通过纳什流形的边缘实现棋盘集的纳什统一
Nash uniformization of chessboard sets by Nash manifolds with corners
AI总结：
本文研究了通过具有边缘的纳什流形对棋盘集进行纳什统一化的方法，并提供了相关理论和结果。
摘要：
Bierstone和Parusiński研究了d维闭合解析集的脱奇异化，特别是d维闭合半代数集。他们的主要工具是Hironaka对实代数集的脱奇异化（用于“统一”闭合半代数集的Zariski闭包）以及Hironaka对非奇异实代数集的嵌入脱奇异化（用于后续统一闭合半代数集边界的Zariski闭包）。在脱奇异化过程中获得的模型，我们称为闭合棋盘集，是（有限）连通分量的闭包的并集，这些连通分量是非奇异实代数集的正常交叉除子的补集的连通分量。d维棋盘集的局部模型是（标准）闭合正交象限的并集，即$\bigcup_{(\varepsilon_1,\ldots,\varepsilon_d)\in{\mathfrak F}}\{\varepsilon_1{\tt x}_1\geq0,\ldots,\varepsilon_d{\tt x}_d\geq0\}\subset{\mathbb R}^d$，其中${\mathfrak F}\subset\{-1,1\}^d$为某些集合。我们研究如何使用具有边缘的纳什流形${\mathcal Q}$对d维闭合棋盘集${\mathcal S}$进行纳什统一化，这些流形具有与${\mathcal S}$相同数量的连通分量（或等价地说，相同数量的不可约分量）。具有边缘的纳什流形是闭合棋盘集，其局部模型是${\mathbb R}^d$或半代数集，类型为$\{{\tt x}_1\geq0,\ldots,{\tt x}_k\geq0\}$，其中$1\leq k\leq d$。更一般地说，棋盘集是半代数集，介于有限个非奇异实代数集的正常交叉除子的补集的连通分量的并集及其闭包之间。我们还为一般的棋盘集${\mathcal S}$提供了一个纳什统一化结果。
英文摘要：
Bierstone and Parusiński studied the desingularization of $d$-dimensional closed subanalytic sets and in particular of $d$-dimensional closed semialgebraic sets. Their main tools are Hironaka's desingularization of real algebraic sets (to `uniform' the Zariski closure of the closed semialgebraic set) and Hironaka's embedded desingularization of real algebraic subsets of non-singular real algebraic sets (to uniform afterwards the Zariski closure of the boundary of the uniformed closed semialgebraic set). The obtained models in the desingularization process, that we call in the following closed chessboard sets, are the closures of (finite) unions of connected components of the complements of normal-crossings divisors of non-singular real algebraic sets. The local models for $d$-dimensional chessboard sets are unions of (standard) closed orthants of ${\mathbb R}^d$, that is, $\bigcup_{(\varepsilon_1,\ldots,\varepsilon_d)\in{\mathfrak F}}\{\varepsilon_1{\tt x}_1\geq0,\ldots,\varepsilon_d{\tt x}_d\geq0\}\subset{\mathbb R}^d$ for some set ${\mathfrak F}\subset\{-1,1\}^d$. We study the Nash uniformization of $d$-dimensional closed chessboard sets ${\mathcal S}$ using Nash manifolds with corners ${\mathcal Q}$ with the same number of connected components as ${\mathcal S}$ (or equivalently the same number of irreducible components). Nash manifolds with corners are closed chessboard set whose local models are either ${\mathbb R}^d$ or semialgebraic sets of the type $\{{\tt x}_1\geq0,\ldots,{\tt x}_k\geq0\}$ for some $1\leq k\leq d$. More generally, a chessboard set is a semialgebraic set in between a finite union of connected components of the complement of a normal-crossings divisor of non-singular real algebraic set and its closure. We also provide a Nash uniformization result for general chessboard sets ${\mathcal S}$.
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