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Hole event for random holomorphic sections on compact Riemann surfaces

Tien-Cuong Dinh, Subhroshekhar Ghosh, Hao Wu

Abstract

Let $X$ be a compact Riemann surface and $\mathcal L$ be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of $\mathcal L^n$ which do not vanish on $D$ for some fixed open subset $D$ of $X$. We prove that as $n$ tends to infinity, the zeros of these sections are equidistributed outside $D$ with respect to a probability measure $ν$. This gives rise to a surprising forbidden set.

Hole event for random holomorphic sections on compact Riemann surfaces

Abstract

Let be a compact Riemann surface and be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of which do not vanish on for some fixed open subset of . We prove that as tends to infinity, the zeros of these sections are equidistributed outside with respect to a probability measure . This gives rise to a surprising forbidden set.
Paper Structure (17 sections, 47 theorems, 219 equations)

This paper contains 17 sections, 47 theorems, 219 equations.

Table of Contents

  1. Introduction and main results
  2. Quasi-potentials
  3. Proof of Theorem \ref{['t:main-1']}
  4. Spaces of holomorphic sections of line bundles
  5. Symmetric space
  6. Jacobian variety
  7. Divisors on compact Riemann surfaces
  8. Admissible metric
  9. Riemann theta divisor
  10. $\mathcal{E}_m$ and $\mathcal{F}_m$
  11. Formula of $\mathscr{A} _m^*(V_n^{{\rm FS}})$
  12. Wasserstein balls and its measures
  13. Regularization of $\mathcal{I}_{\omega, D}$
  14. Large deviations and proof of Theorem \ref{['t:main-2']}
  15. Examples on the Riemann sphere and on a torus
  16. ...and 2 more sections

Key Result

Theorem 1.1

There is a unique probability measure $\nu\in \mathcal{M}(X\setminus D)$ which minimizes the functional $\mathcal{I}_{\omega,D}$ on $\mathcal{M}(X\setminus D)$. Moreover, the following properties hold.

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Lemma 2.6
  • ...and 81 more