Série de Poincaré-Koszul associée aux groupes de tresses pures
Let X be the complement of a complex hypersurface in \({\mathbb{C}}P^ N\). The lower central series \(\Pi_ 1(X,*)=\Gamma_ 0\supset \Gamma_ 1\supset...\supset \Gamma_ n\supset..\). defines the Mal'tsev algebra of X, \(L_ X=\lim_{\leftarrow}(\Gamma_ 0/\Gamma_ j\otimes {\mathbb{Q}})\). A new Lie algebra \(J_ X\), called the holonomy algebra, is associated to X as follows. The dual of the cup product induces \(\delta\) : \(H_ 2(X,{\mathbb{Q}})\to H_ 1(X,{\mathbb{Q}})\otimes H_ 1(X,{\mathbb{Q}})\subset T(H_ 1(X,{\mathbb{Q}})\) which factorises into the free Lie algebra \(L(H_ 1(X,{\mathbb{Q}}))\). Then, \(J_ X\) is defined as the quotient of \(L(H_ 1(X,{\mathbb{Q}}))\) by the ideal generated by \(\delta (H_ 2(X,{\mathbb{Q}})).\) Using the Sullivan theory and the mixed Hodge structure, the author proves that \(J^*_ H\cong L_ X\), when \(J^*_ X\) is the nilpotent completion of \(J_ X\). If \(X={\mathbb{C}}^{n+1}-\Lambda_ n\), where \(\Lambda_ n\) is the union of the hyperplanes \(H_{ij}\) \((x_ i=x_ j)\), then the author constructs a free resolution of \({\mathbb{Q}}\) by \(J_ X\)-modules and proves that there exists canonical isomorphisms \(H^ j(X,{\mathbb{Q}})\cong H^ j(\Pi_ 1(X),{\mathbb{Q}})\cong H^ j(J_ X,{\mathbb{Q}})\).
scientific article; zbMATH DE number 3917255
scientific article