I am currently very confused about the real side of the Ward correspondence. Recall that the Ward correspondence gives a one-to-one correspondence between:


$M$-trivial holomorphic bundles $E$ on $Z$, and,


holomorphic bundles $\hat{E}$ on $M$ with holomorphic connection,


where one has an analytic family $Z\overset{\eta}{\leftarrow} F \overset{\tau}{\to} M$ of compact complex manifolds. $M$-trivial means $\forall t \in M$ the bundle $E\mid\_{X\_t}$ is trivial, where $X\_t:=\eta\left(\tau^{-1}\left(t\right)\right)$. This I would call the complex side of the Ward correspondence and I understand this side.


Now, suppose that $p\colon Z \to M'$ is the twistor space of an anti-self dual manifold $M'$. Then there is an anti-holomorphic involution $\sigma$ on $Z$ that induces an anti-holomorphic involution $\sigma'$ on $M$ such that $X\_{\sigma'\left(t\right)}=\sigma\left(X\_t\right)$ and the fixed point set of $\sigma'$ satisfies $M'\cong M^{\sigma'}$. If one starts with a smooth vector bundle $E'$ on $M'$ equipped with an anti-self dual connection $\nabla'$ then the complexified pullback to $Z$ via $p$ is canonically holomorphic and $M$-trivial. By the Ward correspondence this is the same as a holomorphic vector bundle $\hat{E}$ with holomorphic connection $\nabla$ on $M$. Now one wants to be able to reconstruct $E'$ and $\nabla'$ from $\hat{E}$ and $\nabla$.


When the bundle $E'$ is not complex it is part of the Ward transform to complexify $p^\*E\to Z$. My problem is with the case where $E$ is already complex, since then $\tilde{E}:=p^\*E\to Z$ is also holomorphic and $\hat{E}=\tau\_\*\eta^\*\tilde{E}$ admits a holomorphic connection. Then


$\tau^{\*}\_0\left(\tau\_\*\eta^\*\tilde{E}\mid\_{M^{\sigma'}} \right)\cong \left( \tau^\*\tau\_\*\eta^\*\tilde{E} \right)\mid\_{F\_0} \cong \left( \eta^\*\tilde{E} \right)\mid\_{F\_0} \cong \eta\_0^\*\tilde{E}\cong \tilde{E},$


where $F\_0 = \tau^{-1}\left(M^{\sigma'}\right)$ and $\tau\_0= \tau\mid\_{F\_0}$, $\eta\_0=\eta\mid\_{F\_0}$, note that $\eta\_0$ is a real analytic isomorphism. This real analytic isomorphism above allows to define a real analytic isomorphism $E \to \left(\tau\_\*\eta^\*E\right)\mid\_{M^{\sigma'}}$ by pulling back sections to $\tilde{E}$ and then descending to $\left(\tau\_\*\eta^\*\tilde{E}\right)\mid\_{M^{\sigma'}}$ which is possible since the pullback sections are constant along the fibers of $p$.


But such an isomorphism would spell trouble because of the following argument:


$\hat{E}$ admits a holomorphic connection, meaning if e.g. $M= \text{Gr}\_2\left(\mathbb{C}^4\right)$ then all Chern classes of $\hat{E}$ vanish and thus also all Chern classes of $\hat{E}\mid\_{M^{\sigma'}} \cong E$ vanish. But in this case $M^{\sigma'}\cong S^4$ and there exist anti-self dual bundles on $S^4$ with non-vanishing Chern classes.


Does anyone have an idea what I am doing wrong?



Real part of the Ward correspondence
The isomorphism does not spell trouble, my argument is just flawed. I forgot that one only obtains a holomorphic connection on $M\setminus A$ where $A$ is some hypersurface. This is because an instanton bundle on $\mathbb{CP}^3$ may fail to $\text{Gr}\_2\left(\mathbb{C}^4\right)$-trivial on this hypersurface $A$. Then the Chern class argument breaks down and there is no problem.


