Questions tagged [local-systems]

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9 votes
2 answers
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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}(...
Josh Lam's user avatar
  • 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{ ...
JackYo's user avatar
  • 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 ...
Daniel Litt's user avatar
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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 $\tilde{C}$...
Marco Fava's user avatar
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(...
kindasorta's user avatar
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 ...
kindasorta's user avatar
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 ...
kindasorta's user avatar
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 $...
ali's user avatar
  • 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 ...
Daniel Litt's user avatar
  • 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 $m^{th}$ ...
Martin Skilleter's user avatar
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 ...
W. Rether's user avatar
  • 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 ...
mtraube's user avatar
  • 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 ...
Daniil Rudenko's user avatar
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'...
John Pardon's user avatar
  • 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 ...
Longma's user avatar
  • 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-...
Carlos's user avatar
  • 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 ...
Jaime's user avatar
  • 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 ...
Randy's user avatar
  • 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 ...
budding geometer's user avatar
0 votes
0 answers
118 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. ...
Nicolas Schmidt's user avatar
7 votes
1 answer
1k 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?
hitme's user avatar
  • 71
5 votes
1 answer
745 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, ...
user36461's user avatar
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 ...
Dmitry Vaintrob's user avatar