Contraction centers in families of hyperkähler manifolds

Abstract

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperk\"ahler manifold $M$. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the K\"ahler cone. Homology classes which can possibly be orthogonal to a wall of the K\"ahler cone of some deformation of $M$ are called MBM classes. We prove that all MBM classes of type (1,1) can be represented by rational curves, called MBM curves. All MBM curves can be contracted on an appropriate birational model of $M$, unless $b_2(M) \leq 5$. When $b_2(M)>5$, this property can be used as an alternative definition of an MBM class and an MBM curve. Using the results of Bakker and Lehn, we prove that the diffeomorphism type of a contraction locus remains stable under all deformations for which these classes remains of type (1,1), unless the contracted variety has $b_2\leq 4$. Moreover, these diffeomorphisms preserve the MBM curves, and induce biholomorphic maps on the contraction fibers, if they are normal.