Projective structures, neighborhoods of rational curves and Painlevé equations

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.