Function Theory on Varieties

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Abstract: The main goal of the project is to find the conditions on analgebraic variety that guarantee good estimates for analytic functionson the variety in terms of their values at real points. Particularly,it is concerned with estimates analogous to those of thePhragmen-Lindelof theorem which gives uniform upper bounds forpluri-subharmonic functions that are bounded above at the real pointsof the complex Euclidean space and satisfy an asymptotic linear growthrate. A new geometric condition, that the variety is nearlyhyperbolic, is proposed to characterize such properties. The newcondition gives a precise sense in which the variety has many realpoints, and is intermediate in strength between having a fulldimensional set of real points and admitting a projection map withreal fibers over real points. The project aims to determine the geometric properties ofalgebraic and analytic varieties in Euclidean space that determinewhen the analytic functions on the variety have properties similar tothose of entire functions on Euclidean space. It is motivated byconnections with the properties of solution operators for systems ofconstant coefficient partial differential operators and convolutionoperators on global smooth functions. The goal is to give algorithmsfor deciding whether or not specific operators admit solutions andsolution operators on spaces of infinitely differentiable functionsand distributions. The study uses methods from and contributes newresults to pluri-potential theory, the area whose relationship toanalytic functions of several complex variables is the analogue of therelationship of classical potential theory to the theory of analyticfunctions of one complex variable.