Sur le Th\'eor\`eme Principal de Zariski en G\'eom\'etrie Alg\'ebrique et G\'eom\'etrie Analytique

TL;DR

This paper proves an analogue of Zariski's Main Theorem in complex analytic geometry, establishing conditions under which holomorphic maps are open embeddings, and extends these results to algebraic varieties over characteristic zero.

Contribution

It introduces the Generalized Zariski Main Theorem for analytic spaces and algebraic varieties, filling a gap in complex analytic geometry and linking it with algebraic geometry via GAGA.

Findings

Proves holomorphic maps with discrete fibers are open embeddings.

Establishes flatness and bimeromorphicity as criteria for open embeddings.

Extends results to algebraic varieties over characteristic zero.

Abstract

On Zariski Main Theorem in Algebraic Geometry and Analytic Geometry. We fill a surprising gap of Complex Analytic Geometry by proving the analogue of Zariski Main Theorem in this geometry, i.e. proving that an holomorphic map from an irreducible analytic space to a normal irreducible one is an open embedding if and only if all its fibers are discrete and it induces a bimeromorphic map on its image. We prove more generally the "Generalized Zariski Main Theorem for analytic spaces", which claims that an holomorphic map from an irreducible analytic space to a irreducible locally irreducible one is an open embedding if and only if it is flat and induces a bimeromorphic map on its image. Thanks to the "analytic criterion of regularity" of Serre-Samuel in GAGA [12] and to "Lefschetz Principle", we finally deduce the "Generalized Zariski Main Theorem for algebraic varieties of characteristical zero", which claims that a morphism from such an irreducible variety to an irreducible unibranch one is an open immersion if and only if it is birational and flat. ----- Nous comblons une lacune \'etonnante de la G\'eom\'etrie Analytique Complexe en prouvant l'analogue du Th\'eor\`eme Principal de Zariski dans cette g\'eom\'etrie, c'est-\`a-dire en prouvant que toute application holomorphe d'un espace analytique irreductible dans un espace analytique normal et irreductible est un plongement ouvert si et seulement si toutes ses fibres sont discr\`etes et si elle induit une application bim\'eromorphe sur son image. Nous prouvons plus g\'en\'eralement le ``Th\'eor\`eme Principal de Zariski G\'en\'eralis\'e pour les espaces analytiques'', qui affirme qu'une application holomorphe d'un espace analytique irreductible dans un espace analytique irreductible et localement irreductible est un plongement ouvert si et seulement si elle est plate et induit une application bim\'eromorphe sur son image. Gr\^ace au ``crit\^ere analytique de r\'egularit\'e'' de Serre-Samuel dans GAGA \cite{serre} et au ``Principe de Lefschetz'', nous en d\'eduisons enfin le ``Th\'eor\`eme Principal de Zariski G\'en\'eralis\'e pour les vari\'et\'es alg\'ebriques de caract\'eristique nulle'', qui affirme qu'un morphisme d'une telle vari\'et\'e irreductible dans une autre unibranche est une immersion ouverte si et seulement s'il est birationnel et plat.