Cohomology of the complement of a free divisor

Castro-Jiménez, Francisco-Jesús; Macarro, Luis Narváez; Mond, David

doi:10.1090/s0002-9947-96-01690-x

Cited by 51 publications

(61 citation statements)

References 11 publications

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“…In [6] the logarithmic comparison theorem has been tested for the following non locally quasi-homogeneous plane curve (cf. [9]): D = {f = x ∂ ∂x 1 + (12x…”

Section: 22] the First Author Showed That If D Is A Plane Curve Thementioning

confidence: 99%

Logarithmic cohomology of the complement of a plane curve

Moreno

Mond

Macarro

et al. 2002

Comment. Math. Helv.

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Abstract. Let D, x be a plane curve germ. We prove that the complex Ω • (log D)x computes the cohomology of the complement of D, x only if D is quasihomogeneous. This is a partial converse to a theorem of [5], which asserts that this complex does compute the cohomology of the complement, whenever D is a locally weighted homogeneous free divisor (and so in particular when D is a quasihomogeneous plane curve germ). We also give an example of a free divisor D ⊂ C 3 which is not locally weighted homogeneous, but for which this (second) assertion continues to hold.

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“…In [6] the logarithmic comparison theorem has been tested for the following non locally quasi-homogeneous plane curve (cf. [9]): D = {f = x ∂ ∂x 1 + (12x…”

Section: 22] the First Author Showed That If D Is A Plane Curve Thementioning

confidence: 99%

Logarithmic cohomology of the complement of a plane curve

Moreno

Mond

Macarro

et al. 2002

Comment. Math. Helv.

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“…We now note that since X is a surface, the map Ω • X (log Y) → j * Ω • U is a quasiisomorphism if and only if Y can be written locally as the zero set of a quasihomogeneous polynomial [12,13]. This condition always holds when (X, π) is a projective log symplectic surface, as follows from the classification [41,Section 7] or by examining what happens when we blow up the possible minimal models.…”

Section: Surfacesmentioning

confidence: 99%

Holonomic Poisson Manifolds and Deformations of Elliptic Algebras

Pym¹,

Schedler²

2018

Geometry and Physics: Volume II

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We introduce a natural nondegeneracy condition for Poisson structures, called holonomicity, which is closely related to the notion of a log symplectic form. Holonomic Poisson manifolds are privileged by the fact that their deformation spaces are as finite-dimensional as one could ever hope: the corresponding derived deformation complex is a perverse sheaf. We develop some basic structural features of these manifolds, highlighting the role played by the divergence of Hamiltonian vector fields. As an application, we establish the deformation-invariance of certain families of Poisson manifolds defined by Feigin and Odesskii, along with the "elliptic algebras" that quantize them.

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“…Proposition 6.19. ( Castro-Jiménez, F., Narváez-Macarro, L., & Mond, D. (1996)) Let D be a strongly quasihomogeneous free divisor in the complex manifold X, let U be the complement of D in X, and let j : U → X be inclusion. Then the natural morphism from the complex Ω * X (log D) of differential forms with logarithmic poles along D to R j * C is quasiisomorphism.…”

Section: The Normal Crossing Divisor Casementioning

confidence: 99%