# Lie Groups, Invariant Theory & Complex Geometry

## Accommodation

Here are a couple of hotels that we have used for guests in the pasts and that we got positive feedback on afterwards.

### Motel One

Right in the city center; relatively close to the department, close to the main station and the restaurant where the workshop dinner will be held.### Holiday Inn Express Essen

Directly opposite the department.### Rheinischer Hof

A couple of stations by underground away from the department, but right in the middle of Rüttenscheid, a lively neighbourhood with many cafés, restaurants, bars, etc.## Venue

All talks will place in **Seminar Room WSC-N-U-3.05** in the **Math Department of the University Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen**. Note that this is **not on the main Essen campus** of University Duisburg-Essen.

Maps that show you how to find the seminar room can be found here:

- rough location of math department (look for "Mathematik / Seminarräume / Altendorfer Straße")
- area around the math department (look for "WSC")
- location of seminar room

## Talks - Titles and abstracts

**Michel Brion: Algebraic group actions on normal varieties**

**Abstract:**Let G be a connected algebraic k-group acting on a normal k-variety, where k is a field. We show that X is covered by open G-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a G-linearized vector bundle on an abelian variety, quotient of G. This generalizes a classical result of Sumihiro for actions of connected linear algebraic groups.**Alessandro Ghigi: Automorphisms and measures on a Kähler manifold**

**Abstract:**Let (M,omega) be a Kähler manifold and let K be a compact group that acts on M in a Hamiltonian way. The action of K^\C on M induces an action on the set of probability measures on M. Although this set has no obvious symplectic structure, there is an analogue of the momentum mapping for this action. Indeed many aspects of the momentum mapping (those that relate only to an orbit and its closure) do not need the underlying space to be a smooth or symplectic. This observation allows to establish numerical criteria for stability, semi-stability and polystability in a fairly abstract setting. I will explain this and I will apply it to the action of K^\C on probability measures, getting numerical criteria for stability, semi-stability and polystability of a measure. If time permits I will mention some results on the compactification of the automorphism group using meromorphic maps.**Luca Migliorini: Topological properties of projective maps: supports**

**Abstract:**After discussing some foundational resuts on algebraic maps, in particular stratifications and constructibility, I will define, after Ngo, the notion of support of a map, and give a criterion, due to V. Shende and myself, relying on the notion of Higher discriminant, which turns out to be quite efficient to determine the supports. I will give some examples coming from families of planar curves, where higher discriminants can be determined by deformation theory.**Stefan Nemirovski: Stein domains covered by the ball**

**Abstract:**Starting from the observation that the unit ball is the only complex manifold that can cover both Stein and non-Stein strictly pseudoconvex domains, the talk will discuss the (widely open) problem of characterising these two cases.**Gerry Schwarz: Oka principles and the linearization problem**

**Abstract:**pdf**Andrei Teleman: On the moduli stack of class VII surfaces**

**Abstract:**The most important gap in the Kodaira-Enriques classification table concerns the Kodaira class VII, e.g. the class of surfaces X having kod(X) = -\infty and b_1(X) = 1. The main conjecture which, if true, would complete the classification of class VII surfaces, states that any minimal class VII surface with b_2 > 0 contains b_2 holomorphic curves. A weaker conjecture states that any such surface contains a cycle of curves, and, if true, would complete the classification up to deformation equivalence. In a series of recent articles I showed that, at least for small b_2, the second conjecture can be proved using methods from Donaldson theory. In this talk I will concentrate on minimal class VII surfaces with b_2\leq 2, and I will present recent results on the geometry of the corresponding moduli stacks.**Matei Toma: Moduli spaces of semistable sheaves; an alternative construction method**

**Abstract:**Wall crossing phenomena for Gieseker-Maruyama stability over higher dimensional projective manifolds may give rise to purely irrational walls inside the ample cone. In recent work together with Daniel Greb and Julius Ross we show in the threefold case by use of GIT that projective moduli spaces of semistable sheaves with respect to irrational polarizations exist. In this talk we present an alternative GIT-free construction method of such moduli spaces over projective manifolds which is supposed to generalize to Kaehler manifolds as well. The method uses local moduli and stack theoretical results recently obtained by Alper, Hall and Rhyd. This is part of an ongoing research project together with Daniel Greb and Peter Heinzner.**Tilmann Wurzbacher: (Homotopy) moment maps beyond the symplectic case**

**Abstract:**The quest for a finite dimensional Hamiltonian formulation of classical field theories led to the geometry of multiphase spaces, essentially exterior powers of cotangent bundles. In analogy to the transition from classical mechanics to symplectic geometry, a manifold with a closed non-degenerate differential form of degree two or higher is called multisymplectic. We sketch some of the developements in this area putting special emphasis on the recent notion of a "homotopy co-moment map".

## Schedule

#### Thursday

**10.00 - 11.00:**Nemirovski**11.30 - 12.30:**Migliorini- Lunch break
**14.00 - 15.00:**Schwarz**15.15 - 16.15:**Teleman**17.00 - 18.00:**Ghigi

#### Friday

**09.00 - 10.00:**Toma**10.30 - 11.30:**Brion**11.45 - 12.45:**Wurzbacher

## Workshop Dinner

On **Thursday evening** a **workshop dinner** will take place in Mezzo Mezzo, close to Motel One. The dinner will start around 19.00 h.

#### Registration for workshop dinner

If you want to participate in the workshop dinner, please tell us so by writing an eMail to this eMail address. Dinner excluding drinks will be covered by workshop funds.

**Organisers:**

- Daniel Greb (Essen)
- Frank Kutzschebauch (Bern)
- Christian Miebach (Calais)
- Jörg Winkelmann (Bochum)

**The workshop is supported by:**