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 $\beta₂(\A)\ne 0$ 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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