Mathematics

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.

Outline I. Introduction II. Extension data for a mixed Hodge structure III. Extension data for limiting mixed Hodge structures IV. Period mappings to extension data (A) V. Period mappings to extension data (B) Appendix to Sections IV and V: Examples VI. Local Torelli conditions VII. Global structure of period mappings from complete surfaces

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.

Hodge theory provides a basic invariant of complex algebraic varieties. For algebraic families of smooth varieties the global study of the Hodge structure on the cohomology of the varieties (period mapping) is a much studied and rich subject. When one completes a family to include singular varieties the local study of how the Hodge structures degenerate to limiting mixed Hodge structures is also much studied and very rich. However, the global study of the period mapping at infinity has not been similarly developed. This has now been at least partially done and will be the topic of this talk. Sample applications include: new global invariants of limiting mixed Hodge structures; a generic local Torelli assumption implies that moduli spaces are log canonical (not just log general type); and extension data and asymptotics of the Ricci curvature; a proposed construction of the toroidal compactification of the image of period mapping. The key point is that the extension data associated to a limiting mixed Hodge structure has a rich geometric structure and this provides a new tool for the study of families of singular varieties in the boundary of families of smooth varieties.