Quot schemes and moduli spaces

Abstract

Consider the moduli space M of stable vector bundles over a curve. Verlinde's formulas give the cohomologies of line bundles over M. We develop a novel method to recalculate this. Consider two birationally equivalent spaces, a Quot scheme Q and a projective fibration P, with a PGL(n) action. By lifting the PGL(n) action on P to a line bundle L, the GIT quotient is a fibration over M with an induced line bundle L'. By the [Q,R]=0 theorem, the cohomologies of L' correspond to the invariant part of the cohomologies of L. The Atiyah-Bott localization formula gives the equivariant Euler characteristic of L. The main idea is to apply this formula to Q instead of P. The main conjecture is that we get the same results as using P. We calculate the formula for certain parameters and verify the conjecture for those cases.