On signatures associated with ramified coverings and embedding problems

Abstract

Given a cohomology class ξ∈H 2 (M;Z) there is a smooth submanifold K⊂M Poincaré dual to ξ. A special class of such embeddings is characterized by topological properties which hold for nonsingular algebraic hypersurfaces in CP n . This note summarizes some results on the question: how does the divisibility of ξ restrict the dual submanifolds K in this class ? A formula for signatures associated with a d-fold ramified cover of M branched along K is given and a proof is included in case d=2.