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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Holomorphic shadows in the eyes of model theory


Author: Liat Kessler
Journal: Trans. Amer. Math. Soc. 363 (2011), 3287-3307
MSC (2010): Primary 03C10, 03C98, 32Q65, 32Q60, 53D45
Published electronically: January 6, 2011
MathSciNet review: 2775808
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Abstract: We define a subset of an almost complex manifold $ (M,J)$ to be a holomorphic shadow if it is the image of a $ J$-holomorphic map from a compact complex manifold. Notice that a $ J$-holomorphic curve is a holomorphic shadow, and so is a complex subvariety of a compact complex manifold.

We show that under some conditions on an almost complex structure $ J$ on a manifold $ M$, the holomorphic shadows in the Cartesian products of $ (M,J)$ form a Zariski-type structure. Checking this leads to non-trivial geometric questions and results. We then apply the work of Hrushovski and Zilber on Zariski-type structures.

We also restate results of Gromov and McDuff on $ J$-holomorphic curves in symplectic geometry in the language of shadows structures.


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Additional Information

Liat Kessler
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139
Email: kessler@math.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05219-0
Received by editor(s): May 18, 2009
Received by editor(s) in revised form: October 4, 2009
Published electronically: January 6, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.