The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

Abstract

A central question in arrangement theory is to determine whether the characteristic polynomial Δq of the algebraic monodromy acting on the homology group Hqfalse(F(A),double-struckCfalse) of the Milnor fiber of a complex hyperplane arrangement scriptA is determined by the intersection lattice L(A). Under simple combinatorial conditions, we show that the multiplicities of the factors of Δ1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers βpfalse(scriptAfalse), which in turn are extracted from L(A). When scriptA defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of scriptA to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over double-struckC) for matroids, and estimate the number of essential components in the first complex resonance variety of scriptA. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of prefixSL2false(double-struckCfalse)‐representation varieties, which are governed by the Maurer–Cartan equation.

On the Milnor Monodromy of the Irreducible Complex Reflection Arrangements

“…Several interesting examples have been computed by D. Cohen, A. Suciu, A. Mȃcinic, S. Papadima, M. Yoshinaga, see [4], [30], [21], [31], [32]. When the line arrangement A has only double and triple points, then a complete positive answer is given by S. Papadima and A. Suciu in [26]. However, the determination of the eigenvalues of h 1 in general remains a very difficult question.…”

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 .…”

On Algebraic Surfaces Associated with Line Arrangements

“…One of motivation of our work is to understand whether the Hodge numbers of F are combinatorially determined, one of the main open question in the theory of line arrangements, see [20]. As is explained in Section 8, we have the following formulae…”

“…On the other hand, it follows from [8] that 2q = dim H 1 (F ) =1 , which is known to many line arrangements, see [4]. In fact, in [20], a combinatorial formula for q is given when A has only double or triple points; more examples are given in [24] where q is computed. [25] is a good and recent survey on the monodromy computations and in a recent preprint [9], an effective algorithm to compute q is provided.…”