Vector bundles on projective 3-space

Abstract

On a compact complex manifold X it is an interesting problem to compare the continuous and holomorphic vector bundles. The case of line-bundles is classical and is well understood in the framework of sheaf theory. On the other hand for bundles E with dimE>dimX we are in the stable topological range and one can use K-theory. Much is known in this direction, for example the topological and holomorphic K-groups of all complex projective spaces are isomorphic. This paper deals with what is perhaps the simplest case not covered by the methods indicated above. We shall consider 2-dimensional complex vector bundles over the 3-dimensional complex projective space P3. Our aim is to prove (1.1) Theorem. Every continuous 2-dimensional vector bundle over P3 admits a holomorphic structure. The corresponding result for P2 was proved by Schwarzenberger [13], but this falls into the class of stable problems. In particular 2-dimensional vector bundles over P2 are determined by their Chern classes c 1 , c 2 . This is no longer true on P3 and therein lies the main difficulty and also the interest of this paper. In fact Horrocks in [-10] has already constructed holomorphic (actually algebraic) bundles with arbitrary cl, c 2 subject only to the topologically necessary condition that c~c 2 be even [8; p. 166]. It is not hard to see that, topologically, there are at most two bundles on P3 with given cl, c 2 . The two possibilities arise because the homotopy group n 5 (U(2)) ~ n 5 (S 3) which classifies 2-dimensional bundles over S 6, and acts on the bundles over P3, has order 2. It turns out that there are two sharply different cases depending on the parity of c~. In w 2 we study the case of even c 1 . By tensoring with line-bundles one reduces to the case of cl =0 in which case the structure group is SU(2)~-Sp(1). We view our 2-dimensional complex vector bundle as a quaternion line-bundle and this simplifies the classification because quaternion line-bundles over P3 are already