Almost holomorphic embeddings in Grassmannians with applications to singular simplectic submanifolds

TL;DR

This paper develops approximately holomorphic embeddings of symplectic manifolds into Grassmannians, enabling the construction of complex-like singular submanifolds with broader symplectic types than previous methods.

Contribution

It introduces new embedding techniques into Grassmannians using Donaldson's methods, allowing for the creation of more general singular symplectic submanifolds.

Findings

Constructed asymptotically holomorphic embeddings into Grassmannians.

Controlled embeddings in projective spaces with explicit bounds.

Built singular determinantal submanifolds with diverse symplectic types.

Abstract

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.