martedi' 21-11-2006 (15:00) - sala seminari Emanuele Delucchi (Universita' Zurigo) : Combinatorial remarks on a classical theorem of Pierre Deligne. (Or: The Importance of Being Simplicial)

ABSTRACT: An arrangement is a set of linear hyperplanes in complex n-space. If the defining equations of the planes have real coefficient, the arrangement is called complexified. Deligne proved in 1972 that the complement of a complexified arrangement in n-dimensional complex space is aspherical (i.e., all higher homotopy group vanish) whenever a simple geometric condition ('simpliciality') on the associated real arrangement is satisfied. We will outline Deligne's proof, that consists of two steps. The first step is rather algebraic, the second is more topological in nature. We will approach the first step in Deligne's argument replacing the geometric condition of simpliciality by an equivalent combinatorial condition. Exploiting some combinatorial tools we can prove the converse of Deligne's statement, answering to a question posed by Luis Paris. Moreover, the standard techniques of topological combinatorics allow to state (and reprove) the second step of Deligne's proof in a very quick way. One might then suspect that a weaker condition than simpliciality suffices for the result to hold. We investigate a possible way towards this generalization. No special prerequisites are requested from the audience.

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