Mathematics

I. Introduction II. Construction of a completion of the image of a period mapping III. Curvature properties of the extended Hodge line bundle (A) IV. Curvature properties of the extended Hodge line bundle (B) V. Proof that the extended Hodge line bundle is ample VI. Curvature properties of the Hodge vector bundle

This work initiates a global study of period mappings at infinity, that is motivated by challenges arising when considering the questions: (i) What are natural completions of a period mapping? (ii) What geometric applications do they have?

We give an informal, expository account of a project to construct completions of period maps.

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.

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

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 VIII. Period mappings and the canonical bundle References

Talk at the "Geometry at the Frontier" conference, Pucón, Chile (November 12, 2018), and based in part on joint work in progress with Mark Green, Radu Laza and Colleen Robles.

We give conditions under which natural lines bundles associated with completions of period mappings are semi-ample and ample.

The use of the Hodge line bundle, especially its positivity properties, in algebraic geometry is classical. The Hodge line bundle only sees the associated graded to the limiting mixed Hodge structures (LMHS's) that arise from the singular members in a family of algebraic varieties. This talk will introduce a new type of geometric object associated to the extension data in LMHS's (and not present for general MHS's) whose positivity properties play a central role in the application of Hodge theory to moduli. Their construction will be discussed and illustrated.

It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend to have positivity properties. From these considerations several issues arise: (i) In general for bundles that are semi-positive but not strictly positive; what further natural conditions lead to the existence of sections of its symmetric powers? (ii) In Hodge theory the Hodge metrics generally have singularities; what can be said about these and their curvatures, Chern forms etc.? (iii) What are some algebro-geometric applications of positivity of Hodge bundles? The purpose of these partly expository notes is fourfold. One is to summarize some of the general measures and types of positivity that have arisen in the literature. A second is to introduce and give some applications of norm positivity. This is a concept that implies the di_erent notions of metric semi-positivity that are present in many of the standard examples and one that has an algebro-geometric interpretation in these examples. A third purpose is to discuss and compare some of the types of metric singularities that arise in algebraic geometry and in Hodge theory. Finally we shall present some applications of the theory from both the classical and recent literature.

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.

We shall show that a smooth, quasi-projective variety X has a holomorphically convex universal covering eX when (i) 1(X) is residually nilpotent and (ii) there is an admissable variation of mixed Hodge structure over X whose monodromy representation has a nite kernel, and where in each case a corresponding period mapping is assumed to be proper.

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.