Monodromy of certain Painleve' VI transcendents and reflection groups

B. Dubrovin; M. Mazzocco

Home/Archive/arXiv

arXiv · math/9806056

Monodromy of certain Painleve' VI transcendents and reflection groups

B. Dubrovin·M. Mazzocco

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the global analytic properties of the solutions of a particular family of Painleve' VI equations with the parameters $β=γ=0$, $δ={1\over2}$ and $α$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painleve' VI equation.

Explore related subjects

math.AG math.CA nlin.SI solv-int

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

B. Dubrovin, M. Mazzocco. 1998-06-10. Monodromy of certain Painleve' VI transcendents and reflection groups. https://arxiv.org/abs/math/9806056

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Good minimal models with nowhere vanishing holomorphic $1$-forms

Popa and Schnell show that any holomorphic 1-form on a smooth projective variety of general type has zeros. In this article, we show that a smooth good minimal model has a holomorphic 1-form without zero if and only if it admits an analytic fiber bundle structure over a positive dimensional abelian variety.

math.AG↗

arXiv2026

The automorphism groups of Kummer surfaces in characteristic two and their complex analogues

We calculate the automorphism group of the Kummer surface associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a $K3$ surface. Moreover we discuss the complex $K3$ surfaces with the same Picard lattice as these Kummer surfaces. The paper has two appendices.

math.AG↗

arXiv2026

Algebraic Zhou valuations

In this paper, we generalize Zhou valuations, originally defined on complex domains, to the framework of general schemes. We demonstrate that an algebraic version of the Jonsson--Mustaţă conjecture is equivalent to the statement that every Zhou valuation is quasi-monomial. By introducing a mixed version of jumping numbers and Tian functions associated with valuations, we obtain characterizations of a valuation being a Zhou valuation or computing some jumping number using the Tian functions. Furthermore, we establish the correspondence between Zhou valuations in algebraic settings and their counterparts in analytic settings.

math.AG↗