15
votes
What are examples of D-modules that I should have in mind while learning the theory?
(I'm not an expert, but you don't seem to have gotten any answers so far.)
Let's start from the beginning. Let $A$ be an $n\times n$ matrix of regular functions on a Zariski open subset $X\subset \...
- 34.2k
14
votes
Accepted
Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?
The answer is positive if and only if $\mathbb{A}^1\setminus S$ is an arithmetic curve, i.e., $\pi_1(\mathbb{A}^1\setminus S)\subset \mathrm{Aut}(\mathbb{H}) = PSL_2(\mathbb{R})$ is an arithmetic ...
- 9,417
13
votes
How to compute the cohomology of a local system?
Suppose $X$ is a connected CW complex with fundamental group $\pi:=\pi_1(X,x)$. Then the cellular chain complex $C_*(\widetilde{X})$ of the universal cover is a chain complex of free $\mathbb{Z}\pi$-...
- 35k
9
votes
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A question regarding isomorphism in cohomology for moduli space of stable bundles over a compact Riemann surface
Things are actually simpler. View $\Gamma _n=H^1(X,\mathbb{Z}/n)$ as the group of line bundles $L\in \operatorname{Pic}^{0}(X) $ with $L^{{\tiny \otimes }n}=\mathscr{O}_X$. The
map $N_0(n,k) \times \...
- 37.3k
8
votes
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Degeneration of smooth curves and Picard-Lefschetz formula
Because the symplectic form on $H_1(C_t, \mathbb Z)$ is a perfect pairing, it suffices to check that there is a group homomorphism $H_1(C_t,\mathbb Z) \to \mathbb Z$ that sends $ \gamma$ to $1$, which ...
- 137k
7
votes
How to compute the cohomology of a local system?
In general you should have $H^1(\pi_1(X,x),V)\cong H^1(X,L)\,,$ where on the left we have the group cohomology of $\pi_1(X,x)$ acting on $V$ according to the representation. You will also have an ...
- 1,188
6
votes
Accepted
Thurston universe gates in knots: which invariant is it?
Here is a higher-quality video of the same material. My answer is a more algebraic version of Thurston's presentation, but I will tie this back to Thurston's "intention" at the end.
...
- 26.4k
5
votes
Accepted
Recovering a family of rational functions from branch points
The reason for this phenomenon is that you are afflicted with a serious mathematical condition, that being:
Your monodromy has monodromy.
To be less cryptic, the key thing is that the fundamental ...
- 137k
5
votes
What is the mod l monodromy of a generic trigonal curve?
The paper https://arxiv.org/abs/1403.7399 shows that the monodromy of the moduli stack of trigonal curves of genus g over $\mathbb C$ is as big as possible (equal to $Sp_{2g}(\mathbb Z)$ in the ...
- 1,308
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 ...
- 88.9k
4
votes
Accepted
A question about $p$-adic monodromy of abelian varieties
Let me briefly answer your question. There are two $p$-adic analogues of $R^1\pi_*\mathbb{Q}_\ell/S_0$: a (convergent) $F$-isocrystal $\mathcal E$ and an overconvergent $F$-isocrystal $\mathcal{E}^\...
- 188
4
votes
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Questions about modular forms and the role of monodromy
Monodromy does not appear in such a nice way because $\omega$ is a coherent sheaf, not a locally constant sheaf. Thus there is not necessarily a way of continuing a local section along a path in $\...
- 137k
4
votes
What are examples of D-modules that I should have in mind while learning the theory?
I will edit this question as I learn more, but here's one useful example: Consider the $\mathcal{D}_{\mathbb{A}^1}$-module
$$
\frac{\mathcal{D}_{\mathbb{A}^1}}{\mathcal{D}_{\mathbb{A}^1}(t\partial_t - ...
- 1,706
4
votes
Non semi-simple monodromy in an algebraic family
So it seems passerby's example can be modified to give a projective example.
(Thanks for de Cataldo and Migliorini for some of the following. All mistakes are mine.)
Fix $E$ an elliptic curve and ...
- 5,867
4
votes
Accepted
Which hyperbolic fibered knots have monodromy with a single singularity?
A fibered two-bridge knot with Seifert surface which is a plumbing of figure eight knot Seifert surfaces will have this property. One can follow the prescription of Gabai-Kazez who gave a method to ...
- 66.8k
4
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$ ...
- 27.3k
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 ...
- 54.4k
3
votes
Non semi-simple monodromy in an algebraic family
Won't even an open variety do? Take an elliptic curve $E$ (maybe over a base $B$) and remove the zero-section and another section $s$.
The fiber over $b \in B$ is a twice-punctured complex torus and ...
- 31
3
votes
Accepted
Definition of geometric monodromy
As you say, a Hurewicz fibration defines a map from $\pi_1(B)$ to the automorphisms of the fiber in the homotopy category.
But a topological fiber bundle has more structure than a Hurewicz fibration. ...
- 137k
3
votes
Accepted
Reference result: proof of theorem of Kazhdan-Margulis on monodromy group of a Lefschetz pencil of odd fiber dimenion is "as big as possible"
The proof of this result is worked out in detail in page 250 (theorem 7.5) of the book
Eberhard Freitag, Reinhardt Kiehl "Etale Cohomology and the Weil Conjecture"
You can preview that page in ...
- 17.5k
3
votes
Accepted
Čech cocycles and monodromy
First, one can pull back the Čech cocycle to S^1 and work directly with S^1
instead of X.
Any two open covers have a common refinement,
so it suffices to show that the monodromy map does not change
...
- 36.3k
2
votes
How to compute the cohomology of a local system?
If you're looking for something very concrete in terms of things like explicit boundary map computations, you might be interested in looking at Section 31 of Steenrod's Topology of Fibre Bundles. As ...
- 5,204
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 ...
- 3,029
2
votes
Accepted
How can I determine the monodromy of this variation of mixed hodge structures?
This monodromy is trivial because on the set $t\neq 0$ you can make a change of variables $(x,y,z)\to(x,y,z t^{-1})$, and your ideal becomes $x y (x+y+z)$, so it doesn't depend on $t$. So your family ...
- 3,592
2
votes
Accepted
Reference request: monodromy and isomorphic projections
The comments above are too long, so I am replacing them by an answer. The reference for most such results in algebraic geometry is SGA 7_2. I am sure that the following is in there somewhere.
Your ...
2
votes
The monodromy in the proof of Little Picard via Klein's $J$
The usual proof of Picard's theorem along these lines used another
modular function which is called $\lambda$ and which is related to $J$
by
$$J=\frac{4}{27}\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\...
- 88.9k
1
vote
Accepted
Tightness/Overtwistedness of genus one open book decomposition
Honda–Kazez–Matić and Baldwin proved that tight here is equivalent to right-veering, so you can sit down and work that out. The former paper shows that reducible monodromies always give you a tight ...
- 420
1
vote
Tightness/Overtwistedness of genus one open book decomposition
A general criterion for tightness/overtwistedness of the contact structure in terms of the page and of the monodromy of the open book was given by Andy Wand: "Tightness is preserved
by Legendrian ...
- 2,329
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 ...
- 731
1
vote
Accepted
The holonomy map associated to a mapping torus
If you represent the torus as $\mathbb R^2/\mathbb Z^2$, given a monodromy matrix $A \in GL_2(\mathbb Z)$, we can construct one example of the associated diffeomorphism. The multiplication-by-$A$ map $...
- 137k
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