# Math 245: Equivariant Algebraic Geometry

Winter 2015

Monday Wednesday Friday 1:15-2:05 in 380-F

The goal of this class is to understand as much as possible about Riemann-Roch for Deligne-Mumford stacks, following Dan Edidin. (You can get the paper here.) This will give us an excuse to learn about equivariant techniques, Riemann-Roch, and stacks. In the first week and a half, Dan will outline the proof of Riemann-Roch for Deligne-Mumford stacks (so the punchline will come at the start), making clear what ideas need to be defined and what properties need to be proven in order to make the proof work. (I will serve as TA at this time.) In the rest of the course, we will work through these ideas.

Dan Edidin has a webpage for his notes here.

Prerequisites. You needn't --- and shouldn't --- have any background with equivariant techniques, Grothendieck-Riemann-Roch, or stacks to take this class. In fact, the goal is to learn about these ideas by seeing them in action. You will need comfort with algebraic geometry (the language of schemes), combined with a willingness to work with things you haven't fully mastered. Dan Edidin will also be around for the first two weeks, and is also happy to provide any background you might want.

Intersection theory. We will ned some comfort with the language of intersection theory for varieties/schemes. Reading Chapter 20 of my algebraic geometry notes, for example, would suffice. (We will use the construction of Fulton and MacPherson, as described in Fulton's Intersection Theory, but I can tell you what need during the course.)

Stacks. We will need the language of stacks at the start, but only enough that can be covered in a couple of hours. I gave a crash course on stacks on Tuesday January 6. Some possible references for those seeing stacks for (nearly) the first time (the first three by Edidin):

• What is a stack?.
• Equivariant geometry and the cohomology of the moduli space of curves.
• Notes on the construction of the moduli space of curves.
• Stacks for Everybody, by Barbara Fantechi.
• Picard groups of moduli problems, by David Mumford, and Daniel Litt's exposition thereof (parts one and two).
• Notes on Grothendieck topologies, fibered categories and descent theory, by Angelo Vistoli (lovely comprehensive reference).
• Martin Olsson also has a graduate level textbook on Algebraic spaces and stacks, which is currently circulating in print version.

Course email list. Please be sure you are on the course email list (by asking me).

Instructor: Ravi Vakil (office 383-Q, office hours by appointment). (Guest lectures by Dan Edidin, University of Missouri.)

• Mon. Jan. 5 (Dan Edidin): (first: Ravi on how the course will work) introduction, and plan for first two weeks; Riemann-Roch; Grothendieck groups and their key properties. (notes)
• Tues. Jan. 6: 2-hour crash course in stacks.
• Wed. Jan. 7 (Dan Edidin): Weil divisors, rational equivalence, key properties, Chern classes, Chern character, Todd class, Hirzebruch-Riemann-Roch again (and the genius of Hirzebruch); group actions (beginning). (notes)
• Thurs. Jan. 8 (Dan Edidin), 1:15-2:15 pm: crash course in intersection theory. (notes)
• Fri. Jan. 9 (Dan Edidin): equivariant K-theory (easier), equivariant Chow groups (harder), 5 minutes of quotient stacks. (notes)
• Mon. Jan. 12 (Dan Edidin): coarse moduli spaces; the localization theorem in equivariant K-theory. (notes)
• Wed. Jan. 14 (Dan Edidin): the example of weighted projective spaces. (notes)
• Fri. Jan. 16 (Dan Edidin): trying to naively figure out Riemann-Roch for weighted projective spaces, to figure out "correction" needed. (notes)
• Fri. Jan. 23. (notes by Tony Feng)
• Mon. Jan. 26. (notes by Tony Feng)
• Wed. Jan. 28. (rough notes by Tony Feng)
• Fri. Jan. 30. (rough notes by Tony Feng)
• Mon. Feb. 2 (Arnav Tripathy): application of GRR to moduli of curves; why you would expect the naive statement of Hirzebruch-Riemann-Roch to fail for stacks. (rough notes by Tony Feng)
• Wed. Feb. 4 (Daniel Litt): conceptual review of the Koszul complex; Chern character into Hochschild homology and "formal" version of Grothendieck-Riemann-Roch. (rough notes by Tony Feng)
• Mon. Feb. 9. (rough notes by Tony Feng)
• Wed. Feb. 11. (rough notes by Tony Feng)
• Fri. Feb. 13. (rough notes by Tony Feng)
• Mon. Feb. 16: no class (president's day)
• Fri. Feb. 20.
• Mon. Mar. 9. (rough notes by Tony Feng)
• (Later I might give one-sentence summaries of what was covered in each class.)
Jesse Silliman explains why K_0 = G_0 in good situations.