Holomorphic shadows in the eyes of model theory
HTML articles powered by AMS MathViewer
- by Liat Kessler PDF
- Trans. Amer. Math. Soc. 363 (2011), 3287-3307 Request permission
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.
References
- Eugenio Calabi, Construction and properties of some $6$-dimensional almost complex manifolds, Trans. Amer. Math. Soc. 87 (1958), 407–438. MR 130698, DOI 10.1090/S0002-9947-1958-0130698-7
- M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inv. Math. 82 (1985), 307–347.
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- Robert M. Hardt, Topological properties of subanalytic sets, Trans. Amer. Math. Soc. 211 (1975), 57–70. MR 379882, DOI 10.1090/S0002-9947-1975-0379882-8
- 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, DOI 10.2307/1970547
- 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, DOI 10.1007/BF02921708
- Ehud Hrushovski and Boris Zilber, Zariski geometries, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 315–323. MR 1183999, DOI 10.1090/S0273-0979-1993-00380-X
- Ehud Hrushovski and Boris Zilber, Zariski geometries, J. Amer. Math. Soc. 9 (1996), no. 1, 1–56. MR 1311822, DOI 10.1090/S0894-0347-96-00180-4
- Dusa McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc. 3 (1990), no. 3, 679–712. MR 1049697, DOI 10.1090/S0894-0347-1990-1049697-8
- 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
- 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, DOI 10.1090/coll/052
- 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, DOI 10.2307/2118627
- R. N. Moosa, The model theory of compact complex spaces.
- 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
- David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
- B. Zilber, Model theory and algebraic geometry, Proceedings of 10th Easter conference at Berlin, 1993.
- B. Zilber, Zariski Geometries, http://people.maths.ox.ac.uk/ zilber/
- B. Zilber and Y. Peterzil, Lecture notes on Zariski-type structure, preprint, 1994.
Additional Information
- Liat Kessler
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139
- Email: kessler@math.mit.edu
- Received by editor(s): May 18, 2009
- Received by editor(s) in revised form: October 4, 2009
- Published electronically: January 6, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 3287-3307
- MSC (2010): Primary 03C10, 03C98, 32Q65, 32Q60, 53D45
- DOI: https://doi.org/10.1090/S0002-9947-2011-05219-0
- MathSciNet review: 2775808