Function Theory on Varieties

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract: The main goal of the project is to find the conditions on an algebraic variety that guarantee good estimates for analytic functions on the variety in terms of their values at real points. Particularly, it is concerned with estimates analogous to those of the Phragmen-Lindelof theorem which gives uniform upper bounds for pluri-subharmonic functions that are bounded above at the real points of the complex Euclidean space and satisfy an asymptotic linear growth rate. A new geometric condition, that the variety is nearly hyperbolic, is proposed to characterize such properties. The new condition gives a precise sense in which the variety has many real points, and is intermediate in strength between having a full dimensional set of real points and admitting a projection map with real fibers over real points. The project aims to determine the geometric properties of algebraic and analytic varieties in Euclidean space that determine when the analytic functions on the variety have properties similar to those of entire functions on Euclidean space. It is motivated by connections with the properties of solution operators for systems of constant coefficient partial differential operators and convolution operators on global smooth functions. The goal is to give algorithms for deciding whether or not specific operators admit solutions and solution operators on spaces of infinitely differentiable functions and distributions. The study uses methods from and contributes new results to pluri-potential theory, the area whose relationship to analytic functions of several complex variables is the analogue of the relationship of classical potential theory to the theory of analytic functions of one complex variable.