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A survey on asymptotic equilibrium distribution of zeros of random holomorphic sections

George Marinescu, Duc-Viet Vu

Abstract

This is a survey of results concerning the asymptotic equilibrium distribution of zeros of random holomorphic polynomials and holomorphic sections of high powers of a positive line bundle, as related to the authors' recent work. Our primary focus is on the role of pluripotential theory in this research area.

A survey on asymptotic equilibrium distribution of zeros of random holomorphic sections

Abstract

This is a survey of results concerning the asymptotic equilibrium distribution of zeros of random holomorphic polynomials and holomorphic sections of high powers of a positive line bundle, as related to the authors' recent work. Our primary focus is on the role of pluripotential theory in this research area.

Paper Structure

This paper contains 13 sections, 28 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Bergman kernel functions
  3. Extremal plurisubharmonic functions
  4. Definition and continuity of extremal plurisubharmonic functions
  5. Hölder regularity of extremal psh envelopes
  6. Bernstein-Markov inequality
  7. Polynomial growth of Bergman kernel functions
  8. Zeros of random polynomials of several variables
  9. Almost sure convergence
  10. Expectation of random zeros
  11. Large deviation estimates
  12. Proof of quantitative equidistribution of random zeros
  13. Further discussions and questions

Key Result

Lemma 3.1

The upper semi-continuous regularisation $V_{K,Q}^*$ of $V_{K,Q}$ belongs to $\mathcal{L}(\mathbb{C}^n)$. The function $V_{K,Q}$ is always lower semi-continuous.

Theorems & Definitions (39)

  • Definition 2.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.3
  • Remark 3.4
  • Definition 3.5
  • Theorem 3.6: NCNguyen-envelope
  • Definition 3.7
  • Theorem 3.8: Marinescu-Vu
  • Proposition 3.9
  • ...and 29 more