Almost holomorphic embeddings in Grassmannians with applications to singular symplectic submanifolds

Abstract

In this paper we use Donaldson's approximately holomorphic techniques to build embeddings of a closed symplectic manifold with symplectic form of integer class in the Grassmannians Gr(r, N). We assure that these embeddings are asymptotically holomorphic in a precise sense. We study first the particular case of CPN obtaining control on N and we improve in a sense a classical result about symplectic embeddings. The main reason of our Study is the construction of singular determinantal submanifolds as the intersection of the embedding with certain "generalized Schubert cycles" defined on a product of Grassmannians. It is shown that the symplectic type of these submanifolds is quite more general that the ones obtained by Donaldson and Auroux,as zeroes of "very ample" vector bundles.