Cycles and the Cohomology of Locally Symmetric Spaces

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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.