Workshop on deformation theory in algebraic and differential geometry Berlin 13.-18.12.2007 jointly organised by the projects A2 and A3 of the SFB 647 "Space.Time.Matter." funded by the DFG and the VW Junior Research Group "Special Geometries in Mathematical Physics" funded by the Volkswagen Stiftung

Speakers:

• Dietrich Burde (Wien): Deformations and Degenerations of Lie Algebras
  We will discuss degenerations, contractions and deformations of Lie algebras and algebraic groups. The first task is to give a general definition which covers the known special cases, and which shows how these notions are related. The next question is, how to classify Lie algebra degenerations in certain cases. This involves Lie algebra invariants, which are semi-continuous on the algebraic set of all $n$-dimensional Lie algebra laws, such as the maximal dimension of an abelian subalgebra, or the dimension of the trivial and adjoint cohomology spaces.
• Fabrizio Catanese (Bayreuth): Complex structures and their deformations in the large for tori, torus bundles and nilmanifolds.
  Given an oriented differentiable manifold M of even dimension, one has the infinite dimensional space of almost complex structures on it, and inside it the subspace C(M) of integrable complex structures on M. The Deformations in the large of a complex structure J in C(M) are the complex structures which lie in the same connected component of J inside C(M). The main problem in deformation theory is to understand these deformations in the large, and a prototype theorem is that all deformations in the large of a complex torus yield a complex torus (although there are 'exotic' complex structures on tori, in view of a construction going back to Blanchard and Sommese). To which extent does this result extend to torus bundles and to left invariant complex structures on nilmanifolds G / Gamma ? This question relates to recent work of the author, Frediani,and of Rollenske which is related to the classification of complex structures on nilmanifolds due to several authors (Salamon, Fino, Console, Ugarte, and many others) and says that the answer is positive for certain iterated (complex) torus bundles. These examples include the classical Iwasawa manifold and shed light on the interesting phenomenon of defrmations of complex parallelizable manifolds which are not parallelizable. The methods employed are twofold : local moduli and global limits. Local moduli : any point J of C(M) the Kuranishi slice Kur(J) is dimensionally transverse to the orbit of Diff^o(M), and the Kuranishi equations define a finite dimensional germ of complex analytic space. Taking limits in the large, one has to exploit suitable generalizations and projctions of the classical Albanese maps of compact Kaehler manifolds.
• Jan Christophersen (Oslo)
• Anna Fino (Torino): Dolbeault cohomology, deformation of complex and hermitian structures on nilmanifolds.
  I will discuss how to compute the Dolbeault cohomology of a nilmanifold endowed with a left-invariant complex structure and I will examine deformations of some types of complex structures. Moreover, I will show how the existence of some Hermitian structures, like balanced, strong Kaehler with torsion and Hermitian metrics whose Bismut connection has restricted holonomy in SU(n), is not stable under small deformations on some nilmanifolds.
• Hubert Flenner (Bochum): Moduli for affine surfaces
  Usually there exist no moduli spaces for affine spaces. One of the main obstacles for this are, similarly as in the compact case, affine rulings, which may exist on such surfaces. They are responsible for the non-uniqueness of a completion. We consider in our talk a worst case scenario, namely the Gizatullin surfaces, which even admit many different affine rulings. Gizatullin surfaces are characterized by the property that they admit a completion by a zigzag, that is by a linear chain of rational curves. Their completions are obtained from the quadric $Q={\bf P}^1 \times {\bf P}^1$ by a sequence of blowups in a point $(0,0)$ (in suitable coordinates). We assign to such blowup sequences a so-called configuration invariant that measures a part of the configurations of the blowup centers. The main result is, that in certain cases this invariant serves as a moduli space. As an application we present Gizatullin surfaces with an infinite number of conjugacy classes of $\bf C_+$ and $\bf C^*$-actions. (joint work with M.Zaidenberg and S. Kaliman)
• Sergei Merkulov (Stockholm): Deformation theory via operads
  The theory of operads and props gives a universal approach to the deformation theory of many algebraic and geometric structures. It also gives a conceptual explanation of the wellknown experimental observation that a deformation theory is controlled by a differential graded Lie algebra. We review the main ideas and theorems of this operadic approach to the deformation theory and illustrate it with several applications.
• Sönke Rollenske (Bayreuth): Deformations of Nilmanifolds with left-invariant complex structure
  We will discuss some results concerning small deformations and deformations in the large of nilmanifolds with left-invariant complex structure.
• Jan Stevens (Göteborg): Deformations of nonnormal singularities
  We first discuss how to compute deformations of singularities. Normally one only considers normal singularities (in dimension at least 2). Nonnormal singularities naturally arise by blowing down deformations of resolutions of surface singularities. We illustrate by examples how computations of deformations of nonnormal surface singularities help to find interesting deformations of resolutions.
• Duco van Straten (Mainz)

Date & Location:

December 13: Introductary courses at advanced students' level will be held on the deformation theory of algebraic varieties and nilmanifolds. The talks take place in HS 001 at the Mathematical Institute of Freie Universität Berlin, Arnimallee 3.

December 17-18: The talks take place in the Humboldt-Kabinett at the Mathematical Institute of Humboldt Universität, Rudower Chaussee 25.

A social dinner is taking place on Monday evening (further information to follow)

Registration:

Organisers:

• Simon Chiossi (Humboldt-Universität zu Berlin), sgc@mathematik.hu-berlin.de
• Richard Cleyton (Humboldt-Universität zu Berlin), cleyton@mathematik.hu-berlin.de
• Frederik Witt (Freie Universität Berlin), fwitt@math.fu-berlin.de

All colleagues and students are hereby cordially invited to participate in this workshop!

Poster