Mathematics

To Herb Clemens. Lefschetz wrote that he put the harpoon of topology into the whale of algebraic geometry. Herb but the harpoon of topology into the whale of Hodge theory. Period mappings, or equivalently variations of Hodge structure, have been used both to study families of algebraic varieties and as a subject in its own right.

This lecture will discuss the global structure of period mappings (variation of Hodge structure) defined over complete, 2-dimensional algebraic varieties. Some applications to moduli of general type algebraic surfaces will also be presented.

Hodge theory provides a major tool for the study of moduli. Conversely, moduli have furnished a significant stimulus for the developement of Hodge theory.

The theory of moduli is an important and active area in algebraic geometry. For varieties of general type the existence of a moduli space $\mathcal{M}$ with a canonical completion $\bar{\mathcal{M}}$ has been proved by Koll\'ar/Shepard-Barron/Alexeev. Aside from the classical case of algebraic curves, very little is known about the structure of $\mathcal{M}$, especially it's boundary $\overline{\mathcal{M}} \diagdown\mathcal{M}$. The period mapping from Hodge theory provides a tool for studying these issues. In this talk, we will discuss some aspects of this topic with emphasis on I-surfaces, which provide one of the first examples where the theory has been worked out in some detail. Particular notice will me made of how the extension data in the limiting mixed Hodge structures that arise from singular surfaces on the boundary of moduli may be used to guide the desingularization of that boundary.

Almost all of the deep results in Hodge theory and its applications to algebraic geometry require understanding the limits in a family of Hodge structures. In the literature the proofs of these results frequently use the consequences of the analysis of the singularities acquired in a degenerating family of Hodge structures; that analysis itself is treated as a "black box." In these lectures an attempt will be made to give an informal introduction to the subject of limits of Hodge structures and to explain some of the essential ideas of the proofs. One additional topic not yet in the literature that we will discuss is the geometric interpretation of the extension data in limiting mixed Hodge structures and its use in moduli questions.

From S. S. Chern we learned the importance of curvature in geometry and its special features in the complex case. In this case there are significant geometric and analytic consequences of the curvature having a sign. Both positive and negative curvature have major implications in algebraic geometry and in holomorphic mappings between complex manifolds. The vector bundles (Hodge bundles) and complex manifolds (period domains) that arise in Hodge theory have natural metrics and subsequent curvatures that through the work of very many people over an extended period of time have played a central role in the study of Hodge theory as a subject in its own right and in the applications of Hodge theory to algebraic geometry. Of particular importance are (i) the sign properties of the curvature (positivity of the Hodge bundles and cotangent bundles of period domains); (ii) the result that in the geometric case the non-degeneracy of curvature forms is an algebro-geometric property; (iii) the singularity properties of the curvature, especially that of the Chern forms. Regarding (iii) we note that the essential geometric fact that enables one to control the singularities is a curvature property of the bundles that arise in Hodge theory. The primary purpose of this mainly expository paper is to present some (but not by any means all) of the fundamental concepts and to discuss a few of the basic results in this very active and now vast area of research.

Hodge theory is a subject of both interest in it's own right and as a tool for studying geometric questions. This talk will be mainly concerned with some of the history and uses of Hodge theory: how did it originate and what are some of the ways it helps to address problems in geometry, especially algebraic geometry? Our discussion of both the early history and uses of Hodge theory will be selective, not comprehensive.

The classifcation of algebraic varieties is a central part of algebraic geometry. Among the common tools that are employed are birational geometry, especially using the analysis of singularities, and Hodge theory. The latter is frequently used as a sort of "black box": the cohomology of an algebraic variety has a functorial mixed Hodge structure and the formal properties of the corresponding category have very strong consequences. On the other hand, geometric structures arise naturally in Hodge theory and one aspect of this will be the focus of these lectures. This aspect originates from the fact that mixed Hodge structures have extension data expressed by linear algebra.

The motivation behind this work is to construct a "Hodge theoretically maximal" completion of a period map. This is done up to finite data (we work with the Stein factorization of the period map). The image of the extension is a Moishezon variety that compactifies a finite cover of the image of the period map.