Modular Equalities for Complex Reflection Arrangements

We compute the combinatorial Aomoto--Betti numbers $\beta_{p}(\A)$ of a complex
reflection arrangement. When $\A$ has rank at least 3, we find that $\beta_{p}(\A)\le
2$, for all primes $p$. Moreover, $\beta_{p}(\A)=0$ if $p>3$, and $\beta_{2}(\A)\ne
0$ if and only if $\A$ is the Hesse arrangement. We deduce that the multiplicity
$e_{d}(\A)$ of an order $d$ eigenvalue of the monodromy action on the first
rational homology of the Milnor fiber is equal to the corresponding Aomoto--Betti
number, when $d$ is prime. We give a uniform combinatorial characterization
of the property $e_{d}(\A)\ne 0$, for $2\le d\le 4$. We completely describe
the monodromy action for full monomial arrangements of rank 3 and 4. We
relate $e_{d}(\A)$ and $\beta_{p}(\A)$ to multinets, on an arbitrary arrangement.

2010 Mathematics Subject Classification: Primary 14F35, 32S55; Secondary 20F55, 52C35, 55N25.

Keywords and Phrases: Milnor fibration, algebraic monodromy, hyperplane arrangement, complex reflection group, resonance variety, characteristic variety, modular bounds.

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