Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far-reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one-dimensional bases always admit holomorphic sections.
In recent years, there has been a great effort toward defining suitable higher analogues of the Fano condition, which are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. In this talk, I will discuss a possible notion of higher Fano manifolds in terms of the positivity of higher Chern characters and discuss geometric features of these manifolds.