# Workshop: Riemann-Roch for Deligne-Mumford stacks

U. Görtz, T. Wedhorn

**Date and place**: January 19 + 20, 2013 (Sat + Sun), in Essen

On Saturday, we will start at 13:00, on Sunday we wil start at 9:15.

The workshop will take place at the

Institut für Experimentelle Mathematik

Ellernstr. 29

45326 Essen

(From the main station, take Straßenbahn 106 in direction of Altenessen bf. until the stop Seumannstr., then cross the street and walk down Ellernstr. for about 150 m.)

**Program**: pdf

## Schedule

### Saturday

1.1 | 13:00 | The Riemann-Roch theorem of Baum, Fulton and MacPherson | Timo Keller |

1.2 | 14:15 | Equivariant $K$-theory | Haifeng Wu |

1.3 | 15:45 | Localization in equivariant $K$-theory | Jens Hornbostel |

1.4 | 17:15 | Equivariant Chow groups | Mark Kuschkowitz |

Afterwards: Joint dinner. **Please email ulrich.goertz@uni-due.de, if you want to join the dinner, so that we can make an appropriate reservation.**

### Sunday

2.1 | 9:15 | Chow groups for quotient stacks | Peng Sun |

2.2 | 10:30 | Riemann-Roch for representable morphisms of quotient stacks | Torsten Wedhorn |

2.3 | 12:00 | Riemann-Roch for quotient Deligne-Mumford stacks | Eike Lau |

13:00 | Break | ||

2.4 | 14:00 | Examples | Ulrich Görtz |

The workshop will end around 15:15.

The Riemann-Roch theorem and its generalizations such as the Hirzebruch-Riemann-Roch and the Grothendieck-Riemann-Roch theorems are among the most important results in algebraic geometry. After briefly recalling several versions of the theorem and the required input such as Chern classes and Todd classes, K-groups and Chow groups, the main part of the workshop will focus on the equivariant theory, i.e., on the theory in the presence of the action of a (linear algebraic) group. We will study equivariant K-theory (and localization theorems in equivariant K-theory) and equivariant intersection theory. Most of this is contained in the series [EG1], [EG2], [EG3], [EG4] of Edidin and Graham. The recent preprint [E] by Edidin could serve as a guide. We might also be able to look at the underlying results by Thomason. There are many connections to other topics, and we will certainly look at some interesting examples.

[E] D. Edidin, Riemann-Roch for Deligne-Mumford stacks

[EG1] D. Edidin, W. Graham, Equivariant intersection theory (with an appendix by Angelo Vistoli: The Chow ring of $\mathscr M_2$), Invent. math. 131 (1998), 595-634, cf. arxiv:9609018

[EG2] D. Edidin, W. Graham, Riemann-Roch for equivariant Chow groups, Duke Math.

J. 102 no. 3 (2000), 567-594, cf. arxiv:9905081

[EG3] D. Edidin, W. Graham, Riemann-Roch for quotients and Todd classes of simplicial toric varieties, Special issue in honor of Steven L. Kleiman. Comm. Algebra 31 (2003), no. 8, 3735-3752; cf. arXiv:math/0206116v2

[EG4] D. Edidin, W. Graham, Nonabelian localization in equivariant K-theory and Riemann-Roch for quotients, Adv. Math. 198 (2005), 547-582, cf. arxiv:0411213