Hodge-Theoretical Invariants of Singularities

该提案包括两个部分。两者都集中在Hodge结构的变分理论和解析映射的奇异芽理论上。第一部分是首席调查员和J.Steenbrink工作的继续。他们计算了与曲线奇点有关的谱对，这些奇点的系数在Hodge结构的变化中带有一个阿贝尔单调群。PI计划将这一结果推广到任意维度和更一般的变分(例如，具有可解的单元群的变分)。作为第一步，PI计划调查这种变化的极限混合Hodge结构的存在。在第二节中，PI研究了一个孤立的完全交点的变形，该奇点的判别式是与正规交点的因子。他在零上同调上构造了一个极限混合Hodge结构，并证明了Cattani，Kaplan和Schmid几个结果的局部相似，这些结果对应于幂零轨道定理“代数刻画”的局部相似.PI还建议寻找经典的(全局)幂零轨道定理的几何局部模拟。本研究属于代数几何和奇点理论领域。代数几何是现代数学中最古老的部分之一，但它在过去的三十年里取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，相反，正在这些领域以及物理、理论计算机科学和机器人中找到应用。奇点理论研究用代数几何研究的图形的奇异点。甚至该理论的基本结果在物理学、动力系统理论、混沌和灾变理论，甚至在心理学中都有令人惊讶的应用。

The proposal contains two sections. Both are concentrated on the theory of variations of Hodge structures and singular germs of analytic maps. The first section is a continuation of the work of the principal investigator and J. Steenbrink. They computed the spectral pairs associated with a curve singularity with coefficients in a variation of Hodge structure with an abelian monodromy group. The PI plans to generalize this result for arbitrary dimension and for more general variations (e.g. variations with solvable monodromy group). As a first step, the PI plans to investigate the existence of the limit mixed Hodge structure for such a variations. In the second section, the PI studies a deformation of an isolated complete intersection singularity whose discriminant is a divisor with normal crossings. He constructs a limit mixed Hodge structure on the vanishing cohomology and proves the local analogs of several results of Cattani, Kaplan and Schmid, which correspond to the local analog of the "algebraic characterization" of the Nilpotent Orbit Theorem. The PI proposes to find also the geometric local analog of the classical (global) Nilpotent Orbit Theorem. This research is in the field of algebraic geometry and singularity theory. Algebraic geometry is one of the oldest parts of the modern mathematics, but one which has had a revolutionary flowering in the last thirty years. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics. Singularity theory studies the singular (exotic) points of the figures studied by the algebraic geometry. Even elementary results of the theory have surprising applications in physics, in the theory of dynamic systems, in chaos and catastrophe theory, and even in psychology.