Fekete points and convergence towards equilibrium measures on complex manifolds

By Robert J. Berman, S. Boucksom, David Witt Nyström. 2009

154 citations recorded by the source corpus.

Building on [BB1] we prove a general criterion for convergence of (possibly singular) Bergman measures towards pluripotential-theoretic equilibrium measures on complex manifolds. The criterion may be formulated in terms of the growth properties of the unit-balls of certain norms on holomorphic sections, or equivalently as an asymptotic minimization property for generalized Donaldson L-functionals. Our result settles in particular a well-known conjecture in pluripotential theory concerning the equidistribution of Fekete points and it gives the convergence of Bergman measures towards the equilibrium measure for Bernstein-Markov measures. Applications to interpolation of holomorphic sections are also discussed.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f0becb94cf26e8267afa2d72241aeca80fe4c7b2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry (2018, background): “s that Z X fδPm → Z X fµφ for all f∈ C0(Xan). Here δP denotes the averaging measure over P∈ (XN)an, or rather its image in (Xan)N (see Remark 10.2). The proof of Theorem 10.10 follows the strategy of [BBW11], itself inspired by a variational argument due to Szpiro–Ullmo–Zhang [SUZ97]. Proof. Set for”
Growth of balls of holomorphic sections and energy at equilibrium (2008, background): “The following result is shown in [14], generalising results of Siciak [54].”
Equidistribution of zeros of random holomorphic sections (2013, unknown): “ompact set if VK,q is continuous on X. Throughout this work, we assume that VK,q is continuous and hence VK,q = V∗ K,q.Recall that a suﬃcient condition for continuity of VK,q is local regularity of K([34, 3]). It is well-known that TK,q:= ω+ddcVK,qdeﬁnes a positive closed (1,1) current representing ”
Learning with Hyperspherical Uniformity (2021, background): “Therefore, maximizing det(G) can be approximated by maximizing |det(Ψn)| which is also known as extremal systems [74] or Fekete points [55, 6, 35].”