181 research outputs found
Versal deformation of the analytic saddle-node
We derive simple forms for saddle-node singular points of analytic foliations
in the real or complex plane just by gluing foliated complex manifolds. We give
the versal analytic deformation of the simplest model. We also derive a unique
analytic form for those saddle-node having a central manifold. By this way, we
recover and generalize results earlier proved by J. Ecalle by using mould
theory and partially answer to some questions asked by J. Martinet and J.-P.
Ramis in the 80's
Flat parabolic vector bundles on elliptic curves
We describe the moduli space of logarithmic rank 2 connections on elliptic
curves with 2 poles.Comment: new version: fixed a sign in Proposition 2.
Projective structures and projective bundles over compact Riemann surfaces
A projective structure on a compact Riemann surface X of genus g is given by
an atlas with transition functions in PGL(2,C). Equivalently, a projective
structure is given by a projective sl(2,C)-bundle over X equipped with a
section s and a foliation F which is both transversal to the fibers and the
section s. From this latter geometric bundle picture, we survey on classical
problems and results on projective structures. We will give a complete
description of projective (actually affine) structures on the torus with an
explicit versal family of foliated bundle picture
Projective structures and neighborhoods of rational curves
We investigate the duality between local (complex analytic) projective
structures on surfaces and two dimensional (complex analytic) neighborhoods of
rational curves having self-intersection +1. We study the analytic
classification, existence of normal forms, pencil/fibration decomposition,
infinitesimal symmetries
Holomorphic dynamics, Painlev\'e VI equation and Character Varieties
We study the monodromy of Painlev\'e VI equation from a dynamical point of
view. This is applied to the description of bounded orbits, and to a proof of
the irreducibility of Painlev\'e VI equation in the sens of Casale and
Malgrange. On our way, we compute the entropy of each element of the monodromy
group, and we precise the dictionary between character varieties and Painlev\'e
equations
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