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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$J$-holomorphic curves in almost complex surfaces do not always minimize the genus


Author: G. Mikhalkin
Journal: Proc. Amer. Math. Soc. 125 (1997), 1831-1833
MSC (1991): Primary 57R95, 53C15
MathSciNet review: 1363430
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Abstract: The adjunction formula computes the genus of an almost complex curve $F$ embedded in an almost complex surface $M$ in terms of the homology class of $F$. If $M$ is Kähler (or at least symplectic) and the self-intersection of $F$ is non-negative then the genus of any other surface embedded in $M$ and homologous to $F$ is not less then the genus of $F$ (the proof of this statement (which is a generalization of the Thom conjecture for $\Bbb C P^2$) was recently given by the Seiberg-Witten theory). This paper shows that the extra assumptions on $M$ are essential for the genus-minimizing properties of embedded almost complex curves.


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Additional Information

G. Mikhalkin
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 1A1
Email: mihalkin@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03710-6
PII: S 0002-9939(97)03710-6
Received by editor(s): September 22, 1995
Communicated by: Ronald Stern
Article copyright: © Copyright 1997 American Mathematical Society