Gauss-Manin connections for arrangements. IV: Nonresonant eigenvalues. (English) Zbl 1103.32014
Summary: An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for the hyperplanes. We study the Gauss-Manin connection for the moduli space of arrangements of fixed combinatorial type in the cohomology of the complement with coefficients in the local system determined by the weights. For nonresonant weights, we solve the eigenvalue problem for the endomorphisms arising in the 1-form associated to the Gauss-Manin connection.
[For part I–III of this paper see the authors, [Compos. Math. 136, No. 3, 299–316 (2003; Zbl 1046.32002), Am. J. Math. 127, No. 3, 569–594 (2005; Zbl 1078.32018), and Trans. Am. Math. Soc. 357, No. 8, 3031–3050 (2005; Zbl 1087.32014)].
[For part I–III of this paper see the authors, [Compos. Math. 136, No. 3, 299–316 (2003; Zbl 1046.32002), Am. J. Math. 127, No. 3, 569–594 (2005; Zbl 1078.32018), and Trans. Am. Math. Soc. 357, No. 8, 3031–3050 (2005; Zbl 1087.32014)].
MSC:
| 32S22 | Relations with arrangements of hyperplanes |
| 14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
| 55N25 | Homology with local coefficients, equivariant cohomology |
