非退化局部驯服奇点上的分层Morse临界点与Brasselet数
Stratified Morse critical points and Brasselet number on non-degenerate locally tame singularities
AI总结：
本文证明在非退化局部驯服完备交奇点族上，Brasselet数之差与Milnor纤维正则部分上Morse临界点数目相关，连接了拓扑与几何性质，并导出基于Newton多面体的公式。
摘要：
由Goresky和MacPherson提出的Morse理论的推广是一个里程碑，它彻底区分了奇异空间的拓扑研究与几何研究。设$\{X_t\}_t$是$\mathbb{C}^n$中完备交簇在$0$处的一个合适的芽族，$\{f_t\}_t, \{g_t\}_t$是定义在$X_t$上的非常值多项式函数族。如果对于每个$t$，芽$X_t$、$X_t \cap f_t^{-1}(0)$和$X_t\cap f_t^{-1}(0) \cap g_t^{-1}(0)$都是非退化的、局部驯服的完备交簇，我们证明Brasselet数${\rm B}_{f_t,X_t}(0)$与${\rm B}_{f_t,X_t\cap g_t^{-1}(0)}(0)$之差与$f_t$的Milnor纤维正则部分上在$g_t$的morse化中出现的Morse临界点的数目相关，即使在$g_t$的临界轨迹具有任意维数的情况下也是如此。这一结果将拓扑性质与几何性质联系起来，并使我们能够确定一些有趣的公式，这些公式主要涉及Newton多面体的组合信息。
英文摘要：
The generalization of the Morse theory presented by Goresky and MacPherson is a landmark that divided completely the topological and geo\-me\-tri\-cal study of singular spaces. Let \{$X_t\}_t$ be a suitable family of germs at $0$ of complete intersection varieties in $\mathbb{C}^n$ and $\{f_t\}_t, \{g_t\}_t$ families of non-constant polynomial functions on $X_t$. If the germs $X_t$, $X_t \cap f_t^{-1}(0)$ and $X_t\cap f_t^{-1}(0) \cap g_t^{-1}(0)$ are non-degenerate, locally tame, complete intersection varieties, for each $t,$ we prove that the difference of the Brasselet numbers, ${\rm B}_{f_t,X_t}(0)$ and ${\rm B}_{f_t,X_t\cap g_t^{-1}(0)}(0)$, is related with the number of Morse critical points {on the regular part of the Milnor fiber} of $f_t$ appearing in a morsefication of $g_t$, even in the case where $g_t$ has a critical locus with arbitrary dimension. This result connects topological and geometric properties and allows us to determine some interesting formulae, mainly in terms of the combinatorial information from Newton polyhedra.
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