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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$J$-holomorphic curves in almost complex surfaces do not always minimize the genus
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by G. Mikhalkin PDF
Proc. Amer. Math. Soc. 125 (1997), 1831-1833 Request permission

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
  • Received by editor(s): September 22, 1995
  • Communicated by: Ronald Stern
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 1831-1833
  • MSC (1991): Primary 57R95, 53C15
  • DOI: https://doi.org/10.1090/S0002-9939-97-03710-6
  • MathSciNet review: 1363430