Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds

Let M be a holomorphically symplectic complex manifold, not necessarily compact or quasiprojective, and \(X \subset M\) $X\subset M$ a compact Lagrangian submanifold. We construct a deformation to the normal cone, showing that a neighbourhood of X can be deformed to its neighbourhood in \(T^* X\) $T^{\ast }X$ . This is used to study Lagrangian submanifolds which can be bimeromorphically contracted to a point. We prove that such submanifolds are biholomorphic to \({\mathbb {C}}P^n\) $\mathbb{C}{P}^n$ , and show that a certain neighbourhood of X is symplectically biholomorphic to a neighbourhood of the zero section of its cotangent bundle. This gives a holomorphic version of the Weinstein’s normal neighbourhood theorem.