Forms and complex manifolds

We study the intersection form F_{X} on the second cohomology group H^{2}(X, \mathbb{Z}) of a compact Kähler manifold X of dimension n . Although the structure of F_{X} is relatively well understood in dimensions two and three, much less is known for n \geq 4 . We investigate the fundamental properties of F_{X} in higher dimensions and discuss several applications to birational geometry. Finally, we present a number of open problems concerning the relationship between birational invariants and topological invariants of Kähler manifolds.