Questions tagged [local-systems]
The local-systems tag has no usage guidance.
23
questions
9
votes
2
answers
523
views
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.
$$
\mathfrak{X}_n= \mathrm{Hom}(...
- 222
3
votes
2
answers
271
views
Determine monodromy representation from local system
Let $X$ be a path-connected manifold nice enough such it's universal covering
space $p:\widetilde{X} \to X$ exists, $k$ a field. Then there exist a wellknown
correspondence
$$
\{\textit{linear}\text{ ...
- 479
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 ...
- 21.9k
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 $\tilde{C}$...
- 33
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(...
- 937
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 ...
- 937
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 ...
- 937
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 $...
- 1,033
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 ...
- 21.9k
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 $m^{th}$ ...
- 721
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 ...
- 445
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 ...
- 183
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 ...
- 4,246
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'...
- 18.1k
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 ...
- 159
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-...
- 613
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 ...
- 41
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 ...
- 113
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.
...
- 1,614
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?
- 71
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, ...
- 51
6
votes
1
answer
919
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 ...
- 7,872