Local Geometry of Real and Complex Analytic Mappings

所提出的研究方案涉及实和复解析映射及其像的局部几何。它处于解析几何、交换代数、奇点理论和多元复变函数论的交界处，涉及到各个方向的问题。该计划建立在我目前的NSERC发现基金下所做的非常成功的工作之上。 该计划的两个主要目标是： (A)建立解析映射和多项式映射奇点的可计算分类。 (B)奇异CR几何的发展。 上述长期目标中的第一个是从奇点理论的角度研究映射的局部几何。 我们的主要思想是将对映射几何复杂性的研究归结为对某些代数不变量的自动计算。为了实现这一目标，我们需要开发新的标准，根据与地图相关的某些模块的代数性质来表征地图的几何形状，这可以通过计算机代数方法进行验证。这些标准的例子是我们最近对地图的开放性和平坦性的刻画。我们的方法是研究给定映射的纤维族中的简并(或不连续)。这些通常太微妙了，在代数水平上无法检测到，所以我们需要放大这些不连续，以使它们反映在与地图相关的某些模块的代数属性中。例如，这可以通过传递给地图的纤维力量来实现。 对映射奇点进行分类的局部不变量研究是一个成熟而活跃的研究领域。我们方法的新颖性在于它的有效性，即强调可计算性。我们关于平坦性的有限确定性和其他局部性质的最新结果证明，即使在超越(即非多项式)的情况下，该方法也可能是成功的。 我们的第二个长期目标是将局部解析几何应用于研究复杂环境空间中的真实结构。在非奇异情况下，这一研究领域被称为CR几何，它可以被视为经典的多复变函数分析的一个分支。在这个方案中，我们考虑复空间中的奇异实解析(甚至半解析)对象。这种方法的重要性在于这样一个事实，即在复杂的分析考虑中(例如，作为复数域的边界)，这种奇异集自然地出现。 当然，奇异解析集本身不是CR流形。然而，正如我们最近所展示的那样，他们承认到CR流形的分层具有一些非常好的微分和代数几何性质。这一发现为奇异背景下CR几何的发展奠定了坚实的基础，并允许我们使用半代数和半解析几何的方法。

The proposed research program is concerned with the local geometry of real and complex analytic mappings and their images. It lies at the interface of analytic geometry, commutative algebra, singularity theory and theory of functions of several complex variables, and involves problems in all these directions. The program builds on very successful work done under my present NSERC discovery grant. The two main objectives of the program are: (a) Establishing of a computable classification of singularities of analytic and polynomial mappings. (b) Development of a singular CR geometry. The first of the above long term goals is concerned with the local geometry of mappings from the point of view of singularity theory. Our main idea is to reduce the study of the geometric complexity of a map to an automated calculation of certain algebraic invariants. To achieve this goal, we need to develop new criteria that would characterize the geometry of a map in terms of algebraic properties of certain modules associated with the map, which can be verified by means of computer algebra methods. Examples of such criteria are our recent characterizations of openness and flatness of maps. Our approach is to study degeneracies (or discontinuities) in the family of fibres of a given map. These are often too subtle to be detected on the algebraic level, and so we need to amplify these discontinuities to the extent that they get reflected in algebraic properties of certain modules associated with the map. This can be done, for example, by passing to fibred powers of the map. The study of local invariants classifying the mapping singularities is a well established and active area of research. The novelty of our approach lies in its effectiveness, that is, the emphasis on computability. Our most recent results on finite determinacy of flatness and other local properties prove that this approach may be successful even in the transcendental (i.e., non-polynomial) case. Our second long term goal concerns the application of local analytic geometry to the study of real structures in complex ambient spaces. In the non-singular setting, this area of study is known as CR geometry, which can be viewed as a branch of the classical analysis of functions in several complex variables. In this proposal, we consider singular real analytic (even semianalytic) objects in complex spaces. The importance of this approach lies in the fact that such singular sets appear naturally in complex analytic considerations (e.g., as boundaries of complex domains). Singular analytic sets, of course, are not CR manifolds themselves. However, as we showed recently, they admit a stratification into CR manifolds enjoying some very nice differential and algebro-geometric properties. This discovery forms a firm basis for the development of CR geometry in the singular context, and allows us to use the methods of semialgebraic and semianalytic geometry.