Informal Friday Seminar: Cliff -- On the Kirwan map for moduli stacks of Higgs bundles

To a reductive group G and a smooth projective complex curve C, one associates the
moduli stack of G-Higgs bundles on C.  There is a natural map from the cohomology of the
moduli stack of G-Higgs bundles to the cohomology of the moduli stack of semistable
G-Higgs bundles, known as the Kirwan map.  In the case of G=GL(n), Markman proved that
the Kirwan map is surjective for the component of Higgs bundles of degree d coprime to
n.  By contrast, I explain recent work in which we show (generalizing Hitchin for SL(n))
that the Kirwan map for moduli of G-Higgs bundles fails to be surjective whenever G has
disconnected centre.  This is accomplished via the study of a group action on the moduli
stack and the induced action on its cohomology.  This talk is based on joint work with
Thomas Nevins and Shiyu Shen.  

I will not assume the audience has background knowledge on stacks or Higgs bundles.