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

By B. Z. Shapiro


Abstract. Consider an affine algebraic variety M = C n \ S k i=0 Li, where Li are affine complex hyperplanes. We show that the mixed Hodge structure of M is similar to that of the complex torus C ∗ × · · · × C ∗ i.e. any element in H ∗ (M,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. All cohomology below will have complex coefficients. Theorem. For the complement M(A) to an arbitrary arrangement of affine complex hyperplanes any element of Hi (M) has the Hodge type (i,i). Before the beginning of the proof we want to give some examples and make some remarks. Example 1. Any 1-dimensional complement M(A) to an arrangement is a complex line punctured at several points (p1,...,pk). In this case H1 (M) consists of classes of weight 2 represented by linear combinations of the form dz z−p

Year: 1993
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