论文标题
Berezin-toeplitz操作员，Kodaira地图和随机部分
Berezin-Toeplitz operators, Kodaira maps, and random sections
论文作者
论文摘要
我们研究了大功率$ t_k s_k $ $ l^{\ otimes k} \ to holomorphic积极的hermitian线捆绑的$ t_k s_k $的各个部分的零，而紧凑的k \''ahler complipold $ m $，$ s_k $是$ s_k $的$ s_k $是一个随机的holomorphic部分， berezin-toeplitz运营商，以$ k \ to +\ infty $。特别是，我们计算了这些零分布期望的二阶近似值。 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 $f(x) \neq 0$.这些不同的制度与与$ t_k $相关的归一化fubini-study形式的融合的类似现象有关：它们以$ k \ rightarrow + \ rightarrow + \ iftty $的意义收敛到k \''ahler形式，但不是作为不同的形式（甚至是不同的形式）。这与标准情况$ f = 1 $形成鲜明对比，其中收敛位于$ \ mathscr {c}^{\ infty} $ - 拓扑中。由此，我们能够从$ t_k s_k $的零恢复零集$ f $。
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 $T_k s_k$.
