Curious Hard LefschetzMPIM
by
MPIM, Vivatsgasse, 7 - Seminar Room
Max Planck Institute for Mathematics
The Unterseminar
The cohomology of a smooth complex projective variety $X$ satisfies Poincaré duality, which can be reformulated by saying that the Poincaré polynomial of $X$ is palindromic. A stronger statement, called Hard Lefschetz theorem, is that $H^*(X)$ is naturally an $sl2$-module.
In 2006, Hausel and Rodriguez-Villegas computed the number of $F_q$-points for some character varieties (which are not projective), and observed that the resulting polynomials are palindromic. They suggested that this symmetry is also induced by an $sl2$-action, and dubbed this curious Hard Lefschetz conjecture. It is by now a theorem, proved in a couple of ways. I will start by going through some very explicit examples, then explain some of the ideas that go into the proofs, and if time permits gesture at what this theorem might be telling us.