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.
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We thank Daniel Litt for discussions on Sommese’s conjecture in relation to his work  and Adrian Langer for useful comments. We also thank the referee for useful comments and suggestions.
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The research of the first author was partially supported by NSF Grant DMS-1700769 and by NSF Grant DMS-1440140 while in residence at MSRI in Berkeley during the Spring 2019 semester. The research of the second author was partially supported by a Croucher Foundation Fellowship .
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de Fernex, T., Lau, C.C. Grothendieck–Lefschetz for ample subvarieties. Math. Z. (2021). https://doi.org/10.1007/s00209-020-02693-4
- Ample subvariety
- Picard group
- Abelian variety
Mathematics Subject Classification
- Primary 14C22
- Secondary 14K12