Modular Equalities for Complex Reflection Arrangements

Abstract

We compute the combinatorial Aomoto-Betti numbers $\beta_{p}(\Cal A)$ of a complex reflection arrangement. When $\Cal A$ has rank at least 3, we find that $\beta_{p}(\Cal A)\leq 2$, for all primes $p$. Moreover, $\beta_{p}(\Cal A)=0$ if $p>3$, and $\beta_{2}(\Cal A)\neq 0$ if and only if $\Cal A$ is the Hesse arrangement. We deduce that the multiplicity $e_{d}(\Cal A)$ 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 $e_{d}(\Cal A)\neq 0$, for $2\leq d\leq 4$. We completely describe the monodromy action for full monomial arrangements of rank 3 and 4. We relate $e_{d}(\Cal A)$ and $\beta_{p}(\Cal A)$ to multinets, on an arbitrary arrangement.