Questions tagged [local-systems]

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«Euclidean» local systems

The moduli space of G-local systems on a surface is a fundamental object in mathematics. The cases $G=SU(2)$ and $G=SL_2(\mathbb{R})$ are of particular interest. Consider the group $E$ of isometries ...
6
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0answers
188 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'...
5
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2answers
364 views

Explicit Riemann Hilbert correspondence

For simplicity, we assume that $X=\mathbb P_{\mathbb C}^1-\{s_1, s_2, \dots, s_k\}$ and $\infty \in X$. Consider the trivial bundle $E=\mathcal O_X^r$ with the connection $\nabla$ induced by a ...
7
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0answers
240 views

Category of representations of the path-groupoid

The path-groupoid $\mathcal{P}_1(X)$ of a (smooth) topological space $X$ is a refinement of the fundamental groupoid $\Pi_1(X)$ whose morphisms are given by (piecewise smooth) paths in $X$ up to thin-...
2
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0answers
88 views

Monodromy and simple system of local coefficients

I was interested in the following question: if one has a fibration $F\to E\to B$ there is associated a monodromy map, that is basically an action of the fundamental group $\pi_1(B)$ on the ...
3
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0answers
362 views

Monodromy representations are “quasi-unipotent”

Let $S$ be a smooth complex algebraic variety, let $b$ be a closed point of $X$, and let $f:X\to S$ be a polarized family of smooth projective varieties over $S$. This induces a monodromy ...
1
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0answers
50 views

Restriction of irreducible integrable connections

Let $X$ be a complex analytic manifold and let $\mathcal{M}$ be an integrable connection on $X$, i.e. a $\mathcal{D}_X$-module which happens to be coherent as an $\mathcal{O}_X$-module. If $U$ is any ...
0
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0answers
112 views

Framed braids and local systems

Let me start by admitting that my question is going to be somewhat vague. But hopefully it is one of these vague questions that can be immediately answered by an expert in the appropriate area. ...
6
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1answer
799 views

good reference for the Hitchin fibration

can you please recommend me a good reference to learn about the Hitchin fibration in the language of algebraic geometry?
5
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1answer
524 views

local systems with finite monodromy

This is a question on a sentence in the paper "Faisceaux pervers", p. 163. The say that if $j: U \hookrightarrow X$ is a Zariski open subset and $L$ is a local system on $U$ with finite monodromy, ...
6
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1answer
763 views

What is a higher derived constructible sheaf

Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define the category of "$\infty$-local systems on $X$" to be DG representations of the ring $C_*(\Omega X,k)$ of ...