Abstract
Fano manifolds are complex projective manifolds with positive first Chern class. This condition has far-reaching geometric and arithmetic implications, making Fano manifolds a central topic in algebraic geometry. In recent years, there has been some effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds.

In this talk, I will survey the theory of higher Fano manifolds, defined in terms of positivity of higher Chern characters. I will also present recent joint work with Roya Beheshti, Ana-Maria Castravet, Kelly Jabbusch, Svetlana Makarova, Enrica Mazzon and Nivedita Viswanathan on toric higher Fano manifolds.