Dror Varolin - Minicourse - Lecture 3
Channel: UMD Mathematics
Date: 2017-09-14

Dror Varolin
Variations of Holomorphic Hilbert spaces

Traditional complex analysis focuses on a single space, like a domain in Euclidean space, or more generally a complex manifold, and studies holomorphic maps on that space, into some target space. The typical target space for a domain is the complex plane, but for complex manifolds there is sometimes a need to consider a different target space. The easiest target spaces are line bundles, and then one considers sections rather than functions; the case of the trivial bundle corresponds to studying the graphs of holomorphic functions.

Often the set of all holomorphic sections of a line bundle is simply too vast, so one considers natural subsets of sections. A fruitful approach is to study those sections that are square-integrable with respect to some Hermitian inner product structure and a volume form. The focus on Hilbert spaces was introduced by S. Bergman in the early 20th century, and came into maturity via PDE methods through the works of many distinguished mathematicians including Bochner, Kodaira, Spencer, Morrey, Kohn, H\ormander, Andreotti, Siu, Yau, Bombieri, Demailly and their many students and colleagues since.

With the vast knowledge about the case of a single manifold and Hilbert space in hand, people have begun to examine how these function spaces vary when certain parameters are changed. One could change the complex
structure of the underlying manifold (as was done for compact Riemann surfaces by Teichmuller, and more thoroughly and Ahlfors and his school), and one could also change the Hermitian structure used to define
the Hilbert spaces. Algebraic geometers have been exploring such deformations for a very long time, via the Hodge Theorem and its variants, to great effect, especially under certain strong curvature assumptions. However, as one relaxes these curvature assumptions somewhat, the black box given by the Hodge Theorem cannot be used any longer, and one has to look under the hood.

Perhaps the crowning achievement of these more refined Hilbert spaces methods is the L^2 extension theorem of Ohsawa-Takegoshi; a theorem that gives sufficient, almost necessary conditions for extending weighted square-integrable holomorphic functions from a submanifold. The theorem opened the door for arguments that use induction on dimension, and a major advance was Siu's deformation invariance of plurigenera, which eventually lead to a fundamental breakthrough in the minimal model program of birational geometry.

In 2009, Bo Berndtsson published a paper in the Annals of Mathematics, in which he established two theorems that measure, in a very precise way, the variation of Hilbert spaces of holomorphic sections of line bundles on a complete Kahler manifold. His first theorem, which is stated on the poster for this Atelier, deals with the case of pseudoconvex domains, while his second theorem deals with the case of compact Kahler manifolds. The two theorems have already seen an incredible number of applications, but surely this is just the beginning.

In this segment of the Maryland Analysis and Geometry Atelier, whose acronym---MAGA---is the Hebrew word for `contact', we will make full contact with Berndtsson's first theorem, and some contact with his second theorem. The first lecture will focus on the basics of Hermitian holomorphic geometry. The remaining lectures will focus on Berndtsson's Theorems and a couple of my favorite applications. The most striking application is a new and, in my opinion more conceptual, proof of the Ohsawa-Takegoshi Extension Theorem, due to Berndtsson and Lempert. I will explain the idea of this proof, and also show how the extension theorem is, at least morally if not literally, equivalent to Berndtsson's
Theorems.