The cohomology rings of moduli stacks of principal bundles over curves (Q612848)

In their paper [Phil.\ Trans.\ R.\ Soc.\ Lond.\ A 308, 523--615 (1983; Zbl 0509.14014)], \textit{M. F. Atiyah} and \textit{R. Bott} computed the equivariant cohomology of the moduli space of principal \(G\)-bundles on a Riemann surface \(C\) for any semi-simple complex Lie group \(G\). Their results can be reinterpreted to show that the cohomology ring of the moduli stack of principal \(G\)-bundles on \(C\) is generated by the Künneth-components of the characteristic classes of the universal bundle. In the paper under review, the authors extend this result to the case of smooth projective curves over a field of arbitrary characteristic. The main new ingredient in their proof is a purity result for the cohomology of the moduli stack. To obtain it, they need to show the existence of a projective coarse moduli space for flagged principal bundles in arbitrary characteristic. A flagged principal bundle is a principal bundle with a kind of parabolic structure (a reduction of the structure group to a parabolic subgroup at a finite set of points). The GIT construction of this moduli space occupies the second half of the paper and essentially follows the proof for principal \(G\)-bundles, see e.g.\ \textit{T. L. Gómez, A. Langer, A.H.W. Schmitt, I. Sols} [Adv.\ Math.\ 219, No. 4, 1177--1245 (2008; Zbl 1163.14008)].