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 $T^{\ast }X$. This is used to study Lagrangian submanifolds which can be bimeromorphically contracted to a point. We prove that such submanifolds are biholomorphic to $C{P}^n$, 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.

arXiv comment21 pages, version 1.0.2

References in corpus (2)

• Morse-Novikov cohomology of locally conformally Kähler manifolds
• LCK rank of locally conformally Kahler manifolds with potential

Cited by in corpus (4)

• Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds
• Bimeromorphic geometry of LCK manifolds
• Algebraic cones of LCK manifolds with potential
• Do products of compact complex manifolds admit LCK metrics?