[Paper Review] Rank one local systems on complements of hyperplanes and Aomoto complexes

Morihiko Saito|arXiv (Cornell University)|Jul 1, 2018

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper establishes conditions under which the cohomology of rank one local systems on complements of hyperplane arrangements can be computed via the Aomoto complex, even when the classical non-resonance condition fails. By analyzing Hodge-logarithmic differential forms via embedded resolutions, the authors prove a new criterion for bijectivity of the Aomoto map in degree one, enabling explicit computation of monodromy eigenspaces—demonstrated by proving the vanishing of the $\lambda = \exp(-2\pi i/6)$ eigenspace in the first Milnor cohomology of the $G_{31}$ reflection arrangement without computational tools.

ABSTRACT

We show that the cohomology of a rank 1 local system on the complement of a projective hyperplane arrangement can be calculated by the Aomoto complex in certain cases even if the condition on the sum of the residues of connection due to Esnault et al is not satisfied. For this we have to study the localization of Hodge-logarithmic differential forms which are defined by using an embedded resolution of singularities. As an application we can compute certain monodromy eigenspaces of the first Milnor cohomology group of the defining polynomial of the reflection hyperplane arrangement of type $G_{31}$ without using a computer.

Motivation & Objective

• To extend the applicability of the Aomoto complex to compute cohomology of rank one local systems on hyperplane complements beyond the classical non-resonance condition.
• To overcome the limitation of the condition $\alpha_Z \notin \mathbb{Z}_{>0}$ for dense edges, which restricts the use of Aomoto complexes in complex arrangements.
• To provide a purely analytical method—using Hodge-logarithmic differential forms and embedded resolutions—for computing monodromy eigenspaces in Milnor cohomology.
• To demonstrate the method by proving the vanishing of the $\lambda = \exp(-2\pi i/6)$ eigenspace in the first Milnor cohomology of the $G_{31}$ reflection arrangement without relying on computer algebra.

Proposed method

• The authors use Hodge-logarithmic differential forms defined via embedded resolution of singularities to analyze the cohomology of rank one local systems on the complement of a hyperplane arrangement.
• They introduce a refined criterion for the Aomoto complex to compute cohomology, based on conditions on residues $\alpha_k$ and the structure of singular points.
• The method involves decomposing the arrangement into 'good' and 'bad' indices $I^c$ and $I^b$, where $I^b$ corresponds to points violating the non-resonance condition.
• The key technical tool is the study of the localization of Hodge-logarithmic forms at singular points, particularly those of multiplicity at least 3.
• The proof relies on topological arguments, including the weak Lefschetz theorem and Euler characteristic considerations, to relate dimensions of cohomology groups.
• The method is applied to the $G_{31}$ reflection arrangement by verifying a set of combinatorial and geometric conditions (a)–(d) on a general hyperplane section.

Experimental results

Research questions

• RQ1 Can the Aomoto complex compute the cohomology of rank one local systems on hyperplane complements when the classical non-resonance condition fails?
• RQ2 What geometric and combinatorial conditions on the arrangement and local system connection residues allow the Aomoto complex to still compute the correct cohomology?
• RQ3 Is it possible to compute monodromy eigenspaces of the first Milnor cohomology group for the $G_{31}$ reflection arrangement without computer assistance?
• RQ4 How can Hodge-logarithmic differential forms and embedded resolutions be used to analyze local system cohomology in the presence of resonance?
• RQ5 Can the vanishing of the $\lambda = \exp(-2\pi i/6)$ eigenspace in the first Milnor cohomology of $G_{31}$ be established analytically via the Aomoto complex?

Key findings

• The Aomoto complex computes the first cohomology of a rank one local system on the complement of a projective hyperplane arrangement in $\mathbb{P}^2$ under a new set of conditions (a)–(d), even when the classical non-resonance condition fails.
• The vanishing of the $\lambda = \exp(-2\pi i/6)$ monodromy eigenspace in the first Milnor cohomology of the $G_{31}$ reflection arrangement is established without computer computation.
• The method applies to the $G_{31}$ arrangement by verifying that conditions (a)–(d) hold for a general hyperplane section, including the connectedness of $X^{I^c} \setminus (X^{I^c(0)} \cup X^{I^b})$ and the multiplicity condition on lines.
• The proof relies on the structure of the intersection poset of $G_{31}$, with $|I^b| = 16$ and $|I^c| = 44$, and the existence of at least two ordinary double points on $X_d$.
• The method also applies to $\lambda = -1$, but fails for $\lambda = \exp(\pm 2\pi i/3)$ due to the high number of triple points violating the required conditions.
• The authors confirm that the Aomoto map is bijective in degree one and that the dimensions of the cohomology groups match, validating the method via Euler characteristic arguments.