-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 embedded in an almost complex surface in terms of the homology class of . If is Kähler (or at least symplectic) and the self-intersection of is non-negative then the genus of any other surface embedded in and homologous to is not less then the genus of (the proof of this statement (which is a generalization of the Thom conjecture for ) was recently given by the Seiberg-Witten theory). This paper shows that the extra assumptions on are essential for the genus-minimizing properties of embedded almost complex curves.

**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:
https://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