The Orlik-Solomon model for hypersurface arrangements

Abstract

We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the configuration spaces of points on curves are also studied.for a more precise statement). Here H • (C n \ L) denotes the cohomology of the complement C n \ L with rational coefficients. Theorem 1.1 (Brieskorn-Orlik-Solomon). We have an isomorphism of graded algebrasOne may define an Orlik-Solomon algebra A • (L) for L any hypersurface arrangement inside a complex manifold X. We still have a direct sum decomposition (1), with S • (L) the graded poset of strata of L, as well as product maps (2) and natural morphisms (3). As in the local case, the Orlik-Solomon algebra A • (L) only depends on the poset of strata of L. It is functorial with respect to (X, L) in the sense that any holomorphic map ϕ :• Geometry: the Gysin long exact sequence. For a smooth hypersurface V inside a smooth projective variety X over C, the Gysin morphisms of the inclusion V ⊂ X are the morphisms H k−2 (V )(−1) → H k (X) , where (−1) denotes a Tate twist, obtained as the Poincaré duals of the natural morphisms H 2n−k (X) → H 2n−k (V ) where n = dim C (X). They fit into a long exact sequence, called the Gysin long exact sequence:It is worth noting that the connecting homomorphisms H k (X \ V ) → H k−1 (V )(−1) are residue morphisms, which are easily described using logarithmic forms. We can now state our main theorem (see Theorem 4.8 for more precise statements). Theorem 1.2. Let X be a smooth projective variety over C and L be a hypersurface arrangement in X.(1) For integers q and n let us considerwhere (n − q) is a Tate twist, viewed as a pure Hodge structure of weight q. Then the direct sumhas the structure of a differential graded algebra (dga) in the (semi-simple) category of split mixed Hodge structures over Q. The product in M • (X, L) is induced by the product maps (2) of the Orlik-Solomon algebra and the cup-product on the cohomology of the strata. The differential in M • (X, L) is induced by the natural morphisms (3) and the Gysin morphismsof the inclusions of strata S ⊂ S ′ . The dga M • (X, L) is functorial with respect to (X, L) in the sense explained above.(2) The dga M • (X, L) is a model for the cohomology of X \ L in the following sense: we have isomorphisms of pure Hodge structures over Q L)) which are compatible with the algebra structures, and functorial with respect to (X, L).The precise definition of the Orlik-Solomon model M • (X, L) is given in §4.4. Theorem 1.2 generalizes the case of normal crossing divisors, which is due to P. Del...

Relative cohomology of bi-arrangements

“…In the special case of arrangements, we recover the classical Brieskorn-Orlik-Solomon theorem in the local context, and its global counterpart proved by Looijenga [Loo93] (see also [Dup15]) in the global context.…”

“…Up to a shift, it is the same as the Gysin complex defined in [Dup15]. Dually, the geometric OrlikSolomon bi-complexes for (A , µ) are concentrated in bi-degrees (0, n) with…”

“…which was first defined in [Loo93] and studied in [Dup15] in the context of logarithmic differential forms and mixed Hodge theory.…”

“…We sketch the proof for the case M = ∅ (see [Dup15,Theorem 5.5] for more details); the general case is similar if one uses the complexes of Remark A.3. In this special case, the spectral sequence is Deligne's spectral sequence (A.7) E −p,q 1…”

“…which was first defined in [Loo93] and studied in [Dup15] in the context of logarithmic differential forms and mixed Hodge theory. This is a global generalization of the Brieskorn-Orlik-Solomon theorem [Bri73,OS80], which corresponds to an arrangement of hyperplanes A in X = C n and states that there is an isomorphism…”