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摘要翻译：
{\bf构造。}对于孤立临界值为0的支配多项式映射{$f:k^n\到k^l$}($k$是特征为零的代数闭域），我们构造了一个闭的{\it丛}$g_f\子集t^{*}k^n$。我们将$G_F$限制在$F^{-1}(0)$中的$F$的临界点$sing(F)$上，并将$sing(F)$划分为具有常数维数的$G_F$纤维的点的{It'quasistrata'}。结果表明，当丛的纤维在准层的光滑点处与切空间正交时（例如当$L=1$时），在$F^{-1}(0)$附近存在T-W-a（Thom和Whitney-a）层。而且，只有当$S$对于{T-W-a}分层类是{\bf通用}时，后者才是准层的不可约分量$S$上的正交补，这意味着对于该类中的任何$\{S_J'\}_J$,$\sing(F)=\cup_J S'_J$,存在$S_J'$的分量$S'$，且$S\cap S'$在$S$和$S'$中都是开放和密集的。{\bf结果。}我们证明了只要$G_F$的所有纤维都是准层各自切空间的正交补，就存在仅有通用层的T-W-a分层，然后后者对$Sing(F)$的划分产生最粗的{It通用T-W-a分层}。其关键成分是我们的{bf奇异空间的Sard型定理}，其中奇点被认为是非临界的，因为附近的非奇点是一致非临界的（例如，对于一个支配映射$F:X\到Z$意味着$F$的雅可比矩阵的$l\乘以l$次子的绝对值之和，其中$l=\dim(Z)$不仅不消失，而且用一个正常数与零分开）。
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英文标题：
《Construction of universal Thom-Whitney-a stratifications, their
functoriality and Sard-type Theorem for singular varieties》
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作者：
D.Grigoriev, P.Milman
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最新提交年份：
2009
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Differential Geometry 微分几何
分类描述：Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
复形，接触，黎曼，伪黎曼和Finsler几何，相对论，规范理论，整体分析
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英文摘要：
{\bf Construction.} For a dominating polynomial mapping {$F: K^n\to K^l$} with an isolated critical value at 0 ($K$ an algebraically closed field of characteristic zero) we construct a closed {\it bundle} $G_F \subset T^{*}K^n $. We restrict $ G_F $ over the critical points $Sing(F)$ of $ F$ in $ F^{-1}(0)$ and partition $Sing(F)$ into {\it 'quasistrata'} of points with the fibers of $G_F$ of constant dimension. It turns out that T-W-a (Thom and Whitney-a) stratifications 'near' $F^{-1}(0)$ exist iff the fibers of bundle $G_F$ are orthogonal to the tangent spaces at the smooth points of the quasistrata (e. g. when $ l=1$). Also, the latter are the orthogonal complements over an irreducible component $ S $ of a quasistratum only if $S $ is {\bf universal} for the class of {T-W-a} stratifications, meaning that for any $\{S_j'\}_j $ in the class, $ \Sing (F) = \cup_j S'_j $, there is a component $S' $ of an $ S_j' $ with $S\cap S'$ being open and dense in both $S $ and $ S' $. {\bf Results.} We prove that T-W-a stratifications with only universal strata exist iff all fibers of $G_F$ are the orthogonal complements to the respective tangent spaces to the quasistrata, and then the partition of $\Sing(F)$ by the latter yields the coarsest {\it universal T-W-a stratification}. The key ingredient is our version of {\bf Sard-type Theorem for singular spaces} in which a singular point is considered to be noncritical iff nonsingular points nearby are 'uniformly noncritical' (e. g. for a dominating map $ F: X \to Z $ meaning that the sum of the absolute values of the $l\times l$ minors of the Jacobian matrix of $ F $, where $ l = \dim (Z) $, not only does not vanish but, moreover, is separated from zero by a positive constant).
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PDF链接：
https://arxiv.org/pdf/0811.1373
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