arrangements - J. Math. Kyoto Univ. (JMKYAZ) 47-1 (2007),...

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Unformatted text preview: J. Math. Kyoto Univ. (JMKYAZ) 47-1 (2007), 000–000 Logarithmic sheaves attached to arrangements of hyperplanes By Igor V. Dolgachev ** 1. Introduction Any divisor D on a nonsingular variety X defines a sheaf of logarithmic differential forms Ω 1 X (log D ). Its equivalent definitions and many useful prop- erties are discussed in a fundamental paper of K. Saito [Sa]. This sheaf is locally free when D is a strictly normal crossing divisor, and in this situation it is a part of the logarithmic De Rham complex used by P. Deligne to define the mixed Hodge structure on the cohomology of the complement X \ D . In the theory of hyperplane arrangements this sheaf arises when D is a central arrangement of hyperplanes in C n +1 . In exceptional situations this sheaf could be free (a free arrangement), for example, when n = 2 or the arrangement is a complex reflection arrangement. Many geometric properties of the vector bundle Ω 1 X (log D ) were studied in the case when D is a generic arrangement of hyperplanes in P n [DK1]. Among these properties is a Torelli type theo- rem which asserts that two arrangements with isomorphic vector bundles of logarithmic 1-forms coincide unless they osculate a normal rational curve. In this paper we introduce and study a certain subsheaf ˜ Ω 1 X (log D ) of Ω 1 X (log D ). This sheaf contains as a subsheaf (and coincides with it in the case when the divisor D is the union of normal irreducible divisors) the sheaf of logarithmic differentials considered earlier in [CHKS]. Its double dual is isomorphic to Ω 1 X (log D ). The sheaf ˜ Ω 1 X (log D ) is locally free only if the divisor D is locally formally isomorphic to a strictly normal crossing divisor. This disadvantage is compensated by some good properties of this sheaf which Ω 1 X (log D ) does not posses in general. For example, one has always a residue exact sequence → Ω 1 X → ˜ Ω 1 X (log D ) → ν * O D → , where ν : D → D is a resolution of singularities of D . Also, in the case when D is an arrangement of m hyperplanes in P n , the sheaf ˜ Ω 1 P n (log D ) admits a Received May 7, 2006 Revised December 25, 2006 * University of Michigan ** Research partially supported by NSF grant 0245203 2 Author simple projective resolution → O P n (- 1) m- n- 1 → O m- 1 P n → ˜ Ω 1 P n (log D ) → . In particular, its Chern polynomial does not depend on the combinatorics of the arrangement. This allows us to introduce the notion of a stable (resp. semi-stable, unstable) arrangement and define a map from the space of semi- stable arrangements to the moduli space of coherent torsion-free sheaves on P n with fixed Chern numbers. All generic arrangements are semi-stable (and stable when m ≥ n + 2), and the Torelli Theorem mentioned above shows that the variety of semi-stable arrangements admits a birational morphism onto a subvariety of the moduli space of sheaves. We extend the Torelli theorem proving the injectivity on the set of semi-stable arrangements which contain a...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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arrangements - J. Math. Kyoto Univ. (JMKYAZ) 47-1 (2007),...

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