Geodesic
Modular Equalities for Complex Reflection Arrangements
Documenta mathematica, Tome 22 (2017), pp. 135-150
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We compute the combinatorial Aomoto-Betti numbers βp​(A) of a complex reflection arrangement. When A has rank at least 3, we find that βp​(A)≤2, for all primes p. Moreover, βp​(A)=0 if p>3, and β2​(A)=0 if and only if A is the Hesse arrangement. We deduce that the multiplicity ed​(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 ed​(A)=0, for 2≤d≤4. We completely describe the monodromy action for full monomial arrangements of rank 3 and 4. We relate ed​(A) and βp​(A) to multinets, on an arbitrary arrangement.
DOI : 10.4171/dm/561
Classification : 14F35, 20F55, 32S55, 52C35, 55N25
Mots-clés : hyperplane arrangement, Milnor fibration, complex reflection groups 
@article{10_4171_dm_561,
     author = {Anca Daniela Ma\v{c}inic and Clement Radu Popescu and \c{S}tefan Papadima},
     title = {Modular {Equalities} for {Complex} {Reflection} {Arrangements}},
     journal = {Documenta mathematica},
     pages = {135--150},
     year = {2017},
     volume = {22},
     doi = {10.4171/dm/561},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/561/}
}
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Anca Daniela Mačinic; Clement Radu Popescu; Ştefan Papadima. Modular Equalities for Complex Reflection Arrangements. Documenta mathematica, Tome 22 (2017), pp. 135-150. doi: 10.4171/dm/561

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