Projective ranks of Hermitian symmetric spaces

Abstract

For many mathematicians, the lure of mathematics began with a gradual recognition of the beauty of Euclidean geometry. It attracts us with its simplicity and clarity. Later, with the discovery of Cartesian coordinates, we see the power and precision of algebra in describfng geometric ideas. Such seeds having been sown, I have found myself deeply enthralled for many years now with algebraic geometry. Recently, however, a yearning for a return to a more concrete geometry overcame me. It was a desire to understand the "geometry" in algebraic geometry which has led to a study of those classical examples which seem to be well understood by the old masters. With a background in algebraic groups, I found the symmetric spaces of Cartan to be a good starting point. Currently, the "geometry" in algebraic geometry is represented most effectively by the theory of complex manifolds, many of the most popular of which are projective algebraic varieties. (For one who thought theorems should be proved in a characteristic-free environment, this represents a significant shift in perspective.) The compact Hermitian symmetric spaces can be represented as homogeneous spaces for reductive algebraic groups (so are objects of study in algebraic geometry) and also as homogeneous spaces for compact Lie groups (so are objects of study in Riemannian geometry). Among these spaces are the complex projective spaces and the Grassmann manifolds which play a fundamental role in the study of algebraic manifolds and the geometry on them. This article is concerned with recent characterizations of the Hermitian symmetric spaces using the curvature of the natural K~ihler metrics on each. In particular, the only ones with everywhere positive biholmorphic sectional curvature are the complex projective spaces. A motivating problem for what follows is: Find a similar characterization of Grassmann manifolds in terms of their curvature.