  In this paper we study holomorphic rank two vector bundles on the blow up of
$ {\bf C}^2$ at the origin. A classical theorem of Birchoff and Grothendieck
says that any holomorphic vector bundle on the projective plane ${\bf P}^1$
splits into a sum of line bundles. If $E$ is a holomorphic vector bundle over
the blow up of $ {\bf C}^2$ at the origin, then the restriction of $E$ to the
exceptional divisor is a vector bundle over ${\bf P}^1$ and therefore splits.
Moreover we assume that $E$ is a rank two bundle that has zero first Chern
class. Hence its restriction to the exceptional divisor is of the form $ {\cal
O}(j) \oplus {\cal O}(-j) $ for some integer $j.$ We denote by ${\cal M}_j$ the
moduli space of equivalence classes (under holomorphic isomorphisms) of rank
two holomorphic vector bundles on the blow up of $ {\bf C}^2$ at the origin
whose restriction to the exceptional divisor is $ {\cal O}(j) \oplus {\cal
O}(-j) .$
