Abstract:

I will survey the recent completion (joint with Chen-Liedtke) of the remaining cases of the conjecture that a projective K3 surface contains infinitely many rational curves. As a consequence of this along with the Bogomolov-Miyaoka-Yau inequality and the deformation theory of stable maps, I will explain (joint with Chen) how in characteristic zero one can deduce the existence of infinitely many curves of any geometric genus moving with maximal variation in moduli on a K3 surface. In particular this leads to an algebraic proof of a theorem of Kobayashi on vanishing of global symmetric differentials and applications to 0-cycles.