Homology graph of real arrangements and monodromy of Milnor Fiber

TL;DR

This paper investigates the homology of Milnor fibers for real arrangements, providing an algorithm to compute monodromy eigenvalues and establishing conditions that restrict these eigenvalues to roots of unity.

Contribution

It introduces a new algorithm based on the Salvetti complex for determining monodromy eigenvalues in real arrangements, with conditions limiting eigenvalues to specific roots of unity.

Findings

Eigenvalues are restricted to cubic roots of unity under certain conditions.

The algorithm computes possible monodromy eigenvalues efficiently.

Conditions are provided for eigenvalues of order 3 or 4 when restrictions are not met.

Abstract

We study the first homology group of the Milnor fiber of sharp arrangements in the real projective plane. Our work relies on the minimal Salvetti complex of the deconing arrangement and its boundary map. We describe an algorithm which computes possible eigenvalues of the first monodromy operator. We prove that, if a condition on some intersection points of lines is satisfied, then the only possible non trivial eigenvalues 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.