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How curved is a random complex curve?

Michele Ancona, Damien Gayet

Abstract

In this paper, we study the curvature properties of random complex plane curves. We bound from below the probability that a uniform proportion of the area of a random complex degree $d$ plane curve has a curvature smaller than $-d/8$. Our lower bound is uniform, in the sense that it does not depend on $d$. We also provide uniform upper bounds for similar probabilities. These results extend to random complex curves of projective surfaces equipped with an ample line bundle. This paper can be viewed as a sequel of [1], where other metric statistics were given. On a larger time scale, it joins the general program initiated in [11] of understanding random complex hypersurfaces of projective manifolds.

How curved is a random complex curve?

Abstract

In this paper, we study the curvature properties of random complex plane curves. We bound from below the probability that a uniform proportion of the area of a random complex degree plane curve has a curvature smaller than . Our lower bound is uniform, in the sense that it does not depend on . We also provide uniform upper bounds for similar probabilities. These results extend to random complex curves of projective surfaces equipped with an ample line bundle. This paper can be viewed as a sequel of [1], where other metric statistics were given. On a larger time scale, it joins the general program initiated in [11] of understanding random complex hypersurfaces of projective manifolds.
Paper Structure (9 sections, 14 theorems, 102 equations)

This paper contains 9 sections, 14 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. The standard projective setting
  3. The general setting
  4. Proofs of the theorems
  5. Deterministic preliminaries
  6. The Bargmann-Fock field
  7. Proof of Theorem \ref{['sy']}
  8. Proof of Theorem \ref{['deux']}
  9. Asymptotics of the Bergman kernel

Key Result

Theorem 1.1

There exists $c>0$ such that

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Example 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Lemma 2.1
  • ...and 5 more