Homology graph of real arrangements and monodromy of Milnor fiber

On the Milnor Monodromy of the Irreducible Complex Reflection Arrangements

“…starting from V or f is a rather difficult problem, going back to O. Zariski and attracting an extensive literature, see for instance [2,9,11,19,21,23,24,29,37] for the case n = 2, and some of them dealing only with real line arrangements. In this paper, we take a new look at a method to determine the Alexander polynomial V (t) introduced in [8] and developed in [9,Chapter 6].…”

“…Since A(G) is a central hyperplane arrangement, it has a defining equation f = 0 in C n , where f is a homogeneous polynomial of some degree d. One can associate to this setting the Milnor fiber F (G) of the arrangement A(G). This is a smooth hypersurface in C n , defined by f = 1, and it is endowed with a monodromy morphism h : F (G) → F (G), given by h(x 1 , ..., x n ) = exp(2πi/d) · (x 1 , ..., x n ), see [1,2,3,4,8,10,11,17,21,24,25,27] for related results and to get a feeling of the problems in this very active area. The study of the induced monodromy operator…”

Computing Milnor fiber monodromy for some projective hypersurfaces

“…Example 5.4 (The complex reflection arrangement A (3, 3, 4)). The hyperplane arrangement A (3,3,4) is defined in C 4 by the equation…”

“…A special case of great interest is when f is a product of linear forms, and then V is a hyperplane arrangement A, and the corresponding complement is traditionally denoted by M(A). A lot of efforts were made, in the case of hyperplane arrangements most of the time, to determine the eigenvalues of the monodromy operators (1.1) h m : H m (F, C) → H m (F, C) with 1 ≤ m ≤ n, see for instance [1,2,3,4,8,9,10,12,21,22,32,33,36,45,47]. However, in most of these papers, either only the monodromy action on H 1 (F, C) is considered, or the results are just sufficient conditions for the vanishing of some eigenspaces H m (F, C) λ .…”

Configurations of points and topology of real line arrangements

“…In this case, the real picture of this kind of arrangements determines, not only the fundamental groups [Arv92], or the abelian covers [Hir93], but all the topological information [CR03]. Using this, several methods and algorithms are developed to compute topological invariants of complexified real arrangements (see [Yos13], [Yos15] and more recently [BS17]).…”