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CIMPA SUMMER SCHOOL 2007
Arrangements and Local Systems, Scientific Program



Conferencier: Sergei Yuzvinsky
Titre du cours: Cohomology of Orlik-Salomon Algebras


If $ \ensuremath{\mathcal A}$ is a complex hyperplane arrangement and $ M$ is its complement then a rank 1 local system $ {\cal L}$ on $ M$ is defined by an element $ a\in A_1$ where $ A=\oplus_{p\geq 0}A_p$ is the (graded) Orlik-Solomon algebra of $ \ensuremath{\mathcal A}$. On the other hand, the multiplication by $ a$ in $ A$ defines a cochain complex whose cohomology is denoted by $ H^*(A,a)$. The cohomology $ H^*(M,{\cal L})$ and $ H^*(A,a)$ are closely related to each other and studying these relations is one of the main theme of the arrangement theory.

We propose a short course of lectures exploring $ H^*(A,a)$ that can be given by two or three lecturers. Alternatively it can be given as a survey talk. The syllabus may include the following.

1. Definition of the Orlik-Solomon algebra $ A$ and main properties.

2. Vanishing results for $ H^*(A,a)$ for general position $ a$ (using sheaves on posets).

3. Resonance varieties $ R^p$ where $ H^p(A,a)$ does not vanish.

4. Propagation theorem for $ H^p(A,a)$.

5. Combinatorial description of $ R^1$ using Vinberg's classification of generalized Cartan matrices.

6. Relations of $ R^1$ with geometry of line arrangements; nets and multi-nets in complex projective plane.

7. $ R^1$ and Hesse-like pencils of curves.



Conferencier: Alexander Suciu
Titre du cours: Braid groups, monodromy and topology of arrangement complements



Conferencier: Hiroaki Terao
Titre du cours: What make hyperplane arrangements free?


Although the free arrangements were introduced more than twenty years ago, they are still somewhat mysterious.  Recently algebraic/differential geometric techniques were applied to get a quite intriguing sufficient and necessary condition for a hyperplane arrangement to be free.  The condition remains valid over any field.  As an application, M. Yoshinaga proved the Edelman-Reiner conjecture for the extended Shi/Catalan arrangements.   In this talk, I will discuss how much we can say at this moment about what makes hyperplane arrangements free.  




Conferencier: Dan Cohen
Titre du cours: Cohomology of Local systems


The lecture begins with three equivalent definitions of a local system on the complement of an arrangement. Then homology of the complement with coefficients in the local system is defined, and theorem of Hattori concerning the local system homology of a general position arrangement is stated. Features of Hattori's complex, and similar complexes arising in the context of super solvable arrangements are discussed. The lecture concludes with a discussion of the relationship between the cohomology of the Orik-Solomon algebra and the local system cohomology of the complement.



Conferencier: Michael Falk
Titre du cours: Large Geometry and combinatorics of resonance weights


Abstract (2-3 lectures): Let $ \mathcal A$ be an arrangement of $ n$ hyperplanes in $ \ensuremath{\mathbb{C}}^\ell$ with Orlik-Solomon algebra $ A=\oplus_{i=0}^\ell A^i$. The $ k^{\rm th}$ resonance variety of $ \mathcal A$ is the set $ \ensuremath{\mathcal R}^k(\ensuremath{\mathcal A})$ of one-forms $ a \in A^1$ annihilated by some $ b\in A^k$ which is not a multiple of $ a$. Resonance varieties are relevant to many different aspects of arrangements theory: local system cohomology and hypergeometric functions, Alexander invariants and characteristic varieties, nonlinear fibrations, classification of OS algebras and homotopy types, modules of logarithmic forms.

We will present the current state of knowledge concerning $ R^k(\ensuremath{\mathcal A})$ over various coefficient rings, in terms of the underlying matroid and the geometry of the arrangement. Our main focus will be the case $ k=1$. In this case it is no loss to assume $ \mathcal A$ is a line arrangement in $ {\P }^2.$ Over fields of characteristic zero, the strong results of Libgober and Yuzvinsky, using the Vinberg-Kac classification of Cartan matrices, yield a combinatorial interpretation in terms of multi-nets, and a geometric interpretation in terms of pencils of plane curves that ``interpolate" the arrangement. Over fields of positive characteristic $ \ensuremath{\mathcal R}^1(\ensuremath{\mathcal A})$ is seen to be the ruled variety carried by a special-position intersection of Schubert varieties in the Grassmannian of lines in $ {\P }^n.$ Over rings of non-prime characteristic one sees more strange phenomona which is empirically related to translated tori in characteristic varieties.

For $ k\geq 2$ there are partial generalizations of the results above, mostly involving $ k$-generic arrangements. In addition, we will see a relationship between elements of $ \ensuremath{\mathcal R}^k(\ensuremath{\mathcal A})$ and the critical loci of rational functions with zeros and poles along the arrangement.

The subject is imbued with beautiful examples, which will feature prominently in the lectures.



Conferencier: A. Muhammed Uludag
Titre du cours: Configuration spaces of arrangements


The configuration space of $ m$ ordered points on the projective line $ {\mathbb{P}}^1$ is the pure braid space

$\displaystyle C(1,m):=\{ (p_1, p_2, \dots, p_m) \in ({\mathbb{P}}^1)^m\, :\, p_i\neq p_j \; (i\neq j)\} $

and the configuration space of $ m$ unordered points on the projective line is the quotient of the pure braid space under the obvious action of the symmetric group $ \Sigma_m$ and is called the braid space. The fundamental group of the braid space is the spherical braid group.

The configuration space of $ m$ ordered hyperplanes on the projective space $ {\mathbb{P}}^n$ is the space

$\displaystyle C(n,m):= \{ (p_1, p_2, \dots, p_m) \in ({\mathbb{P}}^n)^m\, :\, p_i\neq p_j \; (i\neq j)\}, $

(where hyperplanes are identified with their duals). For $ n>1$ this space is not very interesting since it is simply connected (we remove just a codimension-2 subset), the analogue of the pure braid group is just the trivial group in this case. The analogue of the braid group is the fundamental group of the quotient $ C(n,m)/ \Sigma_n$, which is just the symmetric group.

In order to get interesting analogues of braid groups ``higher braid groups'' we must impose further restrictions on the admissible sets of hyperplanes. A natural choice is to consider the configuration space of hyperplanes in general position. The corresponding groups were studied by Manin-Schehtman, their presentations were studied by H. Terao and R.J. Lawrence in the pure case. These groups are highly sophisticated.

The lecture will end with the question: Is the generiticity assumption the only plausible restriction to get a good analogue of the braid group?



Conferencier: Eva Maria Feichtner



Conferencier: Daniel Matei "Artin groups, Bestvina-Brady groups and Arrangements of Hypersurfaces ">



A contacter: Graham Denham,

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