Mathematical Sciences: Hodge Theory and L2-Cohomology

这项研究是在局部对称空间和代数几何的超越方法领域。更具体地说，它将集中于关于簇的交同调的霍奇理论。这些结构应该给出更简单的描述，并且它们应该与L2调和形式相对于好的Kahler度量提供的Hodge结构相关。沿着这些主题的一个相关发展是首席研究者1980年猜想的两个证明的出现，该猜想将局部对称簇的通常的L2上同调等同于其Baly-Borel-Satake紧化的交同调。这个问题将得到更多的工作，因为结果的性质将通过将其推广到其自然环境来阐明，这可能是仅具有等阶边界分量的Satake紧凑化。这项工作的本质是研究多项式方程的解集的几何目的。研究了各种不变量，以加深对这些几何对象的理解。反过来，这些课程应该会应用于数学的许多领域，尤其是在几何和分析方面。

This research is in the area of locally symmetric spaces and the transcendental methods of algebraic geometry. More specifically it will concentrate on the Hodge theory for the intersection homology of varieties. These structures should be given simpler descriptions and they should be related to the Hodge structures provided by L2 harmonic forms with respect to good Kahler metrics. A related development along these themes is the appearance of two proofs of the principal investigator's 1980 conjecture equating the usual L2-cohomology of a locally symmetric variety with the intersection homology of its Baily- Borel-Satake compactification. This problem will receive more work for the nature of the result will be elucidated by generalizing it to its natural setting, which is probably to Satake compactifications with only equal-rank boundary components. The nature of this work is in the geometrical end of studying the sets of solutions of polynomial equations. Various invariants are studied which give deep understanding of these geometric objects. These in turn should give applications to many areas of mathematics, most especially in geometry and analysis.