[Paper Review] $β$-nbc bases for cohomology of local systems on hyperplane complements

Michael Falk, Hiroaki Terao|arXiv (Cornell University)|Dec 12, 1994

Algebraic structures and combinatorial models4 citations

TL;DR

This paper constructs explicit bases for the cohomology of local systems on the complement of hyperplane arrangements using β-nbc (no-broken-circuit) bases, under nonresonance conditions. The bases are built from logarithmic differential forms indexed by nbc bases of the arrangement, and the transition matrices between such bases are integer-valued and independent of the local system, providing a canonical combinatorial structure for the cohomology group.

ABSTRACT

We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes $\A$. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement $\A$. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of $\A$. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.

Motivation & Objective

• To provide explicit, combinatorially defined bases for the cohomology of rank one local systems on the complement of hyperplane arrangements.
• To establish that these bases are constructed from logarithmic differential forms and indexed by no-broken-circuit (nbc) bases of the arrangement.
• To show that transition matrices between such bases are integer-valued and depend only on the linear ordering of hyperplanes, not on the local system.
• To demonstrate the existence of monomial logarithmic form bases in special cases.

Proposed method

• The construction uses a linear ordering of the hyperplanes in the arrangement to define β-nbc bases.
• Logarithmic differential forms are used as the building blocks for the cohomology basis.
• The nonresonance condition ensures that the only non-vanishing cohomology group is in a specific degree.
• The basis is indexed by nbc bases of the arrangement, which are combinatorial objects derived from the hyperplane structure.
• Transition matrices between different β-nbc bases are shown to be integer matrices independent of the local system.
• The method combines known results from algebraic geometry, arrangement theory, and cohomological algebra to achieve explicit bases.

Experimental results

Research questions

• RQ1 How can one construct an explicit basis for the cohomology of a local system on the complement of a hyperplane arrangement?
• RQ2 What combinatorial structure governs the indexing of such cohomology bases?
• RQ3 How do different bases constructed via different linear orderings relate to each other?
• RQ4 Under what conditions does the cohomology admit a basis of monomial logarithmic forms?
• RQ5 Are the transition matrices between such bases independent of the local system?

Key findings

• The cohomology group is non-vanishing in only one degree under the nonresonance condition, and explicit bases are constructed for this group.
• The bases are formed from logarithmic differential forms indexed by β-nbc bases of the arrangement.
• The transition matrices between bases constructed from different linear orderings are integer matrices that depend only on the orderings, not on the local system.
• In special cases, such as when the local system is trivial or satisfies certain monodromy conditions, the basis can be chosen from monomial logarithmic forms.
• The construction provides a canonical, combinatorially defined structure for the cohomology, linking topology, algebra, and combinatorics.