# Questions tagged [local-systems]

The local-systems tag has no usage guidance.

23
questions

9
votes

2
answers

523
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### Is a local system on a surface determined by simple closed loops?

Let $\Sigma$ be a closed oriented topological surface of genus $g\geq 2$, and let $\mathfrak{X}_n$ denote the $\mathrm{SL}_n$-character variety of $\pi_1(\Sigma)$, i.e.
$$
\mathfrak{X}_n= \mathrm{Hom}(...

3
votes

2
answers

271
views

### Determine monodromy representation from local system

Let $X$ be a path-connected manifold nice enough such it's universal covering
space $p:\widetilde{X} \to X$ exists, $k$ a field. Then there exist a wellknown
correspondence
$$
\{\textit{linear}\text{ ...

17
votes

0
answers

955
views

### Symmetries of local systems on the punctured sphere

Let $X=S^2\setminus D$, for $D\subset S^2$ some finite set of points, say with $|D|=n\geq 1$. The category of locally constant sheaves of $\mathbb{C}$-vector spaces on $X$ (equivalently, complex ...

1
vote

0
answers

96
views

### Equivariant cohomology with compact support of the generalized Jacobian of a nodal elliptic curve

[Question forwarded from SE for lack of interaction]
Given an geometric genus $1$ curve with $1$ node $C$ (hence with arithmetic genus $2$), its normalization is given by the elliptic curve $\tilde{C}$...

2
votes

0
answers

127
views

### Tangential basepoint of a log singular local system

Consider the Legendre family $f: X\longrightarrow Y = \mathbb{P}^1\setminus\{0,1,\infty\}$, defined over $K = \mathbb{Q}/\mathbb{Q}_p$.
having fibre above $t\in Y$, the elliptic curve $E_t := y^2 = x(...

2
votes

0
answers

200
views

### Monodromy group action on de Rham cohomology

Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...

1
vote

1
answer

316
views

### Higher homotopy local systems

The concept of a local system in algebraic geometry is often described as a locally constant constructible sheaf on a scheme $X$, which is in essence a sheaf whose stalk at a point $x$ comes equipped ...

3
votes

1
answer

256
views

### Projective dimension of group ring

Assume that $G$ is a group and $R$ is a p.i.d. What can we say about the projective dimension of $R[G]$? For example can we say that this dimension is at most $1$ for reductive groups? (I think if $...

15
votes

0
answers

857
views

### Finiteness for motivic local systems

Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$. Say a complex local system $\mathbb{V}$ on $X$ is motivic if there exists a dense Zariski-open subset $U\subset X$, and a smooth proper ...

2
votes

1
answer

176
views

### Character constructed from Kummer local system lifts to representation of algebraic torus

I'm currently reading Mar's and Springer's Character Sheaves. In Chapter 2 (Kummer local systems on tori), they provide a construction of Kummer local systems on a torus $T$ by way of the $m^{th}$ ...

5
votes

1
answer

458
views

### Are equivariant perverse sheaves constructible with respect to the orbit stratification?

[Moved here from MSE]
Consider a variety $X$ over a field $k$ (complex numbers is fine) with the action of a group scheme $G$, and a $G$-equivariant perverse sheaf $F$ over $X$.
Question. Is it true ...

1
vote

1
answer

170
views

### Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann surface

I originally posted the question on math.stackexhange, but there doesn't seem to be an answer. I apalogize in advance for cross posting.
Let $E\rightarrow X$ be a holomorphic vector bundle over a ...

2
votes

0
answers

64
views

### «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 ...

8
votes

0
answers

242
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'...

6
votes

2
answers

558
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 ...

8
votes

0
answers

394
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
votes

0
answers

140
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
votes

0
answers

573
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
vote

0
answers

59
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
votes

0
answers

118
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### 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.
...

7
votes

1
answer

1k
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### 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
votes

1
answer

745
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### 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
votes

1
answer

919
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### 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 ...