On the Hodge conjecture for complete intersections

Abstract

We study the Fano variety of k-planes Ω X ( k ) of a complete intersection X and show the surjectivity of the cylinder homomorphism in this setting, proving along the way the (classical) Hodge conjecture in cases where a numerical condition is satisfied. In particular, a new case is discovered in dimension 6. This work is a generalization of the second author’s paper Motives of some Fano varieties.