that a (complex) manifold of complex dimension n has real dimension 2n, so these higher cohomology groups vanish (for example by comparing them to simplicial (co)homology). The de Rham cycle map also has a down-to-earth explanation: Given a subvariety Y of complex codimension r in a complete variety X of complex dimension n, the real dimension of Y is 2n−2r, so one can integrate any differential (2n−2r)-form along Y to produce a complex number. This induces a linear functional . By Poincaré duality, to give such a functional is equivalent to giving an element of ; that element is the image of Y under the cycle map. References (contains proofs of all of the axioms for Betti and de-Rham cohomology) (idem for l-adic cohomology) Topological methods of algebraic geometry Cohomology theories