Seminar

Sur un problème de noethérianité pour les algèbres enveloppantes.

Jacques ALEV (Univ. Reims)

Dans deux papiers récents S. Sierra et C. Walton ont répondu à la question "élémentaire" de savoir si l'algèbre enveloppante de l'algèbre de Virasoro est noethérienne. Nous allons présenter l'origine et la complexité du problème, puis expliquer la solution en décrivant la stratégie de la preuve finale.

On locally nilpotent derivations of integral domains

Bachar AL HAJJAR (Univ. Dijon)

Linearisation in small dimension

Hanspeter Kraft (Univ. Basel)

We know that every reductive group acting on $\mathbb{A}^2$ is linearisable, as a consequence of the amalgamated product structure of $Aut(\mathbb{A}^2)$. The first counter examples to linearisation appear in dimension $4$, even for finite groups, but dimension $3$ is open. We know that every faithful action of a non-finite reductive group acting on $\mathbb{A}^3$ is linearisable, as a consequence of the joint work with Gerry Schwarz, but for finite groups there are only partial results. The most recent says that a reductive group action on $\mathbb{A}^3$ which admits a semiinvariant coordinate function is in fact linearisable. This is joint work with Gerry Schwarz and Peter Russell. I will describe the state of the art and the new developments, and I will explain the main ideas of the proof which is rather involved.

Holomorphic Linearization

Frank Kutzschebauch (Univ. Bern)

The holomorphic linearization problem asked whether every reductive subgroup $G$ of the group of holomorphic automorphisms of complex affine n-space is conjugated to a subgroup of the general linear group, equivalently: Is every holomorphic action of a reductive group on $\mathbb{C}^n$ linear after a suitable change of coordinates ?
  The first (and only) counterexamples were given by Derksen and the author based on actions whose Luna quotient is not isomorphic to the Luna quotient of a linear representation.
  If two Stein $G$-manifolds are G-biholomorphic, then there Luna quotients are isomorphic. What about the converse? We prove with G. Schwarz and F. Larusson that in the case of isomorphic quotients the obstruction being $G$-biholomorphic is of topological nature. This is a new example of an Oka principle (the homotopy principle in Complex Analysis). In case one of the $G$-manifolds is a linear representation we show that this obstruction in most cases vanishes.

Koszul cohomology and vector bundles

Jan Nagel (Univ. Dijon)

If $X$ is a smooth, projective variety embedded in projective space via a very ample linear system $|L|$, the shape of the minimal resolution of the ideal sheaf of $X$ is governed by the Koszul cohomology groups of the pair $(X,L)$. In this talk we shall discuss the relationship between nonvanishing of Koszul cohomology groups and the existence of suitable vector bundles on $X$.

Do triangular automorphisms form a maximal subgroup ?

Pierre-Marie POLONI (Univ. Bern)

In this talk, we consider the group $G$ of polynomial automorphisms of the affine plane $\mathbf{k}^2$ and its subgroup $B$ of triangular automorphisms: \[B=\{ (x,y)\mapsto (ax+b(y), cy+d) \mid a,c\in\mathbf{k}^*, d\in\mathbf{k}, b(y)\in\mathbf{k}[y] \}.\] We will notably prove the two following results:
 -$B$ is not a maximal subgroup of $G$.
 -$B$ is maximal among the closed subgroups of $G$, when $G$ is seen as an ind-group (infinite dimensional algebraic group).
This is joint work with Jean-Philippe Furter.

The decomposition group of a line

Susanna Zimmermann (Univ. Basel)

The decomposition group of a line $L$ is the group of birational transformations of the plane that preserve $L$. In this talk, we will see that this group is quite similar to the Cremona group of the plane: It is generated by its linear elements and one quadratic map. Moreover, it does not decompose as a non-trivial an amalgam.