Deformations and BBF form on non-Kahler holomorphically symplectic manifolds

Abstract

In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is similar to that of hyperkahler manifolds. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formula for a BG-manifold $M$, showing that there exists a symmetric form q on the second cohomology such that for any $w\in H^2(M)$ one has $\int_M w^{2n}=q(w,w)^n$. This form is a non-Kahler version of the Beauville-Bogomolov-Fujiki form known in hyperkahler geometry.