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Configurations of 2-spheres in the K3 surface and other 4-manifolds

Published online by Cambridge University Press:  24 October 2008

Daniel Ruberman
Daniel Ruberman
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA 02254
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Extract

A current theme in the theory of 4-manifolds is the study of which properties of complex surface are determined the underlying smooth 4-manifold. For instance, the genus of a complex curve in a complex surface is determined by its homology class, via the adjunction formula. Recent work in gauge theory [10–12] has shown that, to a great degree, a similar principal holds for an arbitrary (i.e. not necessarily complex) smooth representative of a 2-dimensional homology class. Another question, still unsolved even in the context of algebraic geometry, is to find the number of disjoint rational curves on a complex surface. The classical case, namely that of hypersurfaces in CP3, has only been settled for degrees d ≤ 6. The papers [1, 2, 4, 8, 14, 15] contain bounds on the number of such curves and constructions of surfaces with many ( — 2)-curves; the last two together establish that 65 is the correct bound in degree 6.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 2 , August 1996 , pp. 247 - 253
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

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