Rational Points, Cohomology and Symmetries of Toric Varieties

by Professor G.I. Lehrer

Institution: University of Sydney
Date: Thu 16th August 2007
Time: 3:15 PM
Location: Elisabeth Murdoch Building, Theatre A, Uni of Melbourne

Abstract: Given a finite group G which acts on an algebraic variety X over a number field, the associated action of G on the de Rham cohomology of X may be studied via the rational points of X over finite fields. The tools linking the subjects involve eigenvalues of Frobenius and the Hodge filtration. I shall discuss some general theorems in this vein (joint work with Mark Kisin). As an application, I shall derive a simple formula for the action of a finite Weyl group on the cohomology of its associated complex toric variety. This formula in turn has several applications, such as the theorem that the reflection representation has multiplicity which depends only on the rank.