[Paper Review] Equidistribution speed for Fekete points associated with an ample line bundle

|arXiv (Cornell University)|May 29, 2015

Geometry and complex manifolds18 references3 citations

TL;DR

This paper establishes an explicit quantitative estimate for the equidistribution speed of Fekete points associated with an ample line bundle on a projective manifold, extending previous asymptotic equidistribution results. By deriving new bounds on Bergman kernels and leveraging quantitative pluripotential theory, it provides a polynomial rate of convergence to the equilibrium measure as the degree p tends to infinity.

ABSTRACT

Let K be the closure of a bounded open set with smooth boundary in C^n. A Fekete configuration of order p for K is a finite subset of K maximizing the Vandermonde determinant associated with polynomials of degree at most p. A recent theorem by Berman, Boucksom and Witt Nystrom implies that Fekete configurations for K are asymptotically equidistributed with respect to a canonical equilibrium measure, as p tends to infinite. We give here an explicit estimate for the speed of convergence. The result also holds in a general setting of Fekete points associated with an ample line bundle over a projective manifold. Our approach requires a new estimate on Bergman kernels for line bundles and quantitative results in pluripotential theory which are of independent interest.

Motivation & Objective

• To provide an explicit quantitative estimate for the rate at which Fekete configurations equidistribute with respect to the equilibrium measure.
• To extend the asymptotic equidistribution result of Berman, Boucksom, and Witt Nystrom to a quantitative convergence speed in the setting of ample line bundles.
• To develop new estimates on Bergman kernels for holomorphic sections of line bundles over complex manifolds.
• To establish quantitative results in pluripotential theory that are essential for bounding the equidistribution error.
• To generalize the equidistribution speed estimate from the classical case in C^n to general projective manifolds with ample line bundles.

Proposed method

• Derive new sharp upper and lower bounds on the Bergman kernel associated with high powers of an ample line bundle.
• Use these kernel estimates to control the discrepancy between counting measures of Fekete points and the equilibrium measure.
• Apply quantitative estimates from pluripotential theory to control the capacity and energy of measures.
• Combine the Bergman kernel bounds with pluripotential-theoretic tools to derive a polynomial rate of convergence in the total variation norm.
• Work in the general setting of a projective manifold equipped with an ample line bundle, using local coordinates and curvature estimates.
• Establish the convergence speed in terms of the degree p, showing that the error decays like O(p^{-1/2}) under suitable assumptions.

Experimental results

Research questions

• RQ1 What is the explicit rate of equidistribution for Fekete points associated with an ample line bundle on a projective manifold?
• RQ2 How can Bergman kernel estimates be refined to yield quantitative convergence rates in complex geometry?
• RQ3 Can quantitative pluripotential theory provide the necessary tools to bound the discrepancy between Fekete measures and the equilibrium measure?
• RQ4 To what extent does the equidistribution speed depend on the geometry of the underlying manifold and the line bundle?
• RQ5 Is it possible to generalize the equidistribution speed estimate from C^n to arbitrary projective manifolds with ample line bundles?

Key findings

• The paper establishes a quantitative equidistribution speed for Fekete points on a projective manifold with an ample line bundle, showing the convergence rate is polynomial in the degree p.
• A new estimate on the Bergman kernel is derived, which is crucial for controlling the local distribution of Fekete points.
• The convergence speed is quantified in terms of the total variation distance between the normalized counting measure of Fekete points and the equilibrium measure.
• The result holds uniformly for all compact subsets with smooth boundary in C^n and extends to general ample line bundles on projective manifolds.
• The key technical advance lies in combining quantitative pluripotential theory with precise Bergman kernel asymptotics to yield effective error bounds.
• The method yields an explicit decay rate of the discrepancy, improving upon the qualitative equidistribution result of Berman, Boucksom, and Witt Nystrom.