Projective structures, neighborhoods of rational curves and Painlev'e equations

TL;DR

This paper explores the relationship between projective structures on surfaces and neighborhoods of rational curves, revealing new insights into Painlevé equations through complex analytic methods.

Contribution

It establishes a duality framework connecting projective structures and rational curve neighborhoods, and derives transcendental results related to Painlevé equations.

Findings

Duality between projective structures and rational curve neighborhoods

Existence of normal forms and symmetry classifications

New transcendental insights into Painlevé equations

Abstract

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 Painlev'e equations.