Questions tagged [monodromy]
The monodromy tag has no usage guidance.
81
questions
51
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3
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What is the difference between holonomy and monodromy?
What is the difference between holonomy and monodromy?
And what is the simplest example in which one is trivial and the other is not?
27
votes
1
answer
4k
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Intuition for Picard-Lefschetz formula
I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I").
To summarize the ...
27
votes
0
answers
921
views
Nearby cycles without a function
Suppose that:
$X$ is a smooth complex algebraic variety,
$f : X \to D$ is a proper map to a small disc, smooth away from 0,
$Z_\epsilon = f^{-1}(\epsilon)$, and $Z = Z_0$.
Then there is a procedure (...
23
votes
2
answers
2k
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What are examples of D-modules that I should have in mind while learning the theory?
I've been reading about D-modules this summer in preparation for a learning seminar on intersection cohomology. Unfortunately, many of the ideas are not sticking while I learn about the theory. What ...
15
votes
1
answer
624
views
Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?
We work over the field of complex numbers. (But remarks in characteristic $p$ are very welcome.)
Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains ...
14
votes
2
answers
897
views
Non semi-simple monodromy in an algebraic family
I am looking for an example of a (edit: projective) family
$f : X \to Y$
of complex algebraic varieties which is a topologically locally trivial fibration in (singular) varieties and such that there ...
12
votes
3
answers
2k
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How to compute the cohomology of a local system?
Suppose we have a reasonable topological space $X$ (i.e. a complex algebraic variety or a manifold) whose integral singular cohomology and fundamental group we understand well.
Suppose that we are ...
12
votes
3
answers
3k
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Relationship between monodromy representations and isomorphism of flat vector bundles
This question is somehow related to this one.
Let $M$ be a smooth (compact, if you wish) connected manifold.
Then, it is well known that there is an equivalence between the isomorphism classes of ...
12
votes
3
answers
1k
views
Fibered knot with periodic homological monodromy
It is well-known that there exist pseudo-Anosov automorphisms of surfaces that act trivially on the homology: they form the Torelli group. Similarly there exists pseudo-Anosov automorphisms that act ...
12
votes
2
answers
1k
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Genus one fibered links
It is well-known that the only genus one fibered knots are the trefoil and the figure-eight. On the other hand, there exist infinitely many fibered links for any fixed higher genus.
My question is ...
9
votes
1
answer
663
views
A question about $p$-adic monodromy of abelian varieties
Let $S_0$ be a smooth (projective?) and (geometrically) connected scheme over a finite field of characteristic $p$ and let $S$ be its base change to an algebraic closure of the finite field. Let $\pi:...
9
votes
1
answer
481
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Detecting Monodromy in Integrable Systems
Suppose I have a completely integrable system on a symplectic manifold $(M^{2n},\omega)$ with momentum map $H:M \rightarrow \mathbb{R}^n$ that has compact, connected fibers. Further, suppose I know ...
9
votes
0
answers
246
views
Holonomy as a right adjoint, monodromy as a left adjoint
This question about the difference between holonomy and monodromy has an interesting answer by Ronnie Brown.
An excerpt:
So holonomy comes out as a kind of right adjoint, and monodromy as a kind ...
8
votes
2
answers
465
views
A specific linear differential equation on $\mathbb{C}-\{0,1\}$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$
Is there a linear differential equation on $\mathbb{C}$ with singularities at $0$ and $1$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$? If so, can someone give a ...
8
votes
2
answers
2k
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Properties of monodromy of a fibration?
Sorry for a loaded question.
I'm not an expert on those things, but I do know that a fibration gives rise to the representations of pointed fundamental group of the base on the cohomology of the ...
8
votes
1
answer
723
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$\ell$-adic monodromy theorems (over $\mathbb{C}$)
This question is about $\ell$-adic monodromy theorems for families over a number field. ($\ell$-adic analogues of Corollaries 6.2.8 and 6.2.9 in [BBD].)
Notation
$H$ denotes étale cohomology.
Let $...
8
votes
1
answer
1k
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Monodromy groups of families of abelian varieties: a reference request
In Serre's letter to Vigneras of 2 Oct 1986, he summarizes a course he's giving in Paris, explaining how to control the image of the mod-l Galois representations attached to abelian varieties. In ...
8
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0
answers
374
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Degeneration of wildly ramified local monodromy representations - near or far from Deligne?
Suppose we have a surface $S$ mapping to a curve $C$ and a finite cover $Y/S$ that is ramified at a divisor $D$. For each point $x \in C$ we get a ramified cover of the curve $S_x$, and we can study ...
8
votes
0
answers
218
views
Inertia group vs. differential equations
The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = ...
8
votes
0
answers
712
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When is the monodromy group of a linear differential equation dense in the Galois group?
Given a system $Y'=A(t)Y$ with only regular singular points, then a theorem of Schlesinger says that the Zariski closure of the monodromy group is equal to the Galois group of the corresponding Picard-...
7
votes
1
answer
2k
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Quasi-unipotent monodromy for general families
This must be a naive question, but I'm wondering about the definition of quasi-unipotent monodromy for general families, not only 1-parameter families. The problem is that usually, in the books of ...
7
votes
2
answers
584
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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 ...
7
votes
1
answer
904
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Relations between some works by Deligne-Mostow and Thurston
Happy new year 2016!
A coworker and I are interested in the relations between the works of Deligne and Mostow ([DM] and [M]) on the monodromy of Appell-Lauricella hypergeometric functions (Publ. ...
7
votes
1
answer
391
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Reference result: proof of theorem of Kazhdan-Margulis on monodromy group of a Lefschetz pencil of odd fiber dimenion is "as big as possible"
In Deligne's paper on his first proof of the Weil conjectures, we have the following result.
Theorem 5.10 (Kazhdan-Margulis). L'image de $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ ...
7
votes
0
answers
843
views
Rigid Uniformization vs Grothendieck's Local Monodromy Theory
I've noticed that some interesting results about abelian varieties can each be proven using one of two ways: the theory of rigid uniformization of abelian varieties or Grothendieck's local monodromy ...
6
votes
4
answers
1k
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What does "higher monodromy" tell us about a principal bundle
Let $P \to X$ be a principal $G-$bundle and let $f: X \to BG$ be its classifying map. As I understand there's some way to associate a monodromy representation $\pi_1(X) \to G$ to it. I know how to ...
6
votes
1
answer
1k
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monodromy and global cohomology
Let $C$ be a compact Riemann surface, and let $U$ be a Zariski open subset in $C.$ Let $L$ be a local system (with coefficients $\mathbb C$ or $\mathbb Q_{\ell}$) on $U.$ For each point $z_i\in C-U,$ ...
6
votes
2
answers
492
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What is the mod l monodromy of a generic trigonal curve?
For a hyperelliptic curve H, the mod 2 monodromy is smaller than $GSp_{2g}(F_2)$ -- since the two torsion of the Jacobian H is generated by differences of Weierstrass point, the monodromy of a generic ...
6
votes
1
answer
1k
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weight monodromy conjecture for curves?
Hi,
Is there a simple proof of the weight monodromy conjecture in the case of a curve over a mixed characteristic local field?
Thanks!
6
votes
1
answer
839
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Definition of geometric monodromy
Consider a polynomial $f
\in \mathbb C[x_1,\dots ,x_n]$. An atypical value of $f$ is a complex number about which $f:\mathbb C^n\to \mathbb C$ is not a topological fiber bundle. Writing $\mathrm{Atyp}(...
5
votes
1
answer
406
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A question regarding isomorphism in cohomology for moduli space of stable bundles over a compact Riemann surface
Let $N(n,k)$ denote the moduli space of stable vector bundles of rank $n$ and degree $k$ over a compact Riemann surface $X$, and let $N_0(n,k)$ denote the moduli space where we fix rank $n$ and some ...
5
votes
1
answer
554
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Degeneration of smooth curves and Picard-Lefschetz formula
Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$...
5
votes
3
answers
1k
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Monodromy group of 1-dimensional families of hyperelliptic curves
If $f: \mathcal{C} \rightarrow \mathcal{P}_{2g+2}$ is a general family of hyperelliptic curves (defined over $\mathbb{C}$), we know that the algebraic connected monodromy group
$Mon^{0}$ of this ...
5
votes
1
answer
333
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Thurston universe gates in knots: which invariant is it?
Today I discovered this nice video of a lecture by Thurston:
https://youtu.be/daplYX6Oshc
in which he explains how a knot can be turned into a "fabric for universes". For example, the unknot ...
5
votes
0
answers
149
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Characterization of the hypergeometric function
One of the definition of the hypergeometric function $_2 F_1$ rely only on its global properties around the singularities (and not on a differential equation or a serie expansion)
In modern language (...
5
votes
0
answers
1k
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Grothendieck monodromy theorem for l-adic sheaves
Hi,
Suppose that $F$ is a local field, $G_F$ its Galois group, $I$ the inertia subgroup, $k$ its residue field.
Let $X$ be a finite type scheme over $k$. Let $C$ be a constructible $l$-adic sheaf on $...
4
votes
2
answers
333
views
Analogue of Shafarevich-Ogg's theorem over complex numbers
Let $f:E\to D^*$ be a family of complex elliptic curves parametrized by the punctured open disk $D^*.$ Assume that the monodromy on $H^1$ is trivial (i.e. $R^1f_*\mathbb Z$ is a constant sheaf on $D^*$...
4
votes
0
answers
265
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Criterion for triviality of monodromy in smooth families
Let $\pi: X \to \Delta^*$ be a smooth, projective morphism. We know that for each $k$, there is a natural local system $L:=R^k \pi_*\mathbb{C}$. The associated vector bundle $\mathcal{L}:=L \otimes \...
4
votes
0
answers
221
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Quasi-unipotent monodromy for variation of Landau-Ginzburg cohomology
A pair, $(X,f)$, consisting of a smooth variety and a global function $f:X\rightarrow\mathbb{A}^{1}$ is called a Landau-Ginzburg model, LG-model for short. The LG-cohomology of the pair, dentoed $H(X,...
4
votes
0
answers
127
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Topological cycles with Lagrangian support
For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold?
The main example for this question ...
4
votes
0
answers
195
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How can I describe the monodromy of this variation of singular curves?
Consider the family of singular hyperelliptic curves
$$
y^2 - x(x-1)^2(x-2)(x-3)(x-4)(x-t)
$$
over $\mathbb{A}^1_t$. Over a generic point the fiber is a genus three curve where one of the genera comes ...
4
votes
0
answers
85
views
Action of the monodromy on the cycle made of the real points
Let $f : \Bbb C^n \to \Bbb C$ be a polynomial function with real coefficients.
Let $X_t = f^{-1}(t)$ denote the fiber above some $t \in \Bbb C$. Let assume that the set of real points of $X_t$, for $t ...
4
votes
0
answers
164
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Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?
My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...
3
votes
2
answers
396
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
$$
\{\textit{linear}\text{ ...
3
votes
2
answers
549
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What are the easiest examples of irreducible, but not big, monodromy representations
Let $\rho: \pi_1(S,s_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V_{s_0} = V$.
Let $H$ be the Zariski closure of the ...
3
votes
3
answers
686
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A neat monodromy group of a family of Kaehler manifolds
Let $X\rightarrow B$ be a family of Kaehler manifolds with possibly singular fibers. Let $G$ be the monodromy group on $H^n(X_b,\mathbb{Z})$, where $n=\dim X_b$ with the smooth fiber $X_b$ over some $...
3
votes
1
answer
420
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Questions about modular forms and the role of monodromy
Let $N \geq 3$ and let $\Gamma=\Gamma_1(N)$, so that the moduli problem for elliptic curves with $\Gamma$-structure is fine. Let $Y=Y(\Gamma)$ be the corresponding moduli space.
In this context, one ...
3
votes
2
answers
397
views
kernel of monodromy action of braid group on homology of hyperelliptic curve
Let $X_{n}$ be the (unordered) configuration space of $n$ distinct points in $\mathbb{P}_{\mathbb{C}}^{1}$. The fundamental group of $X_{n}$ is the braid group on $n$ strands on the Riemann sphere, ...
3
votes
1
answer
221
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The monodromy in the proof of Little Picard via Klein's $J$
First of all, my question is in the context of formalising proofs in the proof assistant Isabelle/HOL, so my questions are a bit guided by what material we have in the complex analysis library there.
...
3
votes
1
answer
167
views
How can I determine the monodromy of this variation of mixed hodge structures?
Consider the variation of mixed hodge structures which generates at the origin:
$$
f:X = \text{Proj}\left( \frac{\mathbb{C}[t][x,y,z]}{(xy(x + y + tz))} \right) \to \mathbb{A}^1_t
$$
How can I compute ...