Riemann-Roch for algebraic stacks III Virtual structure sheaves and virtual fundamental classes, by Roy Joshua

In this paper we apply the Riemann-Roch and Lefschetz-Riemann-Roch theorems proved in our earlier papers in the setting of Bredon-style homology and cohomology theories to define virtual fundamental classes for the moduli stacks of stable curves in great generality and establish various formulae for them (some of them proven elsewhere by other methods). These formulae are derived in a uniform manner using Riemann-Roch starting with various formulae for virtual structure sheaves. Our Riemann-Roch transformations take values in Bredon-style homology theories. Making use of the relationship between Bredon style theories and the more traditional theories our formulae for virtual fundamental classes hold in traditional homology theories for algebraic stacks.

Roy Joshua <joshua@math.ohio-state.edu>