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On vector bundles on algebraic surfaces and 0-cycles

Published online by Cambridge University Press:  22 January 2016

E. Ballico
E. Ballico*
Affiliation:
Department of Mathematics, University of Trento, 38050 Povo (TN), Italy
*
e-mail: (bitnet) ballico itncisca (Decnet) itnvaxi: ballico fax: italy + 461881624
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Let X be an algebraic complex projective surface equipped with the euclidean topology and E a rank 2 topological vector bundle on X. It is a classical theorem of Wu ([Wu]) that E is uniquely determined by its topological Chern classes . Viceversa, again a classical theorem of Wu ([Wu]) states that every pair (a, b) ∈ (H (X, Z), Z) arises as topological Chern classes of a rank 2 topological vector bundle. For these results the existence of an algebraic structure on X was not important; for instance it would have been sufficient to have on X a holomorphic structure. In [Sch] it was proved that for algebraic X any such topological vector bundle on X has a holomorphic structure (or, equivalently by GAGA an algebraic structure) if its determinant line bundle has a holomorphic structure. It came as a surprise when Elencwajg and Forster ([EF]) showed that sometimes this was not true if we do not assume that X has an algebraic structure but only a holomorphic one (even for some two dimensional tori (see also [BL], [BF], or [T])).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 130 , June 1993 , pp. 19 - 23
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

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