Special Hermitian structures and symplectic geometry

Symplectic forms taming complex structures on compact manifolds are strictly related to a special type of Hermitian metrics, known in the literature as “strong Kaehler with torsion” metrics. I will present general results on “strong Kahler with torsion” metrics, their link with symplectic geometry and more in general with generalized complex gometry. Moreover, I will show for certain 4-dimensional non-Kaehler symplectic 4-manifolds some recent results about the Calabi-Yau equation in the context of symplectic geometry.