Projective structures, neighborhoods of rational curves and Painlev'e equations
Abstract
We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having selfintersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries. We deduce some transcendental result about Painlev'e equations.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.07868
 Bibcode:
 2017arXiv170707868F
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Complex Variables;
 Mathematics  Differential Geometry
 EPrint:
 This version is a shortening of previous version v3. Some new results have been added, like prolongation of foliated structure in section 3.4, and non existence of foliated structure for many Painleve equations in section 6