Celebratio Mathematica — Atiyah

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M. F. Atiyah and E. Rees: “Vector bundles on projective 3-space,” Invent. Math. 35 : 1 (1976), pp. 131–153. MR 0419852 article

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 \( \dim E \geq \dim X \) 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 \( P_3 \). Our aim is to prove

Every continuous 2-dimensional vector bundle over \( P_3 \) admits a holomorphic structure.