On the relative cohomology of the Hitchin fibration

This is joint work with Pierre-Henri Chaudouard.

The main tool in Ngô Bao Châu's proof of the Langlands-Shelstad fundamental lemma, is a theorem on the support of the relative cohomology of the elliptic part of the Hitchin fibration.

For $\mathrm{GL}(n)$ and a divisor of degree $>2g-2$, the theorem says that the relative cohomology is completely determined by its restriction to any dense open subset of the base of the Hitchin fibration.

In the talk I would like to present in this particular case, our extension of that theorem to the whole Hitchin fibration, including the global nilpotent cone.