Questions tagged [stratifications]

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9
votes
1answer
172 views

Local topology of Whitney stratified spaces

Let $M$ be a smooth manifold, let $\mathcal{P}$ be a Whitney stratification of $M$ and let $S\subset M$ be a stratum with closure $\overline{S}$. Question: Does there exist an open neighborhood $U\...
3
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2answers
281 views

Piecewise isomorphism versus equivalence in Grothendieck ring

$\DeclareMathOperator\Var{Var}$Let $K_{0}(\Var_{\mathbb{C}})$ be the Grothendieck ring of varieties over $\mathbb{C}$. The class of a variety, $X$, in $K_{0}$ is denoted $[\,X\,]$. If $X$ and $Y$ are ...
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0answers
19 views

Stratification which makes the defining functions isotrivial

Let $0\in X\subset\mathbb{C}^N$ be a germ of complex space and $0\in Z\subset X$ be a closed analytic subset (globally) defined by holomorphic functions $f_1,\dots,f_r$. Is there a complex analytic ...
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59 views

subanalytic realization of smooth abstract stratification

Consider an $C^\infty$ abstract stratification $A$ (in the Thom-Mather sense, see Mather's note). Can we embed $A$ in some $\mathbb{R}^n$ (or in an analytic manifold) as a subanalytic set? If not, ...
7
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1answer
217 views

Non-example for Whitney (a) stratifications

Given a $C^1$ stratification $\mathscr{S}$ of a $C^1$ manifold $M$, we write $N^\ast \mathscr{S}$ for the union of conormals to the strata. The stratification is said to be Whitney (a) if $N^\ast \...
3
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0answers
152 views

Genus two curves on abelian surfaces

Considering a smooth genus two curve $C_2$, let $J(C_2)$ be its Jacobian surface, and take $p \in J(C_2)$ an $m$-torsion point. Let $A = J(C_2)/Z_m$, where $Z_m$ acts by $x \mapsto x+p$. The image of $...
5
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1answer
168 views

Whitney stratification of algebraic varieties

When do the orbits of an action on an algebraic variety make a Whitney stratification?
8
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0answers
213 views

Why mu-stratifications?

In the microlocal theory of sheaves developed by Kashiwara and Schapira, there is the notion of a $\mu$-stratification, which is a stratification satisfying a stronger property ("$\mu$") than Whitney'...
4
votes
1answer
101 views

Image of a quiver variety under natural morphism

We know that the natural morphism $\pi:\mathfrak{M}_{\theta}(Q,\mathbf{v},\mathbf{w})\rightarrow \mathfrak{M}_0(Q,\mathbf{v},\mathbf{w})$ between a smooth and affine quiver variety is not necessarily ...
2
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0answers
156 views

Grothendieck group of constructible sets

Let $K_0$ be the Grothendieck group of complex algebraic varieties. This is the group generated by all complex algebraic varieties, subject to the relations: (i) $[X]=[Y]$ if $X,Y$ are isomorphic, (...
4
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0answers
211 views

A cell decomposition of a CW-complex and, stratification of a topological space

What is the difference between the notion of cell decomposition of a CW-complex, and the notion of stratification of a topological space ? I know that cell decomposition of a CW-complex is usefull to ...
16
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0answers
479 views

Proof of MacPherson's result about set-valued constructible sheaves and exit paths

I'm looking for a proof of a theorem that is attributed to MacPherson. Treumann (Section 1.1 in Exit paths and constructible stacks, 2009) states the theorem as: Theorem 1.2 (MacPherson). Let $(X,S)...
8
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0answers
158 views

Stratification of space of labelled circles in the plane

Consider the space of $n$ round circles in the plane to be the open subset of $\mathbb R^{3n}$: $$C_n = \{ (v_1, v_2, \cdots, v_n, r_1, r_2, \cdots, r_n ) : v_i \in \mathbb R^2, r_i \in (0, \infty) \ ...
1
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1answer
106 views

Confusion about locally cone-like spaces

Definition: A filtered space $X$ of formal dimension $n$ is locally cone-like if for all $i$, $0 \le i \le n$, and for each $x \in X^i - X^{i-1} = X_i$ there is an open neighborhood $U$ of $x$ in $X_i$...
0
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0answers
114 views

Isn't stratification by orbit types actually a stratification by stibilizer types?

I asked this question on Math Exchange but considering the law number of people who viewed the question, I think that the question is difficult enough to post it on math overflow. I hope I am right. ...
18
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1answer
531 views

Local homology of a space of unitary matrices

Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$). Let $$ {\cal D} \subset U(n) $$ denote the subspace of those matrices having a non-trivial $(+1)$-eigenspace. ...
3
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2answers
450 views

Whitney Conditions vs Equisingularity

In studying singular spaces, it is often important to pick an appropriate stratification which encodes the singularity structure. One class of such stratifications are called "Whitney stratifications" ...
3
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1answer
198 views

On the notion of conelike stratified (cs-) space

The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper "Intersection cohomology of cs-spaces and Zeeman's ...
3
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0answers
310 views

Where should I look for computing the intersection homology of projective varieties?

I'm learning about intersection cohomology topologically through MacPherson's "New York Times Article". This is a very nice guide which gives a nice idea on how to use these methods for low-...
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0answers
24 views

Sufficient conditions for a conormal vector to be regular for an orbit stratification

Let a complex reductive group $G$ act on a $\mathbb{C}^{n}$ with finitely many orbits. Let $\mathcal{S}$ be the stratification of $\mathbb{C}^{n}$ according to these orbits. Let $(x,\xi) \in T_S^{*}\...
11
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2answers
324 views

Homotopy property of constructible sheaves on stratified spaces

Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...
8
votes
1answer
427 views

Topology on the space of constructible sheaves

Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
6
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1answer
1k views

Stratification of complex algebraic varieties

Let $V$ be a complex quasi-projective variety, we know from H. Whitney's and B Teissier works on stratifications of algebraic varieties that $V$ has an intrinsic stratification $$X_0\subset X_2\...
4
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0answers
278 views

Stratification of a smooth map

So, this is an exercise. But from math.stackexchange I have been suggested to post this question here. To find the Thom-Boardman stratification of the smooth map $f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+...
7
votes
1answer
470 views

Iterated Milnor fibrations and Thom's a_f condition

Ok so there's a lot of litterature about nearby cycles functor since it was introduced by Grothendieck and Deligne but I couldn't find any clear answer to the following natural question: Problem: Let ...
3
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2answers
284 views

intersection of Whitney stratifications

Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is ...
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0answers
194 views

transverse intersection of Whitney stratifications

Let $M$ be a smooth manifold. If $X$ and $Y$ are two Whitney objects, i.e. subsets with a given Whitney stratification, then $X$ and $Y$ are transverse if each stratum of $X$ is transverse to each ...
2
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0answers
172 views

When a Whitney stratification has no stratum of codimension one?

Let $G$ be a compact Lie group, and $M$ be a smooth $n$-dimensional $G$-manifold which admits an orientation preserving the $G$-action. Then $M$ has a natural Whitney stratification induced by the ...
5
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0answers
512 views

singular support of D-module smooth w.r.t. a stratification

(1) Suppose that $X$ is a smooth complex algebraic variety, stratified by some nice smooth stratification $S$. Let $M$ be a $D$-module on $X$, s.t. its shriek-pullback (or star... whatever is ...
5
votes
1answer
372 views

Is the Alexander-Pontryagin duality applicable to stratified spaces

If $D$ is the discriminant of the space of all planar curves of a fixed degree, and $D'$ is the subspace whose only singularities are nodes or cusps, then is it possible to apply Alexander-Pontryagin ...