Questions tagged [stratifications]
The stratifications tag has no usage guidance.
27
questions
5
votes
0answers
51 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, ...
6
votes
1answer
160 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
votes
0answers
132 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 $...
1
vote
0answers
80 views
Whitney stratification of algebraic varieties
When do the orbits of an action on an algebraic variety make a Whitney stratification?
6
votes
0answers
189 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
78 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
votes
0answers
153 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,
(...
3
votes
0answers
180 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 ...
14
votes
0answers
378 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
votes
0answers
156 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 $\mathbb R^{3n}$:
$$C_n = \{ (v_1, v_2, \cdots, v_n, r_1, r_2, \cdots, r_n ) : v_i \in \mathbb R^2, r_i \in (0, \infty) \ ...
1
vote
1answer
99 views
Confusion about locally cone-like spaces
Definition: A filtered space $X$ of formal dimension $n$ is locally cone-like if for all $i$, $0 \le i \le n$, and for each $x \in X^i - X^{i-1} = X_i$ there is an open neighborhood $U$ of $x$ in $X_i$...
0
votes
0answers
106 views
Isn't stratification by orbit types actually a stratification by stibilizer types?
I asked this question on Math Exchange but considering the law number of people who viewed the question, I think that the question is difficult enough to post it on math overflow. I hope I am right.
...
18
votes
1answer
515 views
Local homology of a space of unitary matrices
Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$). Let
$$
{\cal D} \subset U(n)
$$
denote the subspace of those matrices having
a non-trivial $(+1)$-eigenspace.
...
3
votes
2answers
408 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
votes
1answer
181 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
votes
0answers
296 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-...
1
vote
0answers
22 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
votes
2answers
308 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
408 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
votes
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\...
3
votes
0answers
265 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
444 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
votes
2answers
268 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 ...
0
votes
0answers
189 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
votes
0answers
167 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
votes
0answers
497 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
354 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 ...
