The projective plane, <i>J</i>-holomorphic curves and Desarguesʼ theorem

Siddhartha Gadgil

doi:10.1016/j.crma.2013.10.022

Geometry/Differential topology

The projective plane, J-holomorphic curves and Desarguesʼ theorem

Presented by: Étienne Ghys

Siddhartha Gadgil ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 915-920.

Abstract
Résumé

By a theorem of Gromov, for an almost complex structure J on $C P^{2}$ tamed by the standard symplectic structure, the J-holomorphic curves representing the positive generator of homology form a projective plane. We show that this satisfies the Theorem of Desargues if and only if J is isomorphic to the standard complex structure. This answers a question of Ghys.

Dʼaprès un théorème dû à Gromov, pour toute structure presque complexe J sur $C P^{2}$ qui est compatible avec la structure symplectique standard, les courbes J-holomorphes dans la classe du générateur positif de $H_{2} (C P^{2})$ correspondent à la collection des droites dʼun plan projectif. Nous démontrons que ce plan projectif est arguésien si et seulement si J est isomorphe à la structure complexe standard, répondant ainsi à une question posée par Ghys.

Received: 2010-10-01
Accepted: 2013-10-21
Published online: 2013-11-07

DOI: 10.1016/j.crma.2013.10.022

Author's affiliations:

Siddhartha Gadgil ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

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Siddhartha Gadgil. The projective plane, J-holomorphic curves and Desarguesʼ theorem. Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 915-920. doi : 10.1016/j.crma.2013.10.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.10.022/

References
Cited by

[1] E. Artin Geometric Algebra, John Wiley & Sons, Inc., New York, 1988 (Reprint of the 1957 original)

[2] M. Gromov Pseudoholomorphic curves in symplectic manifolds, Invent. Math., Volume 82 (1985) no. 2, pp. 307-347

[3] L. Pontrjagin Über stetige algebraische Körper, Ann. Math. (2), Volume 33 (1932) no. 1, pp. 163-174

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