The Thullen-type extension theorem for positive holomorphic vector bundles

We present a proof of the following theorem announced by Siu in 1972: Let $X$ be a complex manifold, $Y$ an irreducible hypersurface in $X$, and $G$ an open set containing $X\setminus Y$ and having a non-empty intersection with $Y$. Let $(E,h)$ be a holomorphic Hermitian vector bundle over $G$ with Nakano-positive curvature. Then $E$ extends to $X$ as a reflexive sheaf. The proof follows an original idea of Siu and involves Demailly's extension of Hörmander's $L^2$-technique for non-complete complex manifolds.