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ARRANGEMENTS AND LOCAL SYSTEMS
 MATHEMATICAL RESEARCH LETTERS 7, 299–316 (2000)
, 2000
"... We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Aomoto complex of the arrangement. Using this result, we establish ..."
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We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Aomoto complex of the arrangement. Using this result, we establish the relationship between the cohomology support loci of the complement and the resonance varieties of the OrlikSolomon algebra for any arrangement, and show that the latter are unions of subspace arrangements in general, resolving a conjecture of Falk. We also obtain lower bounds for the local system Betti numbers in terms of those of the OrlikSolomon algebra, recovering a result of Libgober and Yuzvinsky. For certain local systems, our results provide new combinatorial upper bounds on the local system Betti numbers. These upper bounds enable us to prove that in nonresonant systems the cohomology is concentrated in the top dimension, without using resolution of singularities.
Combinatorial and algebraic structures in OrlikSolomon algebras
 European J. Combin
"... The OrlikSolomon algebra A(G) of a matroid G is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing G. On the other hand, some features of t ..."
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The OrlikSolomon algebra A(G) of a matroid G is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing G. On the other hand, some features of the matroid G are reflected in the algebraic structure of A(G). In this mostly expository article, we describe recent developments in the construction of algebraic invariants of A(G). We develop a categorical framework for the statement and proof of recently discovered isomorphism theorems which suggests a possible setting for classification theorems. Several specific open problems are formulated. The OrlikSolomon algebra of a matroid Let G be a simple matroid with ground set [n]: = {1,..., n}. The OrlikSolomon
A.: The determinant of a hypergeometric period matrix
 Houston J. of Math
, 1998
"... The Euler beta function is an alternating product of Euler gamma functions, B(α, β) = Γ(α) Γ(β) ..."
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The Euler beta function is an alternating product of Euler gamma functions, B(α, β) = Γ(α) Γ(β)
GaussManin connections for arrangements
 I Eigenvalues, Compositio Math. 136 (2003), 299–316; MR 2004a:32042
"... Abstract. We construct a formal connection on the Aomoto complex of an arrangement of hyperplanes, and use it to study the GaussManin connection for the moduli space of the arrangement in the cohomology of a complex rank one local system. We prove that the eigenvalues of the GaussManin connection ..."
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Abstract. We construct a formal connection on the Aomoto complex of an arrangement of hyperplanes, and use it to study the GaussManin connection for the moduli space of the arrangement in the cohomology of a complex rank one local system. We prove that the eigenvalues of the GaussManin connection are integral linear combinations of the weights which define the local system. 1.
Selberg Integral and Multiple Zeta Values
"... In this paper, we show that the coefficients of the Talor expansion of Selberg integrals with respect to its exponent variables are expressed as a linear combinations of multiple zeta values. First object we treat is Selberg integral, the peirod integrals of abelian coverings of the moduli spaces of ..."
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Cited by 4 (0 self)
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In this paper, we show that the coefficients of the Talor expansion of Selberg integrals with respect to its exponent variables are expressed as a linear combinations of multiple zeta values. First object we treat is Selberg integral, the peirod integrals of abelian coverings of the moduli spaces of npoints in
On the cohomology of discriminantal arrangements and OrlikSolomon algebras
, 1998
"... We relate the cohomology of the OrlikSolomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The OrlikSolomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the ..."
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We relate the cohomology of the OrlikSolomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The OrlikSolomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in an arbitrary complex rank one local system. We also establish the relationship between the cohomology support loci of the complement of a discriminantal arrangement and the resonant varieties of its OrlikSolomon algebra.
VANISHING AND BASES FOR COHOMOLOGY OF PARTIALLY TRIVIAL LOCAL SYSTEMS ON HYPERPLANE ARRANGEMENTS
, 2005
"... In this paper we prove a vanishing theorem and construct bases for the cohomology of partially trivial local systems on complements of hyperplane arrangements. As a result, we obtain a nonresonance condition for partially trivial local systems. ..."
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In this paper we prove a vanishing theorem and construct bases for the cohomology of partially trivial local systems on complements of hyperplane arrangements. As a result, we obtain a nonresonance condition for partially trivial local systems.
GaussManin connections for arrangements III, Formal connections
 TRANS. AMER. MATH. SOC
, 2004
"... We study the GaussManin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal GaussManin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal conn ..."
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We study the GaussManin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal GaussManin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to GaussManin connection matrices.
GaussManin Connections for Arrangements, II Nonresonant Weights
, 2002
"... We study the GaussManin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a nonresonant complex rank one local system. Aomoto and Kita determined this GaussManin connection for arrangements in general position. We use their results and an algorithm constr ..."
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We study the GaussManin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a nonresonant complex rank one local system. Aomoto and Kita determined this GaussManin connection for arrangements in general position. We use their results and an algorithm constructed in this paper to determine this GaussManin connection for all arrangements.
Stratified Morse Theory in Arrangements
"... For Robert MacPherson on the occasion of his sixtieth birthday Abstract: This paper is a survey of our work based on the stratified Morse theory of Goresky and MacPherson. First we discuss the Morse theory of Euclidean space stratified by an arrangement. This is used to show that the complement of a ..."
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For Robert MacPherson on the occasion of his sixtieth birthday Abstract: This paper is a survey of our work based on the stratified Morse theory of Goresky and MacPherson. First we discuss the Morse theory of Euclidean space stratified by an arrangement. This is used to show that the complement of a complex hyperplane arrangement admits a minimal cell decomposition. Next we review the construction of a cochain complex whose cohomology computes the local system cohomology of the complement of a complex hyperplane arrangement. Then we present results on the GaussManin connection for the moduli space of arrangements of a fixed combinatorial type in rank one local system cohomology. 1.