Cycles and the Cohomology of Locally Symmetric Spaces

本研究计画系关于算术流形之霍奇型定理之证明。“霍奇猜想是数学中最重要的未解决问题之一-它是克莱数学研究所的七个千年问题之一，其中每个问题的解决方案都提供了100万美元的奖金。霍奇猜想只在特殊情况下得到了验证。PI和合作者证明了霍奇猜想的一个基本的新的无限类的例子。这项工作导致制定和证明一个新的改进版本的霍奇猜想在进一步的特殊情况下，一个大的和重要的家庭空间。显然，更一般的霍奇定理应该适用于几何学和群论共同领域中发生的大多数情况。这是这个项目的目标，制定和证明这样的改进版本的霍奇猜想。调查员将继续工作之间的关系循环算术代数的对称空间相关联的正交群和酉群和上同调类在这些空间构造使用韦尔表示。 研究人员和合作者研究了稳定迹公式的使用，以证明在低次情况下，一类特殊的余维k全测地子流形（称为“特殊循环”）的庞加莱对偶（与欧拉/陈类一起）跨越了上述空间的第k上同调的典型被加数，称为特殊（精炼）霍奇被加数。在幺正情形下，对于复双曲空间，他们证明了复单位球的标准算术乘积在远离中维的度数上的霍奇和泰特定理。该项目的目标是制定和证明霍奇猜想的改进版本，适用于与酉群，正交群和可能的辛群相关的所有局部对称空间（不一定是埃尔米特对称）。

This research project is concerned with proving "Hodge type theorems for arithmetic manifolds." The Hodge conjecture is one of the most important unsolved problems in mathematics -- it is one of the seven Millenium Problems of the Clay Mathematical Institute, which offers a one million dollar prize for the solution of each of these problems. The Hodge conjecture has been verified only in special cases. The PI and collaborators proved the Hodge conjecture for a fundamental new infinite class of examples. This work led to the formulation and proof of a new refined version of the Hodge conjecture in further special cases for a large and important family of spaces. It is clear that still more general Hodge conjectures should hold for most of the cases that occur in the area common to geometry and group theory. It is the goal of this project to formulate and prove such refined versions of the Hodge conjecture. The investigator will continue work on the relation between cycles in arithmetic quotients of the symmetric spaces associated to the orthogonal groups and unitary groups and cohomology classes on these spaces constructed using the Weil representation. The investigator and collaborators have studied the use of a stabilized trace formula to prove that in low degrees the Poincare duals of a special class of totally-geodesic submanifolds of codimension k called "special cycles" together with the Euler/Chern class) span a canonical summand of the k-th cohomology of the above spaces called the special (refined) Hodge summand. In the unitary case, for complex hyperbolic space, they obtained proofs of the Hodge and Tate conjectures for the standard arithmetic quotients of the complex unit ball in degrees away from the middle dimensions. The goal of the this project is to formulate and prove refined versions of the Hodge conjecture for all locally symmetric spaces (not necessarily Hermitian symmetric) associated to unitary groups, orthogonal groups, and possibly symplectic groups.