Analytic Continuation of Holomorphic Mappings From Non-minimal Hypersurfaces

TL;DR

This paper investigates the extension of biholomorphic mappings from non-minimal hypersurfaces to hyperquadrics, revealing conditions under which such mappings extend along paths and linking the problem to singular complex ODEs.

Contribution

It establishes conditions for local biholomorphic extension of mappings from non-minimal hypersurfaces and connects the problem to the theory of singular complex ODEs.

Findings

Mappings extend biholomorphically along paths outside a complex hypersurface

Connection between nonminimal hypersurfaces and singular complex ODEs

Extension relies on non-degeneracy conditions

Abstract

We study the analytic continuation problem for a germ of a biholomorphic mapping from a non-minimal real hypersurface $M\subset\CC{n}$ into a real hyperquadric $\mathcal Q\subset\CP{n}$ and prove that under certain non-degeneracy conditions any such germ extends locally biholomorphically along any path lying in the complement $U\setminus X$ of the complex hypersurface $X$ contained in $M$ for an appropriate neighborhood $U\supset X$. Using the monodromy representation for the multiple-valued mapping obtained by the analytic continuation we establish a connection between nonminimal real hypersurfaces and singular complex ODEs.