Here we consider a complex vector bundle $E$ over a complex manifold $X$. We want to study the interaction between two structures

• •

  A hermitian metric on $E$;

• •

  A holomorphic structure on $E$, which can be defined by a $\overline{\partial}$-operator

  $$\overline{\partial}:\Omega^{0}(E)\rightarrow\Omega^{0,1}(E).$$

A basic fact is that given both of these structures there is a unique compatible unitary connection, in the sense that the $\overline{\partial}$-operator is the $(0,1)$-component of the covariant derivative. Now there are two ways of setting up the theory. In the first—the traditional point of view in complex geometry, as is [14] for example—we fix a holomorphic structure and consider the various Hermitian metrics. Then we have, for example, the formula

$$F_{h}=\overline{\partial}(h^{-1}\partial h)$$

(1)

for the curvature tensor in a local holomorphic trivialisation, where the metric is defined by a matrix-valued function $h$. In the second point of view—closer to what one does in general Yang-Mills theory—we fix the Hermitian metric and consider various $\overline{\partial}$-operators. We can identify the set of these operators with the space ${\cal A}$ of unitary connections on $E$. This point of view brings in two infinite dimensional groups. First, the group $U(E)$ of unitary automorphisms of $E$ and second the group $GL(E)$ of general linear automorphisms. Then $GL(E)$ acts on the space of $\overline{\partial}$-operators by conjugation, and hence on the set ${\cal A}$ of connections. The $\overline{\partial}$-operators which define equivalent holomorphic structures are exactly those which are in the same orbit of the $GL(E)$-action.

The advantage of this second point of view comes when studying the “jumping” of holomorphic structures. This arises from the fact that the $GL(E)$ orbits are not usually closed in ${\cal A}$. Fix a Kahler metric on the base space $X$ and use this to define the Yang-Mills functional: the $L^{2}$ norm of the curvature. When one seeks Yang-Mills connections compatible with a given holomorphic structure ${\cal E}$ one attempts to minimise this functional over a $GL(E)$ orbit in ${\cal A}$. But it may happen that there is no minimum, in the simplest case because the infimum is achieved at a point in ${\cal A}$ in the closure but not in the orbit itself. Then one finds a Yang-Mills connection not on the original holomorphic bundle ${\cal E}$, but on another one ${\cal E}^{\prime}$, such that there are arbitrarily small deformations of ${\cal E}^{\prime}$ which are isomorphic to ${\cal E}$. This lies at the root of the solution of the link between Yang-Mills theory and stability of holomorphic bundles expressed by the Kobayashi-Hitchin conjecture [3], [33], [7].