On the GAGA principle for algebraic affine hypersurfaces

Vo Van Tan

doi:10.5802/crmath.254

Abstract

On the GAGA principle for algebraic affine hypersurfaces

Highlights

Unless the contrary is explicitly stated, all C-analytic spaces X are assumed to be equipped with an analytic structural sheaf OX
For any C-algebraic variety X, let us denote by P i c(X ) (resp. by Pi c(X ) = H 1(X, OX∗ )), the algebraic Picard group of X, where X is the C-analytic space associated to X
Assume that a given compact C-analytic space Y is biholomorphic to an underlying topological space of some complete C-algebraic variety Y; since there is a 1-1 correspondence between linear equivalent classes of Cartier divisors and locally free sheaves of rank 1, it follows from Serre G AG A principle, (see e.g. [7, Chapitre XII, Théorème 4.4] that the analytic Picard group Pi c(Y ) and the algebraic Picard group Pic(Y ) are isomorphic

Summary

Introduction

Unless the contrary is explicitly stated, all C-analytic spaces X are assumed to be equipped with an analytic structural sheaf OX. In contrast with Proposition 2, it is known that Xi are not algebraically isomorphic [16, Proposition 3.1] In spite of this fact and against all expectations, we have the following interesting result which was communicated to us by the referee which we gratefully acknowledge. Let kV∗ be the constant sheaf on V associated to k∗, let Gm,V be the units sheaf on V , and let Uk,V := the presheaf cokernel of (kV∗ → Gm,V ) It is known [14, Lemma 2] that (1) Uk,V is a sheaf on V , (2) Pic(V ) = H 1(V, Uk,V ) = H 1(V, Gm.V ), and (3) for a smooth curve B and a Zariski fibration [14, Definition 3] f : E −→ B with fibre F , one has [14, Theorem 5] the following exact sequence. Confronted with this state of affairs, we are looking at a class of affine algebraic hypersurfaces X with dim .X ≥ 3

The non-singular hypersurfaces

Hypersurfaces with isolated singularities

Question

Proper hyperplane sections

The transverse hypersurface sections