Hodge Theory of Nondegenerate Minimal Toric Hypersurfaces

Abstract

In this thesis we study toric hypersurfaces in the context of higher-dimensional algebraic geometry. The topics are quite complicated but restricting to generic situations and almost smooth birational models (minimal models), we are able to get good results. We ask how to calculate invariants like the Plurigenera or the Hodge numbers of toric hypersurfaces. Deforming such hypersurfaces within the surrounding toric variety we study a Kodaira-Spencer map, parameterizing infinitesimal deformations one-to-one and the infinitesimal Torelli theorem, bridging deformation theory and Hodge theory, both by very explicit formulas, though for this part we restrict to surfaces in toric 3-folds.