Degenerations of K3 Surfaces, Gravitational Instantons, and M-Theory
The detailed study of degenerations of K3 surfaces as complex manifolds goes back more than forty years and is fairly complete. Much less is known about the analogous problem in differential geometry of finding Gromov--Hausdorff limits for sequences of Ricci-flat metrics on the K3 manifold. I will review work of H.-J. Hein and G. Chen--X. Chen on gravitational instantons with curvature decay, and describe applications to the K3 degeneration problem. I will also review some very recent work of Hein-Sun-Viaclovsky-Zhang constructing some additional important examples. The known examples are connected to physical dualities, as I shall also explain.