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The mixed Hodge structure of the complement to an arbitrary arrangement of affine complex hyperplanes is pure

Author: B. Z. Shapiro
Journal: Proc. Amer. Math. Soc. 117 (1993), 931-933
MSC: Primary 32S35; Secondary 52B30
MathSciNet review: 1131042
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Abstract: Consider an affine algebraic variety $ \mathcal{M} = {{\mathbf{C}}^n}\backslash \bigcup\nolimits_{i = 0}^k {{L_i}} $, where $ {L_i}$ are affine complex hyperplanes. We show that the mixed Hodge structure of $ \mathcal{M}$ is similar to that of the complex torus $ {{\mathbf{C}}^{\ast}} \times \cdots \times {{\mathbf{C}}^{\ast}}$, i.e., any element in $ {H^{\ast}}(\mathcal{M},{\mathbf{C}})$ has the Hodge type $ (i,i)$. This is another example of the similarity of the properties of complements to arrangements and affine toric varieties.

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Keywords: Arrangements of hyperplanes, mixed Hodge structure
Article copyright: © Copyright 1993 American Mathematical Society