Questions tagged [monodromy]
The monodromy tag has no usage guidance.
33
questions with no upvoted or accepted answers
27
votes
0
answers
924
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 (...
- 5,867
9
votes
0
answers
248
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 ...
- 10.3k
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 ...
- 137k
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] = ...
- 1,986
8
votes
0
answers
713
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-...
- 85.1k
7
votes
0
answers
844
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 ...
- 15.1k
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 (...
- 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 $...
- 2,802
4
votes
0
answers
269
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 \...
- 2,195
4
votes
0
answers
222
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,...
user108998
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 ...
- 41
4
votes
0
answers
197
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 ...
- 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 ...
- 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 ...
- 411
3
votes
0
answers
217
views
Factorization algebras as factorizable cosheaves on the (extended) Ran Space
A basic fact in the theory of factorization algebras is that, to state it in a rough way, the exit path category of the Ran space of a topological manifold $M$ is equivalent to the category consisting ...
- 884
3
votes
0
answers
144
views
Frobenius structure for A_n singularities
I need to compute monodromy matrices $M(v)$, associated to a Frobenius structure for $A_n$ singularity with flat coordinates $v_1,\dots,v_n$, that is, $f(x)=x^{n+1}$. (due to Saito, Dubrovin etc.) ...
- 31
3
votes
0
answers
106
views
Multivalued functions with three independent branches
Let $n$ be a positive integer and $f: \mathbb{C} \rightarrow \mathbb{C}$ be a multivalued function, analytic everywhere except for branch points at $0$, $1$ and $\infty$. Around one of those ...
- 263
3
votes
0
answers
595
views
Monodromy representations are "quasi-unipotent"
Let $S$ be a smooth complex algebraic variety, let $b$ be a closed point of $X$, and let $f:X\to S$ be a polarized family of smooth projective varieties over $S$. This induces a monodromy ...
- 113
3
votes
0
answers
140
views
Does the monodromy of such VHS have to be trivial
Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ...
- 375
3
votes
0
answers
157
views
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 ...
- 1,595
2
votes
0
answers
113
views
Some questions about $\ell$-adic monodromy
I'm stucking on the proof of the Lemma 3.12 of A p-adic analogue of Borel’s theorem.
Here $\mathcal A_{g,\mathrm K}$ is just a shimura variety defined over $\mathbb Z_p$, and full level $\ell$ ...
- 523
2
votes
0
answers
134
views
Steenbrink spectral sequence and modifications of the central fibre
If $f: X \to S$ is a proper map from a complex manifold to a disc, $Y=f^{-1}(0)$ is a divisor with strictly normal crossings and the action of monodromy on $X_t=f^{-1}(t)$ for some (hence any) $t \neq ...
- 4,060
2
votes
0
answers
159
views
Abelian variety corresponding to a vector space
I would like to know what the following statement means:
"Let $B_t$ be the Abelian subvariety in $J_t$ corresponding to the $\mathbb{Q}$-vector subspace $H^1(C_t,\mathbb{Q})_{van}$ in the space $...
- 519
2
votes
0
answers
169
views
Computing monodromy groups of curves over function fields
Suppose I consider a hyperelliptic curve given by an equation such as $y^2 = x^{n} + tx + 1$ or some variation on this (where $t$ is a parameter on $\mathbb P^1$ and this curve is really a surface ...
- 7,646
2
votes
0
answers
166
views
Monodromy group of the generic plane curve
Let's work over $\mathbb{C}$. The degree $d$ curves in $\mathbb{P}^2_{\mathbb{C}}$ are parameterized by a projective space $|\mathcal{O}_{\mathbb{P}^2}(d)|$. Let $U_d\subset |\mathcal{O}_{\mathbb{P}^2}...
user39380
2
votes
0
answers
167
views
Cycle map and flat cycle
Let $\mathcal X\rightarrow C$ be a smooth projective morphism over an open subset of $\mathbb A_k^1$ ($k$ algebraically closed of characteristic $p>0$, one can suppose $C$ to be the spectrum of a ...
- 155
1
vote
0
answers
57
views
Etale local systems and proper base change
I am looking for a reference, or a proof, of the following statement:
Let $f:Y\longrightarrow X$ be a smooth proper map of quasiprojective $K$ schemes, and let $\overline{f}:\overline{Y}\...
- 1,641
1
vote
0
answers
138
views
Monodromy Representation on $H_1$ of Elliptic Curve
I'm reading this post by Charles Siegel on Monodromy Representations
and there is a construction in example a not unterstand.
We look at the family $y^2z=x(x-z)(x-\lambda z)$ of projective elliptic
...
- 6,000
1
vote
0
answers
151
views
Moduli space of genus $g$ curves ${\mathcal{M}_g}$ irreducible by 'Monodromy argument'
I'm reading this post by Charles Siegel on Monodromy Representations
and there is a short remark on the proof of irreducibility of moduli space of genus $g$ curves ${\mathcal{M}_g}$ :
Just look at ${...
- 6,000
1
vote
0
answers
312
views
why is monodromy weight filtration compatible with cup product?
This question is about a statement I took for granted in this question.
If $f: X \to S$ is a moprhism from a complex manifold to a punctured disc then the monodromy operator $T$ is quasi-unipotent, so ...
- 4,060
1
vote
0
answers
107
views
Transverse $S^1$ actions on mapping tori
Up until now I have thought that the existence of a transverse $\mathbb{S}^1$ action on a symplectic mapping torus implies that the mapping torus is trivial. Unfortunately I also came up with a ...
- 959
1
vote
0
answers
244
views
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 ...
- 3,717
1
vote
0
answers
201
views
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 $\...
- 637