Berezin-Toeplitz operators, Kodaira maps, and random sections

TL;DR

This paper investigates the asymptotic behavior of zeros of sections formed by Berezin-Toeplitz operators acting on random holomorphic sections over Kähler manifolds, revealing different regimes depending on the principal symbol's value.

Contribution

It provides a second order approximation of the expected zero distribution of these sections and analyzes the convergence behavior of associated Fubini-Study forms.

Findings

Expected zero distribution exhibits two regimes depending on the principal symbol.

Fubini-Study forms converge to the Kähler form as currents but not as smooth forms.

Zero set of the principal symbol can be recovered from zeros of the sections.

Abstract

We study the zeros of sections of the form $T_k s_k$ of a large power $L^{\otimes k} \to M$ of a holomorphic positive Hermitian line bundle over a compact K\''ahler manifold $M$, where $s_k$ is a random holomorphic section of $L^{\otimes k}$ and $T_k$ is a Berezin-Toeplitz operator, in the limit $k \to +\infty$. In particular, we compute the second order approximation of the expectation of the distribution of these zeros. In a ball of radius of order $k^{-\frac{1}{2}}$ around $x \in M$, assuming that the principal symbol $f$ of $T_k$ is real-valued and vanishes transversally, we show that this expectation exhibits two drastically different behaviors depending on whether $f(x) = 0$ or $f(x) \neq 0$. These different regimes are related to a similar phenomenon about the convergence of the normalized Fubini-Study forms associated with $T_k$: they converge to the K\''ahler form in the sense of currents as $k\rightarrow + \infty$, but not as differential forms (even pointwise). This contrasts with the standard case $f=1$, in which the convergence is in the $\mathscr{C}^{\infty}$-topology. From this, we are able to recover the zero set of $f$ from the zeros of $T_k s_k$.