Admissible local systems for a class of line arrangements

Abstract

A rank one local system L on a smooth complex algebraic variety M is admissible roughly speaking if the dimension of the cohomology groups H m (M, L) can be computed directly from the cohomology algebra H * (M, C).We say that a line arrangement A is of type C k if k ≥ 0 is the minimal number of lines in A containing all the points of multiplicity at least 3. We show that if A is a line arrangement in the classes C k for k ≤ 2, then any rank one local system L on the line arrangement complement M is admissible. Partial results are obtained for the class C 3 .

Admissibility of Local Systems for some Classes of Line Arrangements

“…Namely, for non-negative integer k, the line arrangement A is said to be of type C k if k is the minimal number of lines in A containing all the points of multiplicity at least 3. The following theorem was proved in [17].…”

“…In a recent paper [17], another class of line arrangements A was introduced for which all rank one local systems on the complements are admissible. Namely, for non-negative integer k, the line arrangement A is said to be of type C k if k is the minimal number of lines in A containing all the points of multiplicity at least 3.…”

“…We then prove Theorem 1.4. At the end of Section 2 we give an example of a line arrangement where the results in [7] and [17] cannot be applied, while Theorem 1.4 is useful (Example 2.5).…”

Characteristic Varieties for a Class of Line Arrangements

“…In fact, to determine these varieties, we use a recent result by S. Nazir and Z. Raza [12], saying that in such a situation all rank one local systems are admissible (see Definition 2.1). A consequence of this fact is that properties (i) and (ii) hold, as shown in [5].…”

Resonance varieties and Dwyer–Fried invariants

“…Then Ω 1 r (A) = σ r (R 1 (A, Q)) ∁ , for all 1 ≤ r ≤ n. Proof. As shown by Nazir and Raza in [23], the characteristic variety V 1 (A) of such an arrangement has no translated components. The conclusion follows from Theorem 11.2.…”