This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: J. Math. Kyoto Univ. (JMKYAZ) 471 (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 1forms 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. semistable, unstable) arrangement and define a map from the space of semi stable arrangements to the moduli space of coherent torsionfree sheaves on P n with fixed Chern numbers. All generic arrangements are semistable (and stable when m ≥ n + 2), and the Torelli Theorem mentioned above shows that the variety of semistable 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 semistable arrangements which contain a...
View
Full
Document
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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Math

Click to edit the document details