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

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.