Topology of Algebraic Varieties

A decade ago, John Morgan and I classified all integral weight 3 VHS of Hodge type (1,1,1,1) which can underly a family of Calabi-Yau three folds over the thrice-punctured sphere, subject to conditions on monodromy coming from mirror symmetry. There are 14 generalized hypergeometric $_4F_3$ differential equations of this type. Each can be realized as the Picard-Fuchs differential equation of a family of Calabi-Yau threefolds constructed as hypersurfaces or complete intersections in a Gorenstein toric Fano variety. The 14th case was constructed only recently as a family of singular Calabi-Yau threefolds admitting no projective crepant resolution. Nevertheless, each of these Calabi-Yau three folds admits fibrations by K3 surfaces of high Picard rank, and this fibration structure proved the key to understanding the geometry of the 14th case family. In this talk, we will discuss new results motivated by these Calabi-Yau fibrations. Firstly, in joint work with Andreas Malmendier motivated by the uniform construction of Calabi-Yau threefolds as K3 surface fibrations in the 14 hypergeometric cases, we develop an iterative procedure to construct both periods and Picard-Fuchs differential equations of threefold families starting from extremal rational elliptic surfaces. In particular, this construction gives families of elliptic K3 surface fibered Calabi-Yau threefolds realizing all 60 of the sympletically rigid Calabi-Yau type equations classified by Boegner and Reiter. Secondly, in joint work with Andrew Harder, Andrey Novoseltsev, and Alan Thompson, we fix a lattice-polarized K3 surface type and construct the most general class of Calabi-Yau threefolds admitting K3 fibers of this type, give criteria for smoothness, and compute the Hodge numbers in terms of the geometry of the fibration. We also describe the mirror construction involving smoothings of Fano threefolds intersecting in anticanonical K3 surfaces, and explain the precise relationship between the Landau-Ginzburg models of the Fano threefolds and the K3 fibered Calabi-Yau threefolds mirror dual to these smoothings. This last answers a question of Andrei Tyurin.