×

zbMATH — the first resource for mathematics

Eigenvalues for the monodromy of the Milnor fibers of arrangements. (English) Zbl 1036.32019
Libgober, Anatoly (ed.) et al., Trends in singularities. Basel: Birkhäuser (ISBN 3-7643-6704-0/hbk). Trends in Mathematics, 141-150 (2002).
Let \(l_1,\dots, l_d\) be \(d\) linear forms on \(\mathbb{C}^{n+1}\), \(A\) the cone of the equation \(\prod l_i= 0\) and \({\mathcal A}\) the corresponding arrangement of hyperplanes in \(\mathbb{P}^n\). If \(F_A\) is the Milnor fiber of \(A\) at \(0\), the author describes, in terms of the combinatorics of \({\mathcal A}\), upper bounds for the orders of the eigenvalues of the monodromy acting on \(H_k(F_A, \mathbb{C})\) \((k\leq n-1)\). When \(n= 2\) he also computes the multiplicities of these eigenvalues. He then studies several examples giving in particular some conditions implying that eigenvalues different from 1 can only occur in top dimension.
For the entire collection see [Zbl 0997.00011].

MSC:
32S22 Relations with arrangements of hyperplanes
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
PDF BibTeX XML Cite