Contraction centers in families of hyperkähler manifolds

Selecta Mathematica, New Series. 2021. Vol. 27. Article 60.

Amerik E., Verbitsky M.

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperkähler 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 Kähler cone. Cohomology classes which can possibly be orthogonal to a wall of the Kähler 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. Any MBM curve can be contracted on an appropriate birational model of M, unless 𝑏2(𝑀)⩽5b2(M)⩽5. When 𝑏2(𝑀)>5b2(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 stratified 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 𝑏2⩽4b2⩽4. Moreover, these diffeomorphisms preserve the MBM curves, and induce biholomorphic maps on the contraction fibers, if they are normal.

Keywords: hyperkähler manifolds, birational contractions

Publication based on the results of: Categorical, classical and differential geometry of algebraic varieties (2021)