Rational curves on HK manifolds

By a classical theorem of Bogomolov and Mumford, every ample linear system on a projective K3 surface contains a sum of rational curves. As a natural higher-dimensional analogue, we can ask whether every ample class on a projective hyperkähler (HK) manifold has a multiple containing a sum of uniruled divisors. Such divisors are special: by adjunction, smooth ample divisors on varieties with trivial canonical bundle are of general type, hence not uniruled. The question is also motivated by Voisin’s conjectures on the Chow group of 0-cycles, for which ample uniruled divisors provide the first evidence. In this talk I will present the known results on the known deformation families of HK manifolds, including joint work with A. Grossi and work in progress with A. Grossi and G. Nanni.