Almost holomorphic embeddings in Grassmannians with applications to singular simplectic submanifolds

2000/02/25 by Vicente Muñoz, Fran Presas, Francisco Presas +4
Mathematics · #53D35 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #msc:53D35

paper · pdf · doi:10.48550/arxiv.math/0002212

40 pages, no figures, Latex2e

arxiv created 2000/02/25 · openalex publication_date 2000/02/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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