Teichmuller space for hyperkahler and symplectic structures

الأصل · EN

Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and Diff₀ the connected component of its diffeomorphism group. The quotient S/₀ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifold M are cohomology classes which can be represented by a minimal rational curve on a deformation of M. We determine the Teichmuller space of hyperkahler structures on a hyperkahler manifold, identifying any of its connected components with an open subset of the Grassmannian SO(b₂-3,3)/SO(3)× SO(b₂-3) consisting of all Beauville-Bogomolov positive 3-planes in H²(M, R) which are not orthogonal to any of the MBM classes. This is used to determine the Teichmuller space of symplectic structures of Kahler type on a hyperkahler manifold of maximal holonomy. We show that any connected component of this space is naturally identified with the space of cohomology classes v∈ H²(M,) with q(v,v)>0, where q is the Bogomolov-Beauville-Fujiki form on H²(M,).