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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 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 -holomorphic curves in symplectic geometry in the language of shadows structures.

**[Ca]**Eugenio Calabi,*Construction and properties of some 6-dimensional almost complex manifolds*, Trans. Amer. Math. Soc.**87**(1958), 407–438. MR**0130698**, 10.1090/S0002-9947-1958-0130698-7**[Gr]**M. Gromov,*Pseudo holomorphic curves in symplectic manifolds*, Inv. Math.**82**(1985), 307-347.**[GR]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[Ha]**Robert M. Hardt,*Topological properties of subanalytic sets*, Trans. Amer. Math. Soc.**211**(1975), 57–70. MR**0379882**, 10.1090/S0002-9947-1975-0379882-8**[Hi]**Heisuke Hironaka,*Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II*, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2)**79**(1964), 205–326. MR**0199184****[HLS]**Helmut Hofer, Véronique Lizan, and Jean-Claude Sikorav,*On genericity for holomorphic curves in four-dimensional almost-complex manifolds*, J. Geom. Anal.**7**(1997), no. 1, 149–159. MR**1630789**, 10.1007/BF02921708**[HZ1]**Ehud Hrushovski and Boris Zilber,*Zariski geometries*, Bull. Amer. Math. Soc. (N.S.)**28**(1993), no. 2, 315–323. MR**1183999**, 10.1090/S0273-0979-1993-00380-X**[HZ2]**Ehud Hrushovski and Boris Zilber,*Zariski geometries*, J. Amer. Math. Soc.**9**(1996), no. 1, 1–56. MR**1311822**, 10.1090/S0894-0347-96-00180-4**[McD]**Dusa McDuff,*The structure of rational and ruled symplectic 4-manifolds*, J. Amer. Math. Soc.**3**(1990), no. 3, 679–712. MR**1049697**, 10.1090/S0894-0347-1990-1049697-8**[MS1]**Dusa McDuff and Dietmar Salamon,*Introduction to symplectic topology*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR**1698616****[MS2]**Dusa McDuff and Dietmar Salamon,*𝐽-holomorphic curves and symplectic topology*, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2004. MR**2045629****[MW]**Mario J. Micallef and Brian White,*The structure of branch points in minimal surfaces and in pseudoholomorphic curves*, Ann. of Math. (2)**141**(1995), no. 1, 35–85. MR**1314031**, 10.2307/2118627**[Mo1]**R. N. Moosa,*The model theory of compact complex spaces*.**[Mo2]**Rahim Nazim Moosa,*Contributions to the model theory of fields and compact complex spaces*, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR**2702428****[Mu]**David Mumford,*Algebraic geometry. I*, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties; Grundlehren der Mathematischen Wissenschaften, No. 221. MR**0453732****[Zi1]**B. Zilber,*Model theory and algebraic geometry*, Proceedings of 10th Easter conference at Berlin, 1993.**[Zi2]**B. Zilber,*Zariski Geometries*, http://people.maths.ox.ac.uk/ zilber/**[ZP]**B. Zilber and Y. Peterzil,*Lecture notes on Zariski-type structure*, preprint, 1994.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
03C10,
03C98,
32Q65,
32Q60,
53D45

Retrieve articles in all journals with MSC (2010): 03C10, 03C98, 32Q65, 32Q60, 53D45

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.