Forms and Complex Manifolds

We study the intersection form $F_X$ on the second cohomology group $H^2(X, \mathbb{Z})$ of a compact K\"ahler 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 K\"ahler manifolds.