Complete intersections with middle Picard number 1 defined over \({\mathbb{Q}}\)
Let X be a non-singular complete intersection in a projective space over the complex number field. It is known that the cohomology \(H^{2i}(X,{\mathbb{Q}})\) is generated by the powers of the class of the hyperplane section except for the case \(2i=\dim X\). In this case, the dimension of the space of algebraic cycles of codimension i viewed as a \({\mathbb{Q}}\)-subvectorspace of \(H^{2i}(X,{\mathbb{Q}})\) is called the middle Picard number of X. \textit{P. Deligne} [SGA 7 II, Lect. Notes Math. 340 (1973] showed that X has the middle Picard number 1 if X has a type different from (2;3), (2n;2), (2n;2,2). Here a complete intersection is said to be of type (N;\b{a}), where \b{a}\(=a_ 1,...,a_ d\) is a sequence of positive integers, if it is a complete intersection of hypersurfaces of degree \(a_ 1,...,a_ d\) in \(P^{N+d}\). The paper under review proves that for a type as above, there exists a complete intersection of middle Picard number 1.
scientific article; zbMATH DE number 3927068