Arrangements of lines and monodromy of associated Milnor fibers

作者：Salvetti Mario^*; Serventi Matteo

来源：Journal of Knot Theory and Its Ramifications, 2016, 25(12): 1642014.

DOI：10.1142/S0218216516420141

• Arrangements of hyperplanes
• monodromy
• Milnor fiber

Consider an arrangement A of homogeneous hyperplanes in C-n, with complement M(A). The (co) homology of M(A) with twisted coefficients is strictly related to the cohomology of the Milnor fiber associated to the natural fibration onto C*, endowed with the geometric monodromy. It is still an open problem to understand in general the cohomology of the Milnor fiber, even for dimension 1. In Sec. 1, we show that all questions about the first homology group are detected by a precise group, which is a quotient ot the commutator subgroup of pi(1)(M(A)) by the commutator of its length zero subgroup, which didn't appear in the literature before.
In Sec. 2, we state a conjecture of alpha-monodromicity for the first homology, which is of a different nature with respect to the known results. Let Gamma be the graph of double points of A : we conjecture that if Gamma is connected, then the geometric monodromy acts trivially on the first homology of the Milnor fiber (so the first Betti number is combinatorially determined in this case). This conjecture depends only on the combinatorics of A. We show the truth of the conjecture under some stronger hypotheses.

• 出版日期2016-10