Plurisubharmonic Functions on Algebraic Varieties

代数簇上的多次调和函数

• 批准号: 0070725
• 负责人: B. Alan Taylor
• 金额: $ 9.79万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070725&HistoricalAwards=false
• 关键词: Plurisubharmonic; Functions; Algebraic; Varieties

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.

摘要：经典 Phragmen-Lindelof 定理将极大值原理扩展到无界解析函数，它表明，满足上半平面内渐近指数界和实轴上一致指数界的解析函数实际上满足上半平面内一致指数界。过去三十年的研究表明，n维复欧几里德空间中代数簇的解析函数的类似特征估计的有效性实际上等价于线性常系数偏微分算子的某些性质。 算子的一些这样的性质是实解析函数或Gevrey类空间上的满射性、基本解中的空白的存在性、线性解算子的存在性以及齐次方程的解的连续性质。 虽然这些估计有不同的集合与算子的不同属性相关，但它们在精神上都是相似的。 这项工作的目的是开发一些方法，对满足给定 Phragmen-Lindelof 条件的代数簇进行几何表征。 如果成功，这项工作还应该能够深入了解有关偏微分方程的问题，例如带有锥形缺口的基本解的存在性。这项工作的重点是开发复杂分析工具，可用于回答有关线性偏微分方程的基本问题。 20 世纪 50 年代，Laurent Schwartz 提出了一般线性偏微分方程的基本问题。 它们总是可以解决的吗？ 如果是这样，解决方案是否可以像问题中的数据一样平滑地选择？ 是否存在根本性的解决方案？ 能否用“公式”来求解方程，从而使答案线性依赖于问题的数据？ 大多数这些问题在 20 世纪 50 年代由 Ehrenpreis 和 Malgrange 得到了回答。 然而，当数据为实解析时是否可以选择解为实解析的问题直到 1960 年代末给出第一个反例时才被提出。 1973 年，Hormander 根据给出微分方程的多项式零集上的解析函数的某些不等式的有效性，给出了具有此性质的方程的表征。 1990年，泰勒、梅斯和沃格特回答了关于解的“公式”是否存在的问题，并表明它们也具有一些类似的不等式的特征。 该项目的目的是开发工具，使人们能够确定所需的估计对于给定的偏微分方程是否有效。我们相信，可以开发一种算法来进行验证，并进一步解释满足不等式所必需的相关多项式的零集的几何形状。