Abstract: The Milnor fibration of a complex, projective hypersurface produces a smooth manifold as a regular, cyclic cover of the hypersurface complement. When the hypersurface is a union of complex hyperplanes, the Milnor fibre is part of the study of hyperplane arrangements. In this case, the hypersurface complement is well known and studied. In particular, it is a Stein manifold, a rationally formal space, and it admits a perfect Morse function.

The cohomology and the monodromy of the Milnor fibre can be understood in terms of the cohomology jump loci of the hypersurface complement. For generic hyperplane arrangements, this cohomology and monodromy representation are known and fairly straightforward, although current technique still falls short of being able to describe even the betti numbers in the case of reflection arrangements. Some combinatorial techniques can be used to construct arrangements with Milnor fibres with interesting properties that constrast with the well-behaved nature of the arrangement complements. These include integer homology torsion, non-formality, and non-trivial monodromy representations in all cohomological degrees.

This talk is based on joint work with Alex Suciu.