Berezin–Toeplitz operators, Kodaira maps, and random 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 $\smash{k^{-\frac{1}{2}}}$ 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 $T_k s_k$.