Holomorphic shadows in the eyes of model theory
Author:
Liat Kessler
Journal:
Trans. Amer. Math. Soc. 363 (2011), 32873307
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 to be a holomorphic shadow if it is the image of a holomorphic map from a compact complex manifold. Notice that a 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 on a manifold , the holomorphic shadows in the Cartesian products of form a Zariskitype structure. Checking this leads to nontrivial geometric questions and results. We then apply the work of Hrushovski and Zilber on Zariskitype structures. We also restate results of Gromov and McDuff on holomorphic curves in symplectic geometry in the language of shadows structures.
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 R. Hardt, Topological properties of subanalytic sets, Trans. Amer. Soc. 211 (1975), 5770. MR 0379882 (52:787)
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 H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. (2) 79 (1964), 109326. MR 0199184 (33:7333)
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 H. Hofer, V. Lizan, and J. C. Sikorav, On genericity for holomorphic curves in fourdimensional almostcomplex manifolds, J. Geom. Anal. 7 (1997), no. 1, 149159. MR 1630789 (2000d:32045)
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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/S000299472011052190
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.
