Resonant bands, Aomoto complex, and real 4-nets

Abstract

Introduction The resonant band is a useful notion for the computation of the nontrivial monodromy eigenspaces of the Milnor fiber of a real line arrangement. We develop the resonant band description for the cohomology of the Aomoto complex. As an application, we prove that real 4-nets do not exist. Let us fix some notation: k ∈ Z, k ≥ 3; K a field (generally, R or C) and KP2 the projective plane; A = {H0, . . . ,Hn} a line arrangement in KP2; A = {H1, . . . ,Hn} the affine line arrangement in K2 = KP2 H0 obtained from A; AF2(A) the Orlik-Solomon algebra of A over F2 generated by the symbols e1, . . . , en; For S ⊂ A, consider e(S) := ∑ Hi∈S ei ∈ A 1 F2(A) and η0 := e(A) = ∑n i=1 ei. Definition (k-nets) A supports a k -net structure if and only if there exist a partition A = A1 t · · · t Ak and a finite set of points X ⊂ KP2 such that For all i 6= j , if H ∈ Ai and H ′ ∈ Aj, then H ∩ H ′ ∈ X ; For all p ∈ X and for all i = 1, . . . , k , there exists a unique H ∈ Ai such that p ∈ H. Known facts If k ≥ 5 there does not exist any k -net; There exist infinitely many 3-nets; The Hesse arrangement is the only known 4-net. Theorem 2 (Papadima-Suciu) Consider S A. Then e(S) ∧ η0 = 0 if and only if ∀p ∈ K2 one of the following is satisfied: if |Ap| is odd, then |Ap| = |Sp|; if |Ap| is even then |Sp| is even, where Ap := {H ∈ A | p ∈ H}. Hesse arrangement Consider fλ := λ(x3 + y3 + z3)− 3xyz ∈ C[x, y , z] with λ ∈ C. Now, fλ factors into linear factors if and only if λ ∈ {0, 1, ω, ω2}, where ω = e2πi/3. Consider now A1 := {f0 = 0}, A2 := {f1 = 0}, A3 := {fω = 0} and A4 := {fω2 = 0}. Define the complex arrangement A = A1 t A2 t A3 t A4. This partition give as a 4-net structure on A. Theorem 1 (Papadima-Suciu) Let Q be a defining equation of A over C. Consider F := {(x, y , z) ∈ C3 | Q(x, y , z) = 1} the Milnor fiber of A. If A supports a k -net structure then dim H1(F )e2πi/k ≥ k − 2. Corollary If A has a 4-net structure, then dim H(AF2(A), η0) ≥ 2.