Search results for "MONODROMY"

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

2015

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.

Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type D d

2007

Let Y be a smooth, connected, projective complex curve. In this paper, we study the Hurwitz spaces which parameterize branched coverings of Y whose monodromy group is a Weyl group of type D d and whose local monodromies are all reflections except one. We prove the irreducibility of these spaces when $$Y \simeq \mathbb {P}^{1}$$ and successively we extend the result to curves of genus g ≥  1.

Some Special Foliations

2014

In this chapter we study two classes of ubiquitous foliations: Riccati foliations and turbulent foliations. A section will also be devoted to a very special foliation, which will play an important role in the minimal model theory.

MR 3004007 Reviewed Chretien P. and Matignon M. Maximal wild monodromy in unequal characteristic. Journal of Number Theory (2013) 133, 1389--1408. Re…

2013

Let R be a complete discrete valuation ring of mixed characteristic (0, p) with fraction field K. The stable reduction theorem affirms that given a smooth, projective, geometrically connected curve over K, C/K, with genus \geq 2, there exists a unique finite Galois extension M/K minimal for the inclusion relation such that C_{M}:= C x M has stable reduction over M. A such extension is called monodromy extension of C/K and the Galois group Gal(M/K) is called the monodromy group of C/K. In this paper, the authors study stable models of p-cyclic covers of P^1_K. At first, they work with covers of arbitrarily high genus having potential good reduction. In particular, they determine for such cov…

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}…

The monodromy groups of Dolgachev's CY moduli spaces are Zariski dense

2014

Let $\mathcal{M}_{n,2n+2}$ be the coarse moduli space of CY manifolds arising from a crepant resolution of double covers of $\mathbb{P}^n$ branched along $2n+2$ hyperplanes in general position. We show that the monodromy group of a good family for $\mathcal{M}_{n,2n+2}$ is Zariski dense in the corresponding symplectic or orthogonal group if $n\geq 3$. In particular, the period map does not give a uniformization of any partial compactification of the coarse moduli space as a Shimura variety whenever $n\geq 3$. This disproves a conjecture of Dolgachev. As a consequence, the fundamental group of the coarse moduli space of $m$ ordered points in $\mathbb{P}^n$ is shown to be large once it is not…

Integrable Hamiltonian systems with swallowtails

2010

International audience; We consider two-degree-of-freedom integrable Hamiltonian systems with bifurcation diagrams containing swallowtail structures. The global properties of the action coordinates in such systems together with the parallel transport of the period lattice and corresponding quantum cells in the joint spectrum are described in detail. The relation to the concept of bidromy which was introduced in Sadovski´ı and Zhilinski´ı (2007 Ann. Phys. 322 164–200) is discussed.

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.

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 …