Search results for "MONODROMY"
showing 10 items of 44 documents
Principal Poincar\'e Pontryagin Function associated to some families of Morse real polynomials
2014
It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is an Abelian integral also in non generic cases. In non generic cases it is an iterated integral. Uribe [17, 18] gives in a special case a precise description of the Principal Poincar\'e Pontryagin Function, an iterated integral of length at most 2, involving logarithmic functions with only one ramification at a point at infinity. We extend this result to some non isodromic families of real Morse polynomials.
Irreducibility of Hurwitz spaces of coverings with one special fiber
2006
Abstract Let Y be a smooth, projective complex curve of genus g ⩾ 1. Let d be an integer ⩾ 3, let e = {e1, e2,..., er} be a partition of d and let | e | = Σi=1r(ei − 1). In this paper we study the Hurwitz spaces which parametrize coverings of degree d of Y branched in n points of which n − 1 are points of simple ramification and one is a special point whose local monodromy has cyclic type e and furthermore the coverings have full monodromy group Sd. We prove the irreducibility of these Hurwitz spaces when n − 1 + | e | ⩾ 2d, thus generalizing a result of Graber, Harris and Starr [A note on Hurwitz schemes of covers of a positive genus curve, Preprint, math. AG/0205056].
On Hurwitz spaces of coverings with one special fiber
2009
Let X X' Y be a covering of smooth, projective complex curves such that p is a degree 2 etale covering and f is a degree d covering, with monodromy group Sd, branched in n + 1 points one of which is a special point whose local monodromy has cycle type given by the partition e = (e1,...,er) of d. We study such coverings whose monodromy group is either W(Bd) or wN(W(Bd))(G1)w-1 for some w in W(Bd), where W(Bd) is the Weyl group of type Bd, G1 is the subgroup of W(Bd) generated by reflections with respect to the long roots ei - ej and N(W(Bd))(G1) is the normalizer of G1. We prove that in both cases the corresponding Hurwitz spaces are not connected and hence are not irreducible. In fact, we s…
Finite Braid Groups for the SU(2) Knizhnik Zamolodchikov Equation
1995
We consider the monodromy representations of the mapping class group B 4 of the 2-sphere with 4 punctures acting in the solutions space of the zu(2) Knizhnik-Zamolodchikov equation [3] (note that the monodromy representations of the braid group have a more general geometric definition [4]).
Generalizations of Clausen's formula and algebraic transformations of Calabi-Yau differential equations
2011
AbstractWe provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi–Yau differential equations.
Hurwitz spaces of coverings with two special fibers and monodromy group a Weyl group of typeBd
2012
f! Y; where is a degree-two coverings with n1 branch points and branch locus D and f is a degree-d coverings with n2 points of simple branching and two special points whose local monodromy is given by e and q, respectively. Furthermore the covering f has monodromy group Sd and f. D /\ D fD? where D f denotes the branch locus of f . We prove that the corresponding Hurwitz spaces are irreducible under the hypothesis n2 s r dC 1.
A note on coverings with special fibres and monodromy group $ S_{d}$
2012
We consider branched coverings of degree over with monodromy group , points of simple branching, special points and fixed branching data at the special points, where is a smooth connected complex projective curve of genus , and , are integers with . We prove that the corresponding Hurwitz spaces are irreducible if .
On Shimura subvarieties generated by families of abelian covers ofP1
2018
We investigate the occurrence of Shimura (special) subvarieties in the locus of Jacobians of abelian Galois covers of P1 in Ag and give classifications of families of such covers that give rise to Shimura subvarieties in the Torelli locus Tg inside Ag. Our methods are based on Moonen–Oort works as well as characteristic p techniques of Dwork and Ogus and Monodromy computations.
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…
Branch Points of Algebraic Functions and the Beginnings of Modern Knot Theory
1995
Many of the key ideas which formed modern topology grew out of “normal research” in one of the mainstream fields of 19th-century mathematical thinking, the theory of complex algebraic functions. These ideas were eventually divorced from their original context. The present study discusses an example illustrating this process. During the years 1895-1905, the Austrian mathematician, Wilhelm Wirtinger, tried to generalize Felix Klein's view of algebraic functions to the case of several variables. An investigation of the monodromy behavior of such functions in the neighborhood of singular points led to the first computation of a knot group. Modern knot theory was then formed after a shift in mat…