# Moscow Mathematical Journal

Volume 23, Issue 1, January–March 2023  pp. 59–95.

Projective Structures, Neighborhoods of Rational Curves and Painlevé equations

Authors:  Maycol Falla Luza (1) and Frank Loray (2)
Author institution:(1) UFF, Universidad Federal Fluminense, rua Mário Santos Braga S/N, Niterói, RJ, Brasil
(2) Univ Rennes 1, CNRS, IRMAR, UMR 6625, F-35000 Rennes, France

Summary:

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. We deduce some transcendental result about Painlevé equations. Part of the results were announced in Comptes rendus in 2016; an extended version is available at https://arxiv.org/pdf/1707.07868v3.pdf.

2020 Math. Subj. Class. 53B05, 32G13, 34M55.

Keywords: Foliation, projective structure, rational curves.