The genus-minimizing property of algebraic curves

Author:
P. B. Kronheimer

Journal:
Bull. Amer. Math. Soc. **29** (1993), 63-69

MSC (2000):
Primary 57R57; Secondary 14J99, 57R40, 58D27

MathSciNet review:
1193539

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Simon K. Donaldson,*Complex curves and surgery*, Inst. Hautes Études Sci. Publ. Math.**68**(1988), 91–97 (1989). MR**1001449****[2]**S. K. Donaldson,*Polynomial invariants for smooth four-manifolds*, Topology**29**(1990), no. 3, 257–315. MR**1066174**, 10.1016/0040-9383(90)90001-Z**[3]**Michael H. Freedman and Frank Quinn,*Topology of 4-manifolds*, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR**1201584****[4]**P. B. Kronheimer, papers in preparation.**[5]**P. B. Kronheimer and T. S. Mrowka,*Gauge theory for embedded surfaces. I*, Topology**32**(1993), no. 4, 773–826. MR**1241873**, 10.1016/0040-9383(93)90051-V**[6]**John W. Morgan,*Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues*, Topology**32**(1993), no. 3, 449–488. MR**1231956**, 10.1016/0040-9383(93)90001-C**[7]**John W. Morgan and Tomasz S. Mrowka,*A note on Donaldson’s polynomial invariants*, Internat. Math. Res. Notices**10**(1992), 223–230. MR**1191573**, 10.1155/S1073792892000254**[8]**Kieran G. O’Grady,*Algebro-geometric analogues of Donaldson’s polynomials*, Invent. Math.**107**(1992), no. 2, 351–395. MR**1144428**, 10.1007/BF01231894

Retrieve articles in *Bulletin of the American Mathematical Society*
with MSC (2000):
57R57,
14J99,
57R40,
58D27

Retrieve articles in all journals with MSC (2000): 57R57, 14J99, 57R40, 58D27

Additional Information

DOI:
https://doi.org/10.1090/S0273-0979-1993-00399-9

Article copyright:
© Copyright 1993
American Mathematical Society