Grothendieck--Lefschetz for ample subvarieties

Abstract

We establish a Grothendieck--Lefschetz theorem for smooth ample subvarieties of smooth projective varieties over an algebraically closed field of characteristic zero and, more generally, for smooth subvarieties whose complement has small cohomological dimension. A weaker statement is also proved in a more general context and in all characteristics. Several applications are included.

v4: to appear in Math. Z. v5: 19 pages; an error in Theorem B about torsion in positive characteristics, kindly pointed out by Emelie Arvidsson and Lena Ji, is fixed; applications are unaffected