Higher Fano manifolds

Fano manifolds are smooth complex projective varieties having ample anti-canonical class. They form a very special class of varieties. For instance, they are rationally connected, i.e., any two points can be connected by a rational curve.

It is by now well known that proper families of rationally connected varieties over curves always have sections. Recently there has been much effort into finding suitable conditions on the general member of proper families over surfaces that guarantee the existence of rational sections. In this context de Jong and Starr introduced notions of rationally simply connectedness. Conjectural examples of rationally simply connected varieties are higher Fano manifolds. These are Fano manifolds whose Chern characters satisfy suitable positivity conditions.

In this talk I will present a joint work with Ana-Maria Castravet on higher Fano
manifolds. We describe some special aspects of the geometry of higher Fano manifolds and start their classification. Our approach is via rational curves of minimal degree.