论文标题
Chern类别和正常矩阵在拓扑空间上的单一等效性
Chern classes and unitary equivalence of normal matrices over topological spaces
论文作者
论文摘要
本文继续作者在矩阵与拓扑空间（空间上矩阵）复杂值函数中的条目中的单一等效性问题的问题继续进行。具体而言，我们在这里考虑了在共享一个共同特征多项式的空间上对成对矩阵的单一等效性问题，该空间可以全球化为不同的线性因子。我们表明，这种矩阵是可以对角线化的，并且仅当其特征捆绑的第一个Chern类都消失并得出作为一个应用程序时，所有此类矩阵都超过$ \ Mathbb {c} p^m $都是可对角线的。接下来，给定一个在$ c（x）[X）[λ] $中的CW复合物$ x $和多项式$ $ $ $ $ $ $ $ $ $ $ $ $ $ $，我们证明，具有$μ$作为特征多项式的单一等价矩阵的数量仅取决于空间$ x $，并且在其中估计了一些估计，我们可以在其中估计了一些估计。如果$ x $最多是三个尺寸的CW综合体，则我们证明了$ n \ times n $ n $普通矩阵的单一等价类别具有特征性多项式$μ$的元素与组$ $（h^2（x））^{n-1} $之间的两者。最后，当$ x $是一种平滑的多种流形，我们限制在带有光滑条目的矩阵时，我们构建了一个de rham的共同体学类别，该类别的不变是对单一等效性的障碍。
This paper continues the authors' work on the question of unitary equivalence of matrices with entries in the complex-valued functions of a topological space (matrices over spaces). Specifically, we here consider the question of unitary equivalence for pairs of normal matrices over a space that share a common characteristic polynomial that can be globally factored into distinct linear factors. We show that such a matrix is diagonalizable if and only if the first Chern classes of its eigenbundles all vanish and derive as an application that all such matrices over $\mathbb{C}P^m$ are diagonalizable for $m > 1$. Next, given a CW complex $X$ and a polynomial $μ$ in $C(X)[λ]$ that globally splits into distinct linear factors, we prove that the number of unitary equivalence classes of matrices with $μ$ as a characteristic polynomial depends only on the space $X$ and the degree of $μ$, and we give some estimates on how many unitary equivalence classes there can be. In the case that $X$ is a CW complex of dimension at most three, we demonstrate a bijection between the unitary equivalence classes of $n \times n$ normal matrices with characteristic polynomial $μ$ and elements of the group $(H^2(X))^{n-1}$. Finally, when $X$ is a smooth manifold and we restrict to matrices with smooth entries, we construct a de Rham cohomology class whose nonvanishing is an obstruction to unitary equivalence.
