Tamed symplectic cones of compact Hermitian-symplectic manifolds

In this paper we consider the tamed symplectic cone of a compact Hermitian-symplectic manifold. First, we prove a Poincar, duality theorem for the tamed symplectic cone. Second, we study the stability of Hermitian-symplectic metrics under the deformation of complex structures and show that the tamed symplectic cones are invariant under the parallel transport with respect to the Gauss-Manin connection.