Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0273-0979-1993-00399-9
MathSciNet review: 1193539
Full-text PDF Free Access

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.


References [Enhancements On Off] (What's this?)

  • [1] S. K. Donaldson, Complex curves and surgery, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 91-97. MR 1001449 (90k:57041)
  • [2] -, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257-315. MR 1066174 (92a:57035)
  • [3] M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton Univ. Press, Princeton, NJ, 1990. MR 1201584 (94b:57021)
  • [4] 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)
  • [6] J. W. Morgan, Comparison of the Donaldson invariants of algebraic surfaces with their algebro-geometric analogues, Topology (to appear). MR 1231956 (94m:57066)
  • [7] 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)
  • [8] K. G. O'Grady, Algebro-geometric analogues of Donaldson's polynomials, Invent. Math. 107 (1992), 351-395. MR 1144428 (93d:14060)

Similar Articles

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

American Mathematical Society