Positivity of vector bundles and Hodge theory

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.