Rationally Connected Varieties and Fundamental Groups

Abstract

The Lefschetz hyperplane theorem says that if X ⊂ ∙ N is a smooth projective variety over ℂ and C ⊂ X is a smooth curve which is a complete intersection of hyperplanes with X then $$ \pi _1^{top}\left( C \right) \to \pi _1^{top}\left( X \right) $$ ((1.1)) is surjective, where \( \pi _1^{top} \) is the topological fundamental group. Later a quasiprojective version of this result was also established (see, for instance, [13] for a discussion and further generalizations). This says that if X 0 ⊂ X is open and C ⊂ X as above is sufficiently general, then $$ \pi _1^{top}\left( {C \cap {X^0}} \right) \to \pi _1^{top}\left( {{X^0}} \right) $$ ((1.2)) is surjective.KeywordsOpen SubsetFundamental GroupIrreducible ComponentFinite IndexRational CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.