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The genus-minimizing property of algebraic curves
Author(s):
P. B.
Kronheimer
Journal:
Bull. Amer. Math. Soc.
29
(1993),
63-69.
MSC (2000):
Primary 57R57;
Secondary 14J99, 57R40, 58D27
MathSciNet review:
1193539
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Abstract:
A viable and still unproved conjecture states that, if X is a smooth algebraic surface and C is a smooth algebraic curve in X, then C realizes the smallest possible genus amongst all smoothly embedded 2-manifolds in its homology class. A proof is announced here for this conjecture, for a large class of surfaces X, under the assumption that the normal bundle of C has positive degree.
References:
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- [1]
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- P. B. Kronheimer, papers in preparation.
- [5]
- P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. I, II, Topology (to appear). MR 1241873 (94k:57048)
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- J. W. Morgan, Comparison of the Donaldson invariants of algebraic surfaces with their algebro-geometric analogues, Topology (to appear). MR 1231956 (94m:57066)
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- J. W. Morgan and T. S. Mrowka, A note on Donaldson's polynomial invariants, Internat. Math. Res. Notices, no. 10 (1992), 223-230. MR 1191573 (93m:57032)
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Additional Information:
DOI:
10.1090/S0273-0979-1993-00399-9
PII:
S 0273-0979(1993)00399-9
Copyright of article:
Copyright
1993,
American Mathematical Society
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