Search results for "MONODROMY"

MR 3219513 Reviewed Venkataramana T. N. Monodromy of cyclic coverings of the projective line. Invent. Math., 197 (2014), 1–-45. (Reviewer Francesca V…

2014

Let $d \geq 2$ and $n \geq 1$ be integers and $P_{n+1}$ be the pure braid group on $n + 1$ strands. In this paper, the author studies the image of $P_{n+1}$ under the monodromy action on the homology of a cyclic covering of degree $d$ of the projective line. More precisely, let $k_{1}, \ldots, k_{n + 1}$ be integers such that $1 \leq k_{i} \leq d - 1$ and gcd$(k_{i}, d) = 1$ for each $i$. Moreover, let $a_{1}, \ldots, a_{n + 1}$ be distinct points of the complex plane and $C$ be the space of points in $\mathbb{C}^{n + 1}$ with all distinct coordinates. Let us denote by $X_{a, k}$ the affine curve defined by the equation $$ y^{d} = (x - a_{1})^{k_{1}} (x - a_{2})^{k_{2}} \cdots (x - a_{n +1}…

Frobenius polynomials for Calabi–Yau equations

2008

We describe a variation of Dwork’ s unit-root method to determine the degree 4 Frobenius polynomial for members of a 1-modulus Calabi–Yau family over P1 in terms of the holomorphic period near a point of maximal unipotent monodromy. The method is illustrated on a couple of examples from the list [3]. For singular points, we find that the Frobenius polynomial splits in a product of two linear factors and a quadratic part 1− apT + p3T 2. We identify weight 4 modular forms which reproduce the ap as Fourier coefficients.

Unirationality of Hurwitz spaces of coverings of degree <= 5

2011

Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ be the Hurwitz space which parametrizes coverings $p:X \to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal to $S_d$, and $det(p_{*}O_X/O_Y)$ isomorphic to a fixed line bundle $A^{-1}$ of degree $-e$. We prove that, when $d=3, 4$ or $5$ and $n$ is sufficiently large (precise bounds are given), these Hurwitz spaces are unirational. If in addition $(e,2)=1$ (when $d=3$), $(e,6)=1$ (when $d=4$) and $(e,10)=1$ (when $d=5$), then these Hurwitz spaces are rational.

Reflexions on Mahler: Dessins, Modularity and Gauge Theories

2021

We provide a unified framework of Mahler measure, dessins d'enfants, and gauge theory. With certain physically motivated Newton polynomials from reflexive polygons, the Mahler measure and the dessin are in one-to-one correspondence. From the Mahler measure, one can construct a Hauptmodul for a congruence subgroup of the modular group, which contains the subgroup associated to the dessin. In brane tilings and quiver gauge theories, the modular Mahler flow gives a natural resolution of the inequivalence amongst the three different complex structures $\tau_{R,G,B}$. We also study how, in F-theory, 7-branes and their monodromies arise in the context of dessins. Moreover, we give a dictionary on…

Manifestation of Hamiltonian Monodromy in Nonlinear Wave Systems

2011

International audience; We show that the concept of dynamical monodromy plays a natural fundamental role in the spatiotemporal dynamics of counterpropagating nonlinear wave systems. By means of an adiabatic change of the boundary conditions imposed to the wave system, we show that Hamiltonian monodromy manifests itself through the spontaneous formation of a topological phase singularity (2 - or -phase defect) in the nonlinear waves. This manifestation of dynamical Hamiltonian monodromy is illustrated by generic nonlinear wave models. In particular, we predict that its measurement can be realized in a direct way in the framework of a nonlinear optics experiment.

Irreducible components of Hurwitz spaces of coverings with two special fibers

2013

In this paper we prove new results of irreducibility for Hurwitz spaces of coverings whose monodromy group is a Weyl group of type B_d and whose local monodromies are all reflections except two.

Small $C^1$ actions of semidirect products on compact manifolds

2020

Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eige…

Rotation Forms and Local Hamiltonian Monodromy

2017

International audience; The monodromy of torus bundles associated with completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article, we give a general approach to the computation of monodromy that resembles the analytical one, reducing the problem to the computation of residues of polar 1-forms. We apply our technique to three celebrated examples of systems with monodromy (the champagne bottle, the spherical pendulum, the hydrogen atom) and to the case of non-degenerate focus-focus singularities, re-obtaining the classical results. An advantage of this approach …

On proper branched coverings and a question of Vuorinen

2022

We study global injectivity of proper branched coverings from the open Euclidean n$n$-ball onto an open subset of the Euclidean n$n$-space in the case where the branch set is compact. In particular, we show that such mappings are homeomorphisms when n=3$n=3$ or when the branch set is empty. This gives a positive answer to the corresponding cases of a question of Vuorinen. Peer reviewed

Corrigendum to “The monodromy groups of Dolgachev's CY moduli spaces are Zariski dense” [Adv. Math. 272 (2015) 699–742]

2015