Hsueh-Yung Lin: On algebraic approximations of compact Kähler threefolds

In complex geometry, compact Kähler manifolds are natural generalizations of projective varieties from the point of view of the Hodge theory and the deformation theory. Given a compact Kähler manifold X, the so-called Kodaira problem asks whether X has an (arbitrarily small) deformation to a projective variety. For surfaces such a deformation always exists (Kodaira, Buchdahl), but in each dimension greater than 4 there are compact Kähler manifolds which do not even have the homotopy type of a projective variety (Voisin). For threefolds the Kodaira problem remains open and will be the main focus of the talk. We will first present a way to understand the geometry of compact Kähler threefolds through the minimal model program and then discuss recent progress on the existence of algebraic approximations of these varieties.