On a classical problem about Chern numbers

Generalising a question of Hirzebruch, Kotschick asked the following: which Chern numbers are determined up to finite ambiguity by the underlying smooth manifold? Together with S. Schreieder, I treated this question in dimension higher than 3. After explaining such results, I will talk about the 3 dimensional case, where it is believed that Chern numbers are bounded. Results in this direction have been obtained in a recent preprint with P. Cascini, where tools from the Minimal Model Program have been used, combined with topology's and arithmetic's techniques.