# Workshop: The Decomposition Theorem

The theory of perverse sheaves and one of its crowning achievements, the decomposition theorem, are at the heart of a revolution which has taken place over the last thirty years in algebra, representation theory and algebraic geometry. The decomposition theorem is a powerful tool for investigating the topological properties of proper maps between algebraic varieties and is one of the deepest known facts relating their homological, Hodge-theoretic and arithmetic properties.

By now there are three known approaches to the decomposition theorem: the original one, due to A. Beilinson, J. Bernstein, P. Deligne and O. Gabber, via the arithmetic properties of varieties over finite fields, the one of M. Saito via mixed Hodge modules, and the approach of De Cataldo and Migliorini, via classical Hodge theory. Each approach highlights different aspects of this important theorem.

The goal of this workshop is the study of the decomposition theorem, of some applications, and of the proof of De Cataldo and Migliorini.

**Dates and places:**

- November 13, Essen,
- December 4, Paderborn,
- January 22, Düsseldorf.

**Program:** pdf

**Organizers:** Ulrich Görtz, Torsten Wedhorn

## Third meeting: January 22, Mathematisches Institut, Düsseldorf

**Address:** Heinrich-Heine-Universität Düsseldorf, Mathematisches Institut, Düsseldorf

**Lecture hall:** 5G (Building 25.22)

### Schedule

11:00 Introduction

11:15 – 12:15 Toric varieties, Ulrich Terstiege (Essen)

12:25 – 13:25 Examples of semismall morphisms I: The Springer resolution, Christian Kappen (Essen)

14:30 – 15:30 Examples of semismall morphisms II: Hilbert schemes of points, Lutz Hille (Münster)

15:45 – 17:00 Ngô’s support theorem, Martin Kreidl (Essen)

## Second meeting: December 4, Institut für Mathematik, Paderborn

**Address:** Universität Paderborn, Institut für Mathematik, Warburger Str. 100, 33098 Paderborn

**Lecture hall:** A3.301 (Gebäude A, 3. Stock)

### Schedule

11:30 Introduction

11:45-13:00 Statement of the Main Theorem, Elena Fink (Paderborn)

14:00-15:00 The decomposition theorem for semismall morphisms, Daniel Wortmann (Paderborn)

15:15-16:15 Proof for semismall morphisms, Ralf Kasprowitz (Paderborn)

16:30-17:30 Sketch of proof in the general case, Eike Lau (Bielefeld)

## First meeting: November 13, Institut für Experimentelle Mathematik, Essen

**Address:** Institut für Experimentelle Mathematik, Ellernstr. 29, 45326 Essen.

**How to get there:** From Essen Hauptbahnhof, take Straßenbahn 106 (direction Altenessen) to the stop **Seumannstr.** The IEM is very close to the tram stop.

Google maps

### Schedule

10:30 Introduction

10:45 — 11:45 **Hodge theory**, Christian Kappen (Essen)

12:00 — 12:45 **Intersection complexes**, Jörg Schürmann (Münster)

13:45 — 15:15 **Perverse sheaves**, Ulrich Görtz (Essen)

15:45 — 16:45 **Perverse filtration**, Sven Meinhardt (Bonn)