The monodromy tag has no wiki summary.

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### Monodromy of a punctured disc

I meet the following problem which I think related to the monodromy:
Let $D: = \{z \mid |z|<1 \}$ be a disc, and $U \to D$ be a variety fibred over $D$. For each point $t \in D \backslash \{0\}$, ...

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### Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?

Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...

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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 ...

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### 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, ...

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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 ...

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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 ...

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### Monodromy along strata of a pushforward

Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived.
Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...

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### The Fuchsian monodromy problem

I want to understand the argument being made from equation 6.1 to 6.5 in this paper between pages 27-28
6.2, 6.4 and 6.5 are completely out-of-the-blue to me and I have no clue as to from where they ...

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### Monodromy action on the local system $R^2\phi_*\mathbb{Z}$

Let $\phi: \mathcal{X}\rightarrow B$ be a family of complex manifolds i.e. $\phi$ is a proper submersive holomorphic morphism, i write $X_b$ for the fiber of $b\in B$.
Suppose ...

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### 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] = ...

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### 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 ...

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### lifts of maps to $\mathcal{M}_{1,1}$

Hi,
here's there's a construction about elliptic curves that I do not completely understand. Suppose I consider the two following families of elliptic curves over $\mathbb{C}^*$.
The first, which I ...

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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 ...

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### An analog of Picard-Lefschetz theory for finite coverings in lieu of embeddings

Suppose that $f\colon X\to \mathbb P^N$ is a finite morphism, where $X$ is a smooth projective variety over $\mathbb C$. Then one may consider monodromy of the (singular) cohomology of the subvariety ...

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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 ...

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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!

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### 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 ...

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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 ...

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### Quasi-unipotent monodromy for general families

This must be a naive question, but I'm wondering about the definition of the quasi-unipotent monodromy for general (not only 1-parameter families). The problem is that usually in the books of ...

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### Irreducibility of monodromy of eigenspaces of families of cyclic coverings

In the article "La conjecture de Weil", Deligne proves that for the primitive cohomology of a universal family $f:X \rightarrow S$ for $M_{d,n}$ the moduli stack of hypersurfaces of degree $d$ in ...

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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 ...

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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 ...

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### 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 ...

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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 ...

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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 $z_i\in C-U,$ ...

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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 ...

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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 ...