Introduction to stacks (MRI-Masterclass 2010)

Contact: J. Heinloth
Time: Monday 15:00-17:00
Place: Science Park 904, Amsterdam
Room: D1.112 (Science Park, Amsterdam) from Sept 6 to Oct. 18 and Room A1.14 form Nov 1 to Dec 13.

Overview

Moduli spaces are the spaces parameterizing all objects of a specified type. In the master class you will see a lot of examples, like the space of all curves of a given genus. Unfortunately, often such parameterspaces do not really exsist, or if they do they do not have all the propertis we would like them to have. For example, being a parameter space of curves should imply that there is a family of curves parameterized by the points of the space, but this often fails to be the case.
Stacks were introduced to handle this problem. From a simplified point of view they provide a notion of space in which the points may have symmetries and at the same time a large amount of geometric notions still makes sense for these spaces.
The aim of the seminar is to learn what stacks are, why we would like to have them and to get a feeling for how one can work with stacks. You can have a look at a preliminary program.

How will we proceed?

In the first meeting I will give an overview of the seminar and we will distribute the talks.
During the first part of the seminar we will have a talk every monday, split into two parts, prepared by a team of two participants. All of you have different backgrounds and we have to make use of this, so in the first meeting we will form teams of 4 students, who prepare two consecutive meetings. The talks should be two times 40 minutes. At the beginning of each meeting we will briefly discuss the exercise of the previous week. In particular a part of the task of the preparation is to design an exercise for each talk, to collect the homework, grade it and discuss solutions on the next meeting.
In the second part of the seminar we will try to get to more advanced topics. The larger teams of the first part will prepare one of these talks together.
At first sight, the existing literature often looks quite abstract. Therefore it is extremely important to clarify the theory by explaining examples carefully. After all these are the applications we want to have. Also the plan is to have a two-fold approach: We first look at the analytic category, where many technicalities disappear and then fill in what is needed to handle algebraic stacks. I hope that this makes the theory more transparent.

Schedule and Program

Monday September 6, Introduction and distrubution of the talks.
Monday September 13 A very short introduction to sheaves and algebraic geometry (Peter (Peter's notes.)and Dan (Dan's notes.))
September 20 (Can we skip this meeting and have a Seminar on October 25 instead ?)
September 27 Moduli spaces I: Categories, representable functors, examples (Sara and ?)(Exercise for this week)
October 4 Moduli spaces II: Necessary conditions for existance - gluing and descent (Marcelo and Gereon Notes with 3 exercises and a solution for exercise 3 by Egbert)
October 11 Stacks: Pseudo-functors and fibered categories (Andrea (Andrea's notes with one exercise (Proposition 2.1) and Peter(Peter's notes with one exercise))
October 18 Algebraic stacks: The 2-Yoneda Lemma and representable morphisms (Johan his notes with an exercise and Egbert Egbert's notes (expanded version!))
October 25 10:15-12:15 Room D1.114 Differentiable & algebraic stacks: Properties and examples (Camila, Camillo and Yuri,An exercise(definitions added 28.10) )
November 1 Quotients and Groupoids: An equivalent description of stacks (Amir, Boris Boris's notes and Leila, an Exercise)
November 8 More properties of stacks, sheaves and vector bundles (Alberto Alberto's notes, Sander and Alex, an Exercise)
November 15 Elliptic curves again: Mumford's article [Mu]( Notes of Boris, Gereon and Marcello )
November 22 Sheaves and cohomology for stacks: Definition and examples
November 29 The Grothendieck-Lefschetz trace formula - topological and etale (Notes of Andrea and Camila (topological Lefschetz theorem),Notes of Sara on Etale cohomology, Notes of Alexey on the Weil conjectures, Notes of Yuri on Examples of point counting)
December 6 Behrend's trace formula. (Notes of Dan on the Leray spectral sequence)
December 13 Application: Computing cohomology by equivariant point-counting

Literature

[F] B. Fantechi, Stacks for everybody. European Congress of Mathematics, Vol. I (Barcelona, 2000), 349--359, Progr. Math., 201, Birkhäuser, Basel, 2001.
[V] A. Vistoli: Notes on Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, 1--104, Math. Surveys Monogr., 123, Amer. Math. Soc., Providence, RI, 2005., arXiv:math/0412512v4
[Bookproject] Algebraic stacks, book project by Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche and Andrew Kresch, can be found here.
[deJong] The stacks project. This is a book project, initiated by A. J. de Jong, to which many authors contributed. This ongoing projcet aims to contain everything you could want to know about stacks. It is growing. (Do not print it, unless you are sure you want to. The file is huge.) http://math.columbia.edu/algebraic_geometry/stacks-git/book.pdf.
[Moe] I. Moerdijk, Introduction to the language of stacks and gerbes, arXiv:math/0212266v1
[Mu] D. Mumford, Picard groups of moduli problems, 1965 Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) pp. 33--81 Harper Row, New York
[Be] K. Behrend, The Lefschetz trace formula for algebraic stacks. Invent. Math. 112 (1993), no. 1, 127--149.
[LMB] G. Laumon and L. Moret-Bailly. Champs algebriques, volume 39 of Ergeb- nisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 2000
[HLectures] J. Heinloth, Lectures on the moduli stack of vector bundles on a curve
[HDiff] J. Heinloth, Notes on differentiable stacks

Last modified: 2.9.2010, Jochen Heinloth (email: use @uva.nl with J.Heinloth)