Questions tagged [monodromy]

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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?
James Propp's user avatar
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27 votes
1 answer
4k views

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 ...
user84144's user avatar
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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 (...
Geordie Williamson's user avatar
23 votes
2 answers
2k views

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 ...
54321user's user avatar
  • 1,706
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 ...
Neeroen's user avatar
  • 153
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 ...
Geordie Williamson's user avatar
12 votes
3 answers
2k views

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 ...
Kim's user avatar
  • 4,034
12 votes
3 answers
3k views

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 ...
diverietti's user avatar
  • 7,852
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 ...
Pierre Dehornoy's user avatar
12 votes
2 answers
1k views

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 ...
Pierre Dehornoy's user avatar
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:...
Pol van Hoften's user avatar
9 votes
1 answer
481 views

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 ...
Jeremy's user avatar
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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 ...
Arrow's user avatar
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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 ...
Erfan Salavati's user avatar
8 votes
2 answers
2k views

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 ...
Ilya Nikokoshev's user avatar
8 votes
1 answer
723 views

$\ell$-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 $...
jmc's user avatar
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8 votes
1 answer
1k views

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 ...
JSE's user avatar
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8 votes
0 answers
374 views

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 ...
Will Sawin's user avatar
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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] = ...
Jakob's user avatar
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8 votes
0 answers
712 views

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-...
Gjergji Zaimi's user avatar
7 votes
1 answer
2k views

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 ...
Jack's user avatar
  • 637
7 votes
2 answers
584 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
  • 169
7 votes
1 answer
904 views

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. ...
Elbabak's user avatar
  • 347
7 votes
1 answer
390 views

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 $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ ...
user avatar
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 ...
David Corwin's user avatar
  • 15.1k
6 votes
4 answers
1k views

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 ...
Saal Hardali's user avatar
  • 7,549
6 votes
1 answer
1k views

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 $z_i\in C-U,$ ...
shenghao's user avatar
  • 4,195
6 votes
2 answers
492 views

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 ...
David Zureick-Brown's user avatar
6 votes
1 answer
1k views

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!
Nicolás's user avatar
  • 2,802
6 votes
1 answer
839 views

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}(...
Arrow's user avatar
  • 10.3k
5 votes
1 answer
406 views

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 ...
Hajime_Saito's user avatar
5 votes
1 answer
554 views

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 $\pi^{-1}(0)$...
Jana's user avatar
  • 2,022
5 votes
3 answers
1k views

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 ...
Jack's user avatar
  • 637
5 votes
1 answer
333 views

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 ...
Andrea Marino's user avatar
5 votes
0 answers
149 views

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 (...
Jeannette's user avatar
  • 263
5 votes
0 answers
1k views

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 $...
Nicolás's user avatar
  • 2,802
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 $D^*$...
shenghao's user avatar
  • 4,195
4 votes
0 answers
265 views

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 \...
user45397's user avatar
  • 2,195
4 votes
0 answers
221 views

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,...
user avatar
4 votes
0 answers
127 views

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 ...
D. L Garcia's user avatar
4 votes
0 answers
195 views

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 ...
54321user's user avatar
  • 1,706
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 ...
Lierre's user avatar
  • 1,044
4 votes
0 answers
164 views

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 ...
Idempotent's user avatar
3 votes
2 answers
395 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
  • 541
3 votes
2 answers
549 views

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 ...
Pieter C.'s user avatar
3 votes
3 answers
686 views

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 $...
Ariel's user avatar
  • 31
3 votes
1 answer
420 views

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 ...
xlord's user avatar
  • 643
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, ...
Jeff Yelton's user avatar
  • 1,308
3 votes
1 answer
221 views

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. ...
Manuel Eberl's user avatar
  • 1,181
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 ...
54321user's user avatar
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