Extension de fibres vectoriels et profondeur

TL;DR

This paper extends Grothendieck's Lefschetz theorems to vector bundles on quasi-projective varieties, comparing algebraic and analytic cases, and explores vector bundles with connections for stronger results.

Contribution

It adapts Grothendieck's methods to quasi-projective varieties and investigates vector bundles with connections, providing new insights beyond classical projective cases.

Findings

Extension of Lefschetz theorems to quasi-projective varieties

Comparison between algebraic and complex analytic vector bundles

Results on vector bundles with connections

Abstract

In his book SGA2, A. Grothendieck proved Lefschetz theorems, in particular for the Picard group. To some extent he was able to deal with vector bundles instead of line bundles. Here we use his methods in order to study vector bundles on quasi-projective varieties instead of projective ones (as Grothendieck did). In particular we look at the case of complex algebraic varieties where we can compare with the complex analytic category. Finally we study vector bundles with a connection where stronger results can be achieved.