Hodge-Theoretical Invariants of Singularities

该提案包括两部分。两者都集中于霍奇结构的变异理论和解析图的奇异胚芽。第一部分是主要研究者和J. Steenbrink工作的延续。他们计算了与曲线奇点相关的光谱对，其系数与Hodge结构的变化具有阿贝尔单群。PI计划将这个结果推广到任意维度和更一般的变化（例如，具有可解单群的变化）。作为第一步，PI计划调查这种变化的极限混合霍奇结构的存在性。在第二节中，PI研究了一个孤立的完全相交奇点的变形，其判别式是一个法交的除数。他在消失上同调上构造了一个极限混合Hodge结构，并证明了Cattani、Kaplan和Schmid的几个结果的局部类似，这些结果对应于幂零轨道定理的“代数表征”的局部类似。PI还提出寻找经典（全局）幂零轨道定理的几何局部类似。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，但在过去的三十年里，它有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。奇点理论研究的是代数几何中图形的奇异点。即使是理论的基本结果在物理学、动力系统理论、混沌和突变理论，甚至在心理学中也有惊人的应用。

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.