Summer school: Intersection theory on schemes and stacks

The first 8 hours will be devoted to intersection theory on schemes with the goal of proving Baum-Fulton-Macpherson’s Riemann-Roch theorem for singular schemes. We will begin with the definition and basic properties of Chow groups as well as the construction of Segre and Chern classes. Next, the deformation to the normal cone will be presented, which will allow the construction of the pullback along a regular embedding and the definition of the intersection product on smooth varieties. We will then give a proof of the Grothendieck-Riemann-Roch theorem and finally develop the machinery necessary for the proof of the Riemann-Roch theorem for singular schemes. This includes K-theory, bivariant Chow groups and localized Chern characters.

In the second 8 hours we will extend this theory to stacks, with focus on quotient stacks and equivariant methods. Of particular importance in the equivariant theory are localization theorems for both equivariant Chow groups and equivariant K-theory. The localization theorem for K-theory plays a fundamental role in the proof of the Riemann-Roch theorem for Deligne-Mumford stacks as will be explained in the lectures. Finally, we will discuss the virtual fundamental class and virtual localization formula of Graber and Pandharipande.


Lecturers:

Dan Edidin (Univ. of Missouri)
Andrew Kresch (Univ. of Zurich)
Amalendu Krishna (Tata Institute of Fundamental Research)


Organizer:

Bumsig Kim (KIAS)