Deformations of log-symplectic structures

Mărcuţ, Ioan; Torres, Boris Osorno

doi:10.1112/jlms/jdu023

Cited by 32 publications

(23 citation statements)

References 12 publications

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“…The main insight behind the above results is that a stable generalized complex structure is equivalent to a complex log symplectic form, a complex 2-form with a type of logarithmic singularity along the divisor D and whose imaginary part defines an elliptic symplectic form, which is a symplectic form but for a Lie algebroid which we introduce called the elliptic tangent bundle. This approach, analogous to that taken in holomorphic log symplectic geometry [17] as well as in the recent development of real log symplectic geometry [26,24,31,32,7], justifies the intuition that a stable generalized complex structure is a type of singular symplectic structure, and it allows us to apply symplectic techniques such as Moser interpolation. For this reason, we carefully develop the theory of logarithmic and elliptic forms associated to smooth codimension 2 submanifolds.…”

Section: Introductionmentioning

confidence: 68%

“…As observed in [32], we may factor the morphism (4.2) through the inclusion of forms into logarithmic forms, and the resulting morphism…”

Section: Real Poisson Deformationsmentioning

confidence: 99%

“…The L ∞ quasi-isomophism used in Section 4.1 then also factors through the inclusion of forms into logarithmic forms, inducing the same map on Maurer-Cartan elements (4.4), but for B ∈ Ω 2 (M, log Z). Composing these two quasi-isomorphisms, we render the deformation complex formal, leading to the identification made in [32] of the local moduli space of log symplectic structures with an open neighbourhood of [π −1 ] in the second logarithmic cohomology group, which by the theorem of Mazzeo-Melrose [33] is canonically isomorphic to H 2 (M, R) ⊕ H 1 (Z, R).…”

Section: Real Poisson Deformationsmentioning

confidence: 99%

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Stable generalized complex structures

Cavalcanti

Gualtieri

2017

Proc. London Math. Soc.

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A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi-Yau structure. We introduce a Lie algebroid which allows us to view such structures as symplectic forms. This allows us to construct new examples of stable structures, and also to define period maps for their deformations in which the background three-form flux is either fixed or not, proving the unobstructedness of both deformation problems. We then use the same tools to establish local normal forms for the degeneracy locus and for Lagrangian branes. Applying our normal forms to the four-dimensional case, we prove that any compact stable generalized complex 4-manifold has a symplectic completion, in the sense that it can be modified near its degeneracy locus to produce a compact symplectic 4-manifold

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Section: Introductionmentioning

confidence: 68%

“…As observed in [32], we may factor the morphism (4.2) through the inclusion of forms into logarithmic forms, and the resulting morphism…”

Section: Real Poisson Deformationsmentioning

confidence: 99%

Section: Real Poisson Deformationsmentioning

confidence: 99%

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Stable generalized complex structures

Cavalcanti

Gualtieri

2017

Proc. London Math. Soc.

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“…A thorough study of stable generalized complex manifolds was initiated in . Stable generalized complex structures can be seen as the generalized geometric analogue of a log‐symplectic structure, which is a type of mildly degenerate Poisson structure that has recently received a lot of attention .…”

Section: Introductionmentioning

confidence: 99%

Fibrations and stable generalized complex structures

Cavalcanti

Klaasse

2018

Proc. London Math. Soc.

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A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic techniques adapted to Lie algebroids, and use these to construct stable generalized complex structures out of log-symplectic structures. In particular we introduce the notion of a boundary Lefschetz fibration for this purpose and describe how they can be obtained from genus one Lefschetz fibrations over the disc. ContentsLet X be a 2n-dimensional manifold equipped with a closed three-form H ∈ Ω 3 cl (X). Recall that the double tangent bundle TX := T X ⊕ T * X is a Courant algebroid whose anchor is the projection p : TX → T X. It carries a natural pairing V + ξ,This takes (TX, H) to (TX, H + dB), leading to theŠevera class [H] ∈ H 3 (X; R) determining TX up to Courant isomorphism. The Courant automorphisms of TX are generated by the diffeomorphisms and closed B-field transformations.Definition 2.1. A generalized complex structure on (X, H) is a complex structure J on TX that is orthogonal with respect to ·, · , and whose +i-eigenbundle is involutive underThere is an alternative definition of a generalized complex structure using spinors. To state it, recall that sections v = V + ξ ∈ Γ(TX) of the double tangent bundle act on differential forms via Clifford multiplication, given byDefinition 2.2. A generalized complex structure on (X, H) is given by a complex line bundle K J ⊂ ∧ • T * C X pointwise generated by a differential form ρ = e B+iω ∧ Ω with Ω a decomposable complex k-form, satisfying Ω ∧ Ω ∧ ω n−k = 0, and such that dρ + H ∧ ρ = v · ρ for any local section ρ ∈ Γ(K J ) and some v ∈ Γ(TX).Both definitions are related using that K J = Ann(E J ) is the annihilator under the Clifford action of E J , the +i-eigenbundle of J . The bundle K J is called the canonical bundle of J . For later use, we further introduce the analogue of a Calabi-Yau manifold in generalized geometry. Denote by d H = d + H∧ the H-twisted de Rham differential.Example 2.4. The following provide examples of generalized complex structures on (X, 0).• Let ω be a symplectic structure on X. Then K Jω := e iω defines a generalized complex structure J ω . • Let J be a complex structure on X with canonical bundle K J = ∧ n,0 T * X. Then K JJ := K J defines a generalized complex structure J J . • Let P ∈ Γ(∧ 2,0 T X) a holomorphic Poisson structure with respect to a complex structure J. Then K JP,J := e P K J defines a generalized complex structure J P,J .The automorphisms J ω , J J and J P,J are given by, with π = Im(P ):We next introduce the type of a generalized complex structure J , which colloquially provides a measure for how many complex directions there are. The type is an integer-valued upper semicontinuous function on X whose parity is locally constant.Definition 2.5. Let J be a generalized complex structure on X. The type of J is a m...

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“…[16,38]. For ∧ • T X , this can be found, for example, in [50,Lemma 3]. Alternatively, one can show that the corresponding D-modules are isomorphic: at any point of Y \ Y sing , Weinstein's splitting theorem [64] reduces the problem to the case of Poisson surfaces, where it follows by a direct calculation using Proposition 3.2, and the fact that all three D-modules have length two.…”

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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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Deformations of log-symplectic structures

Abstract: We describe the space of Poisson bivectors near a log-symplectic structure up to small diffeomorphisms.

Cited by 32 publications

References 12 publications

Stable generalized complex structures

Stable generalized complex structures

Fibrations and stable generalized complex structures

Holonomic Poisson Manifolds and Deformations of Elliptic Algebras

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