It was conjectured  that any symplectic structure
on a K3 surface is compatible with a Kahler
structure. Surprisingly, for K3 orbifolds this is
false. The "conifold transform" on a K3 surface takes a
pair (symplectic structure, Lagrangian sphere) to a
symplectic structure on an orbifold K3 surface with a
single double point. Using this construction, we are able
to show that an orbifold K3 surface admits a symplectic
structure not compatible with any Kahler structure. On the
way towards this result, we demonstrate existence of
Lagrangian spheres in K3 surfaces which are not
Hamiltonian isotopic to special Lagrangian submanifolds.
This is a work in progress, joint with Michael Entov.