Inequalities for Chern numbers of Fano varieties and varieties of general type

As an application of his famous inequality for stable bundles on surfaces, F. A. Bogomolov proved in 1978 that the self-intersection of the canonical class of a smooth Fano threefold $X$ with $\operatorname{Pic}(X)=Z$ is bounded by 72. This bound is not optimal, as we know from the classification of smooth Fano threefolds suggested by G. Fano and proved by V. A. Iskovskikh. I will tell how one can improve this bound to the optimal value of 64. Moreover, these methods can be applied to locally deformable varieties of general type. The talk is based on a joint work with F. A. Bogomolov and E. Lukzen.