Inductive freeness of Ziegler's canonical multiplicity

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The theory of hyperplane arrangements has been a driving force in mathematics for many decades. It naturally lies at the crossroads of algebra, combinatorics, algebraic geometry, and topology.Frequently, questions relating to reflection arrangements, consisting of the reflecting hyperplanes of an underlying reflection group, arose first for symmetric groups, then were extended to all Coxeter groups and finally embraced the entire class of complex reflection groups. Subsequently, the properties in question were investigated further within the wider class of restrictions of complex reflection arrangements. A prime example of this phenomenon is the question about the topological nature of the complement of the union of the hyperplanes in a complex reflection arrangement.In his seminal work from 1989, Ziegler showed that the restriction to any hyperplane of a free hyperplane arrangement endowed with the natural multiplicity is then a free multiarrangement. In 2018 Hoge and the PI investigated the stronger freeness property of inductive freeness for these canonical free multiarrangements and classified all reflection arrangements with the property that the induced Ziegler multiplicity is inductively free on any restriction.The principal goal of this research proposal is to prove the precise analogue of Ziegler's theorem for this stronger property of inductive freeness in complete generality: If A is inductively free, then so is the Ziegler restriction of A to every hyperplane of A.A second related conjecture we aim to show is the following, where we do not require that the underlying arrangement itself is inductively free: if A and the deletion A' with respect to some hyperplane H are both free and the restriction of A to H itself is inductively free, then so is the Ziegler multiplicity on the restriction of A to H.