Almost holomorphic embeddings in Grassmannians with applications to singular simplectic submanifolds

Details Almost holomorphic embeddings in Grassmannians with applications to singular simplectic submanifolds Almost holomorphic embeddings in Grassmannians with applications to singular simplectic submanifolds

2000-02-25

arxiv.org/abs/math/0002212v1

Mathematics

Differential Geometry

40 pages, no figures, Latex2e

Scientific 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 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.

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