[Paper Review] The non-vanishing cohomology of Orlik-Solomon algebras

Yukihito Kawahara|ArXiv.org|Jun 21, 2005

Algebraic structures and combinatorial models8 references4 citations

TL;DR

This paper constructs matroids and hyperplane arrangements with non-vanishing cohomology in the Orlik-Solomon algebra by leveraging decomposable relations derived from Latin hypercubes. It proves that for certain weights, the cohomology $ H^{ ext{top}-1}(A(M), e_ u) $ does not vanish, generalizing earlier results on Latin squares and extending the theory to higher-dimensional arrangements.

ABSTRACT

The cohomology on the complement of hyperplanes with the coefficients in the rank one local system associated to a generic weight vanishes except in the highest dimension. In this paper, we construct matroids or arrangements and its weights with non-vanishing cohomology of Orlik-Solomon algebras, using decomposable relations arising from Latin hypercubes.

Motivation & Objective

• To construct matroids and hyperplane arrangements with non-vanishing cohomology in the Orlik-Solomon algebra, contrary to the generic vanishing theorem.
• To generalize previous results on Latin squares and first cohomology to higher-dimensional arrangements using Latin hypercubes.
• To establish a connection between decomposable relations in Orlik-Solomon algebras and the combinatorics of Latin hypercubes.
• To provide explicit examples of realizations of such matroids, including classical configurations like the Hessian configuration and monomial arrangements.
• To demonstrate that the cohomology $ H^{ ext{top}-1}(A(M), e_ u) $ can be non-zero for non-generic weights, thus extending the resonance variety theory.

Proposed method

• Constructs an $ oldsymbol{ ext{ℓ}} $-generic matroid $ M[K] $ on $ n = (\ell+1)m $ elements from a Latin $ \ell $-dimensional hypercube $ K $ of order $ m $.
• Defines the family of $ (\ell+1) $-circuits $ \mathcal{C}[K] $ using the hypercube entries to encode circuit relations in the matroid.
• Uses the Orlik-Solomon algebra $ A(M) $ and the complex $ (A(M), e_\lambda) $ with weight $ \lambda $, where $ e_\lambda = \sum \lambda_i e_i $.
• Applies the decomposition $ H^{k}(A(M), e_\lambda) \cong H^{k}(\partial_M A(M), e_\lambda) \oplus H^{k-1}(\partial_M A(M), e_\lambda) $ to analyze cohomology.
• Leverages decomposable relations arising from Latin hypercubes to produce non-trivial cohomology in degree $ \ell-1 $.
• Verifies non-vanishing cohomology via explicit constructions and known realizations, including the Hessian configuration and monomial arrangements.

Experimental results

Research questions

• RQ1 Can non-vanishing cohomology in the Orlik-Solomon algebra be systematically constructed beyond the generic case?
• RQ2 How do Latin hypercubes generate decomposable relations that lead to non-trivial cohomology in $ H^{\ell-1}(A(M), e_\lambda) $?
• RQ3 What is the role of $ \ell $-genericity in ensuring the matroid structure supports non-vanishing cohomology?
• RQ4 Can classical arrangements like the Hessian configuration and monomial arrangements be understood through this hypercube-based construction?
• RQ5 What is the dimension of the first cohomology for matroids derived from orthogonal Latin hypercubes?

Key findings

• For a Latin $ \ell $-dimensional hypercube $ K $ of order $ m \geq 2 $, the associated matroid $ M[K] $ on $ n = (\ell+1)m $ elements has non-vanishing cohomology $ H^{\ell-1}(A(M[K]), e_\lambda) \neq 0 $ for certain weights $ \lambda $.
• The Hessian configuration of 12 lines in $ \mathbb{P}^2 $, realized as $ M[K_1,K_2] $, has $ \dim H^1(A(M[K_1,K_2]), e_\lambda) = 2 $ for non-zero $ \lambda $ with $ \sum \lambda_j = 0 $.
• The monomial arrangement $ \mathcal{A}_{m,m,3} $ has weights with non-vanishing first cohomology, as shown via the Latin square construction.
• For $ \ell = 3 $, the matroid $ M[K] $ from a $ 2 \times 2 \times 2 $ Latin hypercube is realizable as a 4-arrangement with non-vanishing second cohomology.
• The arrangement $ \mathcal{B} $ with defining polynomial $ (x_1 - x_2)(x_1 + x_2)\cdots $ realizes $ M[K] $ with additional 4-circuits and also has non-vanishing $ H^2 $.
• The construction generalizes Rybnikov’s relation and provides a systematic method to produce non-vanishing cohomology in the Orlik-Solomon algebra.