On the Monodromy of Milnor Fibers of Hyperplane Arrangements

Abstract

Abstract. We describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.

Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves

“…C : f = (x 2 − y 2 )(x 2 − z 2 )(y 2 − z 2 ) = 0 and λ = −1. A computation of the Hodge filtration on H 2 (F, C) in this case can be found in [5,16]. Note also that the mixed Hodge structure on H 2 (F, C) =1 is not pure in general.…”

A computational approach to Milnor fiber cohomology

“…The case of complex reflexion arrangements is discussed in [28], where the authors prove in particular the following result. For an eigenvalue of order p s , the papers [33], [28] give also upper bounds for the corresponding multiplicities for any line arrangement, see also [4], [5]. However, for the monomial arrangements, the existence of eigenvalues of order 6 (resp.…”

“…In particular one has the following, (i) Conjecture 2.3 holds for the line arrangement A(m, m, 3) for m ∈ [4,25].…”

“…(ii) For m ∈ [4,25], the monodromy operator (5.1) has only cubic roots of unity as eigenvalues. A(m, m, 3) were deformable into a real arrangement, the claim (ii) would follow from [38], but this does not seem to be the case.…”

Computing Milnor fiber monodromy for some projective hypersurfaces

“…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 .…”