9
votes
Accepted
Are equivariant perverse sheaves constructible with respect to the orbit stratification?
Every complex of sheaves has a unique maximal open subset on which it is locally constant, because if it is locally constant on two open sets, it is locally constant on their union.
Let $U$ be then ...
- 148k
5
votes
Explicit Riemann Hilbert correspondence
It depends on what is called "explicit". If $k>2$, monodromy representation is a transcendental function of the $A_j$ and $s_j$. When $d=0$, it was expressed as an everywhere convergent power ...
- 91.8k
5
votes
Higher homotopy local systems
Infinity local systems have been defined and studied by various people. For instance, Lurie gives the following definition in some notes Algebraic K-theory and Manifold Topology, notes for course Math ...
- 9,627
5
votes
Accepted
Determine monodromy representation from local system
Here is a way to fill in the details. For simplicity, write $I = [0,1]$ for the interval, and $\exp \colon I \to S^1$ for the function $x \mapsto e^{2\pi i x}$. So let $\gamma = \exp \colon I \to S^1$ ...
- 28.7k
4
votes
Accepted
Is a local system on a surface determined by simple closed loops?
$\def\ZZ{\mathbb{Z}}\def\FF{\mathbb{F}}$This is an attempt to fix a (currently deleted) answer of Will Sawin's. Will tried to show that the claim was false for $n = 3^g$. I can show, by a similar ...
- 156k
4
votes
Is a local system on a surface determined by simple closed loops?
The n=2 case is implied by the proof of Theorem 2.1 in:
https://arxiv.org/abs/0901.1402
For the one-holed torus and n=3, the result is in my thesis.
- 8,529
3
votes
Determine monodromy representation from local system
In general, I would say that there is no way around the fact that the answer uses the fact that a sheaf on $[0,1]$ is constant.
Does this help? Instead of the pushforward of the constant sheaf $k$ of ...
- 55.9k
2
votes
Accepted
Flatness of "derived local system sheaves"
Let me first give a simple answer to a related question: If you want to make a sheaf whose value at a point $x$ is $H^1( G_S, \mathcal F_x)$, this won't work and a useful such sheaf cannot be ...
- 148k
2
votes
Accepted
Monodromic but not equivariant sheaves and Braden's theorem
Any $\mathbb G_m$-invariant stratification such that the natural map from the fundamental group of $\mathbb G_m$ to the fundamental group of each stratum is trivial has the property that objects ...
- 148k
2
votes
Explicit Riemann Hilbert correspondence
You are correct that the monodromy representation is given by the $T_i$. To address your concerns about this being unique up to isomorphism, notice that a change of basis of $\mathcal{O}_X^r$ induces ...
- 3,079
1
vote
Projective dimension of group ring
There are the inequalities
$$\max\{\textrm{gl.dim}(R),\textrm{cohom.dim}_R(G)\}\le\textrm{gl.dim}(RG)\le\textrm{gl.dim}(R)+\textrm{cohom.dim}_R(G).$$
The right hand inequality is often realized (as an ...
- 1,638
1
vote
Accepted
Character constructed from Kummer local system lifts to representation of algebraic torus
For anyone interested: it turns out that the fundamental group in question is the étale fundamental group, which is defined as a projective limit along the automorphism groups of a pro-representing ...
- 721
1
vote
Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann surface
I think I found the relation. The paper A Simple Algorithm for Cyclic Vectors by N. Katz seems crucial. The cyclic vector lemma is mostly stated for differential fields (see e.g. section 2 in Galois ...
- 183
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