Christian Liedkte: Curves on K3 Surfaces

Abstract

A (complex and projective) K3 surface is a complex 2-dimensional algebraic manifold that is simply connected and with trivial first Chern class. A long-standing conjecture, which is sometimes attributed to Bogomolov, states that K3 surfaces contain infinitely many rational curves, that is, it predicts the existence of infinitely many distinct non-trivial and algebraic/holomorphic morphism from the projective line to it. In my talk, I will explain why this conjecture is interesting and in particular, the relation it has to other fields, such as, arithmetic geometry, algebraic geometry, differential geometry, and string theory. Finally, I will report on the state of this conjecture from the existence of the first rational curve (due to Bogomolov–Mori–Mukai–Mumford), via important special cases (notably in work of Chen, Bogomolov–Hassett–Tschinkel, and Li–Liedtke) until the recent full establishment of the conjecture (due to Chen–Gounelas–Liedtke).