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Almost holomorphic embeddings in Grassmannians with applications to singular symplectic submanifolds

Muñoz, Vicente and Presas , Francisco and Sols Lucía, Ignacio (2002) Almost holomorphic embeddings in Grassmannians with applications to singular symplectic submanifolds. Journal fur die Reine und Angewandte Mathematik, 547 . pp. 149-189. ISSN 0075-4102

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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.

Item Type:Article
Uncontrolled Keywords:Grassmannians; Symplectic manifold; Compatible almost complex structure
Subjects:Sciences > Mathematics > Differential geometry
Sciences > Mathematics > Geometry
ID Code:17153

V. I. Arnold, S. M. Gusein-Zadeh,A. N.Varchenko,Singularities of smooth mappings, Monographs in

Mathematics, Birkhauser, Boston 1982.

D. Auroux, Asymptotically holomorphic families of symplectic submanifolds, Geom. Funct. Anal. 7 (1997),971–995.

D. Auroux, Symplectic 4-manifolds as branched coverings of CP2, Invent. Math. 139 (2000), 551–602.

D. Borthwick, A. Uribe, Nearly Kahlerian embeddings of symplectic manifolds, Asian J. Math. 4 (2000),599–620.

S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44 (1996), 666–705.

S. K. Donaldson, Lefschetz fibrations in Symplectic Geometry, Doc. Math. Extra Vol. ICM 98 II (1998),309–314.

S. K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Diff. Geom. 53 (1999), 205–236.

P. Grifths, J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York 1978.

M. Gromov, Partial Differential Relations, Springer Ergeb. Math. (3) 9 (1986).

J. Harris, L. Tu, Chern numbers of kernel and cokernel bundles, Invent. Math. 75 (1984), 467–475.

A. Ibort, D. Martınez, F. Presas, On the construction of contact submanifolds with prescribed topology, J.Diff. Geom. 56 (2000), 235–283.

D. McDu¤, D. Salamon, Introduction to symplectic topology, Oxford Math. Monogr., Oxford Univ. Press,New York 1995.

R. Paoletti, Symplectic subvarieties of projective fibrations over symplectic manifolds, Ann. Inst. Fourier Grenoble 49 (1999), 1661–1672.

F. Presas, On the ampleness of the prequantizable line bundle in symplectic and contact geometry, Ph. D.Thesis, Universidad Complutense de Madrid, 2000.

G. Tian, On a set of polarized metrics on algebraic manifolds, J. Diff. Geom. 32 (1990), 99–130.

D. Tischler, Closed 2-forms and an embedding theorem, J. Di¤. Geom. 12 (1977), 229–235.

J. A. Vogelaar, codimension-two subvarieties, Proefscrift. Rijksuniversitteit te Leiden, 1978.

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