AI 中文总结
研究紧致复流形上由特定空间定义的格拉斯曼嵌入中诱导格拉斯曼流形陈类形式的渐近展开，利用其首项渐近和相关理论，证明随机截面线性相关轨迹上积分流的收敛性，行列式情形下给出基于威沙特分布的方法及方差估计。
AI 中文摘要
对于由空间$H^0(X,L^p\otimes E)$定义的格拉斯曼嵌入，其中$L$是紧致复流形上的正线丛，$E$是全纯向量丛，我们证明了诱导格拉斯曼流形陈类形式的完整渐近展开并明确计算了首项系数。作为一阶渐近和丁与西博尼的亚纯变换理论的应用，我们证明在紧致凯勒流形上，几个随机截面线性相关的轨迹上的归一化积分流几乎必然收敛到正线丛曲率形式的相应幂次，并给出收敛速度的定量估计。此外，在行列式情形下，我们还基于威沙特分布提出了一种替代方法及方差估计。
英文摘要
For the Grassmannian embeddings defined by the spaces $H^0(X,L^p\otimes E)$, where $L$ is a positive line bundle and $E$ is a holomorphic vector bundle over a compact complex manifold, we prove a complete asymptotic expansion of the induced Grassmannian Chern forms and compute the first coefficients explicitly. As an application of the first-order asymptotics and of the theory of meromorphic transforms by Dinh and Sibony, we prove that on a compact Kähler manifold, the normalized currents of integration over the loci where several random sections become linearly dependent converge almost surely to the corresponding power of the curvature form of the positive line bundle, with a quantitative estimate for the speed of convergence. Moreover, in the determinant case, we additionally present an alternative method based on the Wishart distribution, together with variance estimates.
Comments31 pages
