Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds

Abstract

Let $M$ be a holomorphically symplectic complex manifold, not necessarily compact or quasiprojective, and $X \subset M$ 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 $T^*X$. This is used to study Lagrangian submanifolds which can be bimeromorphically contracted to a point. We prove that such submanifolds are biholomorphic to $CP^n$, 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.