Let Symp be the (infinitedimensional) space
of all symplectic forms, and Teich:=Symp/Diff_0
its quotient by the group of isotopies
(that is, by the connected component of the
group of diffeomorphisms). It was proven
by Moser that Teich, known as "Teichmuller
space of symplectic structures", is a smooth
manifold. Nevertheless, its structure is
still very much mysterious. Denote by Teich_k
a subset of Teich consisting of all symplectic
structures which are Kahler for some complex structure.
By Kodaira, under additional nonrestrictive assumptions,
the manifold Teich_k is open in Teich.
I will describe the structure of Teich_k on a hyperkahler
manifold, and explain how it can be used to
solve problems of symplectic packing. These
is a joint work with Ekaterina Amerik.
