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$J$-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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  • 1. Friedrich Hirzebruch and Heinz Hopf, Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten, Math. Ann. 136 (1958), 156–172 (German). MR 0100844
  • 2. W. C. Hsiang and R. H. Szczarba, On embedding surfaces in four-manifolds, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 97–103. MR 0339239
  • 3. M. Kervaire, J. Milnor, On 2-spheres in 4-manifolds, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1651-1657. MR 24A:2968
  • 4. G. Mikhalkin, Surfaces of small genus in connected sums of 𝐶𝑃² and real algebraic curves with many nests in 𝑅𝑃², Real algebraic geometry and topology (East Lansing, MI, 1993) Contemp. Math., vol. 182, Amer. Math. Soc., Providence, RI, 1995, pp. 73–82. MR 1318732, 10.1090/conm/182/02088
  • 5. V. A. Rohlin, Two-dimensional submanifolds of four-dimensional manifolds, Funkcional. Anal. i Priložen. 5 (1971), no. 1, 48–60 (Russian). MR 0298684

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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
Received by editor(s): September 22, 1995
Communicated by: Ronald Stern
Article copyright: © Copyright 1997 American Mathematical Society