a smooth ( p, q ) form Φ denote by Φ C α the sum of C α -norms of the coeﬃcients in a ﬁxed atlas. k k Following [16], for α > 0 we deﬁne a distance function on C k by where Φ is a smooth bidegree ( m − k, m − k ) form on”