Plurisubharmonic Functions on Algebraic Varieties

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ABSTRACT: The classical Phragmen-Lindelof theorem extends the maximum principleto unbounded analytic functions by showing that an analytic function thatsatisfies an asymptotic exponential bound in the upper half plane and auniform bound on the real axis in fact satisfies a uniform exponentialbound in the upper half plane. Research of the past three decades hasshown that the validity of estimates of a similar character for analyticfunctions on algebraic varieties in n-dimensional complex Euclidean spaceare in fact equivalent to certain properties of linear constantcoefficient partial differential operators. Some such properties of theoperators are surjectivity on the space of real analytic functions orGevrey classes, the existence of lacuna in fundamental solutions, theexistence of linear solution operators, and continuation properties ofsolutions of the homogeneous equation(s). While there are different setsof these estimates associated to the different properties of the operator,they all are similar in spirit. The aim of this work is to developmethods that give a geometric characterization of the algebraic varietiesfor which a given Phragmen-Lindelof condition is satisfied. Ifsuccessful, the work should also give insight into questions about thepartial differential equations such as the existence of fundamentalsolutions with cone-shaped lacuna. This work is focused on developing tools in complex analysis thatcan be used to answer basic questions about linear partial differentialequations. In the 1950's, Laurent Schwartz formulated such fundamentalproblems for general linear partial differential equations. Are theyalways solvable? If so, can the solution be chosen as smooth as the datain the problem? Do fundamental solutions exist? Can the equations besolved with a "formula", so that the answer depends linearly on the dataof the problem? Most of these questions were answered in the 1950's byEhrenpreis and Malgrange. However, the question of whether the solutioncould be chosen to be real analytic when the data is real analytic wasopen until the late 1960's when the first counter examples were given. In1973, Hormander gave a characterization of the equations with thisproperty in terms of the validity of certain inequalities for analyticfunctions on the zero set of the polynomial giving the differentialequation. In 1990, Taylor, Meise, and Vogt answered the question aboutthe existence of "formulas" for the solution and showed they were alsocharacterized in terms of some similar inequalities. The aim of thisproject is to develop tools that allow one to decide whether or not therequired estimates are valid for a given partial differential equation. Webelieve that it is possible to develop an algorithm that will make theverification, and further, will explain the geometry of the zero set ofthe associated polynomial that is necessary for the inequalities to besatisfied.