Questions tagged [singular-support]
The singular-support tag has no usage guidance.
14
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Sheaf with no singular support in $\mathbb{R}$-direction decomposes as an external tensor product
Consider $X$ to be a smooth manifold. Denote by $p_1, p_2$ respectively the projections $X\times Y$ to $X$ and $Y$ respectively. Recall the definition of microsupport/singular support of a sheaf (...
- 71
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Jet at a singular point or a submanifold
Let $M$ be a smooth manifold, $p\in M$ and $f\in C^\infty(M\setminus\{p\})$.
We will say that $f$ has a power-law singularity at $p$ of order $\eta$ if for every smooth immersion $\gamma:(-1,1)\to M$ ...
- 1,488
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480
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Two notions of singular support?
Arinkin-Gaitsgory have defined the notion of singular support for any quasismooth $Y$
$$\text{SS}(\mathcal{F})\ \subseteq\ \text{Sing}(Y)$$
and $\mathcal{F}$ any ind-coherent sheaf, where $\text{Sing}(...
- 5,651
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258
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Relation between characteristic cycle and singular support of constructible sheaf
Let $M$ be a real analytic manifold. Let $F$ be an object of the bounded derived category of sheaves on $M$ with real constructible cohomology sheaves. Let $CC(F)$ denote the characteristic cycle of $...
- 21.4k
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Is the cone $\Sigma(T)$ orthogonal to the singular support of a distribution?
Hello I am totally new to microlocal analysis and I have a question. Is the cone $\Sigma(T)$ orthogonal to the singular support of a distribution?
$$\xi \notin \Sigma(T) \iff \exists V\ conic\ ...
- 97
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1
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Stratification along which a constructible complex is smooth
Let $X$ be a smooth complex algebraic variety. A constructible complex $F$ on $X$ has a singular support $SS(F)\subset T^*X$. Assume you are given a stratification of $X$ such that $SS(F)$ is the ...
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Is the characteristic cycle map for perverse sheaves injective?
Let $X$ be a smooth irreducible complex variety. Is the characteristic cycle map from the Grothendieck group of perverse sheaves (with complex coefficients) on $X$ to the free abelian group generated ...
- 333
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Singular Radon probabilities on $[0,1]^d$. Is conditioning on $x_i = \alpha$ well-defined?
The question:
Let $\pi$ be a Radon probability measure on $[0,1]^d$, $2\leq d < \omega$, that is singular (w.r.t. to the $d$-dimensional Lebesgue measure).
Suppose that for $i\in \{1,\dots,d\}$ and ...
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Limit involving singular kernel: $\lim_{s\to 1}(1-s)\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{d+2s}} d y. $
Let $\Omega\subset \Bbb R^d$ be a bounded $C^1$ domain. Let $u:\Bbb R^d\to \Bbb R$ be a function in $C^2_b(\Bbb R^d)$. I would like to compute the following limit: for $x\in \partial \Omega$
$$L= \...
- 1,043
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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'...
- 18.6k
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Wavefront set and Duhamel's principle
Consider the Cauchy problem:
$$
\frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0,
$$
where $A$ has real principal ...
- 43
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Interesting (non) examples of singular support
I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $...
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Is every continuous microlocal operator a pseudo-differential operator?
Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space.
Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal.
By being microlocal I mean that the wave ...
- 8,056
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Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.6k