MBM loci in families of hyperkahler manifolds and centers of birational contractions

Abstract

An MBM class on a hyperkahler manifold M is a second cohomology class such that its orthogonal complement in H^2(M) contains a maximal dimensional face of the boundary of the Kahler cone for some hyperkahler deformation of M. An MBM curve is a rational curve in an MBM class and such that its local deformation space has minimal possible dimension 2n-2, where 2n is the complex dimension of M. We study the MBM loci, defined as the subvarieties covered by deformations of an MBM curve within M. When M is projective, MBM loci are centers of birational contractions. For each MBM class z, we consider the Teichmuller space $Teich^{min}_z$ of all deformations of M such that $z^{\bot}$ contains a face of the Kahler cone. We prove that for all $I,J\in Teich^{min}_z$, the MBM loci of (M, I) and (M,J) are homeomorphic under a homeomorphism preserving the MBM curves, unless possibly the Picard number of I or J is maximal.