| Starting from an old problem posed by Hirzebruch, I will report on the following question: if c is the first Chern class of a complex projective 3-fold X, is there a topological bound for c^3? I will discuss some recent progress based on the Minimal Model Program and explain how the birational properties and the topological properties of X interact. The cubic form arising from the cup product on X will reveal itself a key object for our investigation. This is a joint work with P. Cascini. |