Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Holomorphic bundles on complex manifolds with boundary   Let $\Omega$ be a complex manifold, and let $X\subset \Omega$ be an open
submanifold whose closure $\bar X$ is a (not necessarily compact) submanifold
with smooth boundary. Let $G$ be a complex Lie group, $\Pi$ be a differentiable
principal $G$-bundle on $\Omega$ and $J$ a formally integrable bundle almost
complex structure on the restriction $\bar P:= \Pi|_{\bar X}$. We prove that,
if the boundary of $\bar X$ is strictly pseudoconvex, $J$ extends to a
holomorphic structure on the restriction of $\Pi$ to a neighborhood of $\bar X$
in $\Omega$. This answers positively and generalizes a problem stated in the
article "Boundary value problems for Yang-Mills fields" by S. Donaldson. We
obtain a gauge theoretical interpretation of the quotient
$\mathcal{C}^\infty(\partial \bar X,G)/\mathcal{O}^\infty(\bar X,G)$ associated
with any compact Stein manifold with boundary $\bar X$ endowed with a Hermitian
metric. For a fixed differentiable $G$-bundle $\bar P$ on a complex manifold
$\bar X$ with non-pseudoconvex boundary, we study the set of formally
integrable almost complex structures on $\bar P$ which admit formally
holomorphic local trivializations at boundary points. We give an example where
a "generic" formally integrable almost complex on $\bar P$ admits formally
holomorphic local trivializations at no boundary point, whereas the set of
formally integrable almost complex structures which admit formally holomorphic
local trivializations at all boundary points is dense.

36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph