Extension of holomorphic vector bundles across a totally real submanifold

Let ϵ be a Hermitian-holomorphic vector bundle with compatible Chern connection ▽, defined on the complement of Rⁿ inside a polydisc Δ ⊇ Cⁿ. The main result of this notes provides a necessary and sufficient condition for unique extension of ϵ to Δ, namely that the curvature F_▽, when contracted with a non-vanishing holomorphic vector field ξ, is ∂-exact. Applications to removable singularities for anti-self-dual connections across a real time line in C² and solutions of the static monopole equation across a point in R³ are also given as corollaries.