Homology graph of real arrangements and monodromy of Milnor Fiber

Abstract

We study the first homology group H 1 (F, C) of the Milnor fiber F of sharp arrangements A in P 2 R . Our work relies on the minimal complex C * (S(A)) of the deconing arrangement A and its boundary map. We describe an algorithm which computes possible eigenvalues of the monodromy operator h 1 of H 1 (F, C). We prove that, if a condition on some intersection points of lines in A is satisfied, then the only possible non trivial eigenvalues of h 1 are cubic roots of the unity. Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to appear in cases in which this condition is not satisfied.

Computing Milnor fiber monodromy for some projective hypersurfaces

“…Regarded as an arrangement in P 3 , this arrangement is known to be free with exponents (d 1 , d 2 , d 3 ) = (3,3,5), see [35]. Running the algorithm described above and assuming Conjecture 3.8 true, we get (5.6) Sp 0 P (f ) = t 4 12 + 4t .…”

“…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…”

“…Regarded as an arrangement in P 3 , this arrangement is known to be free with exponents (d 1 , d 2 , d 3 ) = (3,5,7), see [35]. Running the algorithm described above and assuming Conjecture 3.8 true, we get (5.10) Sp 0 P (f ) = t 4 16 + 4t 19 16 + 47t 20 16 + +44t 21 16 + 39t 22 16 + 32t 23 16 + 25t 24 16 + 16t 25 16 + 9t 26 16 + 4t 27 16 + t 28 16 .…”

“…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 the topology of the Milnor fibration of a hyperplane arrangement

“…Now, if both z and wz belong to F, then Qpzq " Qpwzq " 1, and so w n " 1. Thus, the restriction (4) π : FpA q Ñ UpA q is the orbit map of the free action of the geometric monodromy on FpA q. Hence, the Milnor fiber FpA q may be viewed as a regular, cyclic n-fold cover of the projectivized complement UpA q, see for instance [43,10,52].…”