  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
embeddings in the projective space $CP^N$ obtaining control on N. The main
reason of our study is the construction of singular determinantal submanifolds
as the intersection of the embedding with certain ``generalized Schur 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 Auroux as
zero sets of approximately holomorphic sections of ``very ample'' vector
bundles.
