Chern classes for the moduli space of bundles on a curve -- Chris Woodward

The moduli space of semistable G-bundles, G semisimple, on a smooth projective curve is a normal projective variety, and in good cases smooth. What can one say about its Chern classes, in particular, which ones are non-vanishing? Other than the first (doesn't vanish) and last Chern class (does vanish), very little was known for arbitrary G. I will describe a vanishing result for top Chern classes c_j, j \in ((dim(G)-rank(G))(g-1),dim(G)(g-1)]. For G= SL(2) this was a conjecture of Newstead and Ramanan, proved by Gieseker, and for G= SL(3) it was proved by Kiem and Li. The result in general is an application of a localization theorem which applies to the moduli stack, which is a K-theory version of what Witten called "nonabelian localization". (This is joint work with Constantin Teleman and can be found in math.AG/0312154)