Oka-Grauert principle, up to the boundary Let $Z\subset \mathbb{C}^n$ a domain of holomorphy with smooth boundary $\partial Z$ and closure $\bar Z$. There is a natural notion of holomorphic vector bundle over $\bar Z$, given in terms of transition functions $(U_{\alpha\beta}:h_{\alpha\beta}\rightarrow GL(m,\mathbb{C}))$ which are holomorphic in $U_{\alpha\beta}\backslash \partial Z$ and smooth up to the boundary.
Suppose $E\rightarrow \bar Z$ is a holomorphic vector bundle, which is trivial as continuous vector bundle. The Oka-Grauert principle then implies that $E\vert_{Z}$ is holomorphically trivial, but does not say anything about whether there exists a global holomorphic frame that extends smoothly up to $\partial Z$. As Donaldson remarks in this paper from 1992, 'the result is almost certainly true', but like him I have 'unfortunately not been able to find such a result in the literature' (he gives an ad hoc proof in $n=2$).
In this paper by Leiterer (1990) there is actually almost the right thing: Theorem 10.1 gives the result for holomorphic vector bundles, which are continuous up to $\partial Z$. That means in the situation above we obtain a global frame that extends continuously to $\partial Z$, but not smoothly.
Questions.

*

*Has somebody since made the effort to write up some sort of Oka-Grauert principle for holomorphic vector bundles that are smooth up to the boundary?

*Is Leiterer's paper (or his original article in German, referenced in there) the best reference for the continuous case?

*Are there nice elementary approaches for special cases of $Z$'s? (E.g. Donaldson gives an argument for $n=2$ and for $Z$ homeomorphic to a ball.)

 A: This is not a full answer, but sketches an approach that works for $n=1$ and likely generalises to higher dimensions with a finite amount of work. Note however, that for $n=1$ there is also an easy shortcut that I will explain first:
Shortcut for $n=1$: In one complex dimension, any holomorphic vector bundle on $\bar Z$ can be extended to a neighbourhood of $\bar Z$ in $\mathbb C$. One can then apply the classical Oka-Grauert principle on the neighbourhood and restrict the resulting trivialisation to $\bar Z$. It will automatically be smooth. (In higher dimensions this does not work, because holomorphic vector bundles cannot simply be extended, as there is a nontrivial integrability condition that the extension would need to satisfy).
Overview of the general approach: The main idea is to view the Oka-Grauert principle as transitivity of a certain group action and try to prove this using an inverse function theorem. This requires a priori estimates for a $\bar \partial$-problem on manifolds with boundary. If $n=1$ these should become standard elliptic estimates. For $n\ge 2$ one has to work harder, but all the right tools should be contained in Hamilton's series of papers on "Deformation of complex structures on manifolds with boundary."

*

*The group action: Let us write $X=\bar Z$ for the closure of $Z$ and assume that this is a compact manifold with smooth boundary. A holomorphic vector bundle structure on the trivial bundle $X\times \mathbb C^m$ is a $(0,1)$-form $\alpha\in \Omega^{0,1}(X,\mathbb C^{m\times m})$, understood to be smooth up to the boundary, that satisfies the integrability condition
$$ 
F_\alpha = \bar \partial \alpha + \alpha\wedge\alpha = 0.
$$
The holomorphic vector bundle $E_\alpha=(X\times \mathbb C^m,\alpha)$ is trivial, if there exists a function $f\in C^\infty(X,GL(m,\mathbb C))$ such that $\bar \partial f + \alpha f = 0$. Note that $\mathbb G = C^\infty(X,GL(m,\mathbb C))$ is a group that acts on the vector space $\mathcal X = \Omega^{0,1}(X,\mathbb C^{m\times m})$ from the right by the rule
\begin{equation}
\alpha.f=f^{-1}(\bar \partial f + \alpha f).
\end{equation}
One can check that this action leaves the subspace $\mathcal X_0=\{\alpha \in \mathcal X: F_\alpha =0\}$ invariant. Further, the assertion that all vector bundles $E_\alpha$ (with $\alpha\in \mathcal X_0$) are trivial is equivalent to the group $\mathbb G$ acting transitively on $\mathcal X_0$.


*The linearisation: To show transitivity one can consider the derivative of $\mathbb G \ni f\mapsto \alpha.f$ for fixed $\alpha \in \mathcal X_0$, which is a linear map
$$
 D_\alpha \colon T_{\mathrm{Id}} \mathbb G=C^\infty(X,\mathbb C^{m\times m})\rightarrow T_{\alpha}\mathcal X_0\subset \Omega^{0,1}(X,\mathbb C^{m\times m}).
$$
If $D_\alpha$ is onto and we have a nice function theorem at our disposal, then we can conclude that the map $f\mapsto \alpha.f$ is locally surjective near $\mathrm{Id}$, which is to say that small perturbations of $\alpha$ within $\mathcal X_0$ stay within the same $\mathbb G$-orbit. If this is true for all $\alpha \in \mathcal X_0$ and additionally $\mathcal X_0$ is connected, then there can only be one orbit and thus the action has to be transitive. We can describe $D_\alpha$ more explicity: Define the holomorphic vector bundle $E_\alpha^{\mathrm{ad}}$ as the bundle $X\times \mathbb C^{n\times n}$ with holomorphic structure induced by the $\bar \partial$-operator
$$
\bar \partial_{\alpha}^{\mathrm{ad}}\colon \Omega^{p,q}(X,E_\alpha^\mathrm{ad}) \rightarrow \Omega^{p,q+1}(X,E_\alpha^\mathrm{ad}),\quad \beta \mapsto   \bar \partial \beta  + [\alpha,\beta].
$$
One can then identify (at least formally)
$$
T_{\mathrm{Id}}\mathbb G\cong \Omega^0(X,E_\alpha^{\mathrm{ad}}),~ T_{\alpha}\mathcal X_0 = \mathrm{ker}(\bar \partial_{\alpha}^{\mathrm{ad}} )\subset  \Omega^{0,1}(X,E^{\mathrm{ad}}_\alpha),\quad D_\alpha= \bar \partial_{\alpha}^{\mathrm{ad}},
$$
such that the surjectivity of $D_\alpha$ is equivalent to the vanishing result
$$
H^{0,1}(X,E_\alpha^{\mathrm{ad}})=0,
$$
where the cohomology is defined in terms of the $\bar \partial_\alpha^{\mathrm{ad}}$-complex for differential forms that are smooth up to the boundary.
This vanishing result is also mentioned as Proposition 4 in Donaldson's paper from the question above.


*The inverse function theorem: If we don't want to leave the smooth setting, we can try to apply Theorem 2.4.1 in Hamilton's paper on the Nash-Moser inverse function theorem. This requires that $\mathbb G$, $\mathcal X_0$ and the action itself are smooth tame and that $D_\alpha$ has a tame right inverse $R_\alpha$. Note that $R_\alpha$ ought to map any $\beta\in\mathrm{ker}~\bar \partial_{\alpha}^{\mathrm{ad}}\subset \Omega^{0,1}(X,E_\alpha^{\mathrm{ad}})$ to a solution $u\in \Omega^{0}(X,E_\alpha^{\mathrm{ad}})$ of $\bar \partial_{\alpha}^{\mathrm{ad}}u=\beta$ and tameness means that
$$
    \Vert R_\alpha \beta \Vert_{s} \lesssim \Vert \beta \Vert_{s+r} \tag{$\dagger$}
$$
for all $s\in \mathbb N$ and some fixed $r\in \mathbb N$, where $\Vert \cdot \Vert_s$ denotes, say, a Sobolev norm of order $s$. The fact that we can indeed solve the equation $\bar \partial_{\alpha}^{\mathrm{ad}}u=\beta$ just comes from  $H^{0,1}(X,E_\alpha^{\mathrm{ad}})=0$, but the estimates require extra work.



*

*In general, it is not clear to me whether $\mathcal X_0$ is indeed a manifold at all and one might have to use a a more complicated inverse function theorem, such as Theorem 3 in Hamilton's deformation paper (with $Q$ being the map $\alpha \mapsto F_\alpha$).


*The use of the Nash-Moser theorem can probably be avoided by working instead with $\mathbb G^{s+1}$ and $\mathcal X^s$ defined as above, but now containing elements of finite regularity $s+1$ and $s$, respectively. The resulting gauge $f$ can then be shown to be smooth using elliptic regularity.


*The case $n=1$ Here the integrability condition disappears and the linearised problem reduces to the following: given $\alpha\in \mathcal X_0=\mathcal X\cong C^\infty(X,\mathbb C^{m\times m})$, find a solution operator $R_\alpha \colon C^\infty(X,\mathbb C^{m})\rightarrow C^\infty(X,\mathbb C^{m})$ such that $u=R_\alpha \beta$ solves $(\partial_{\bar z} + \alpha)u=\beta$ and such that tame estimates as in $(\dagger)$ are satisfied. Let $\Delta_\alpha=(\partial_{z}+\bar \alpha)(\partial_{\bar z} +\alpha)$, then $\Delta_\alpha u = (\partial_z +\bar \alpha)\beta$, so $u$ can be found by solving a (perturbed) Laplace equation and the required estimates should all be standard.

