Stable generalized complex structures

Cavalcanti, Gil R.; Gualtieri, Marco

doi:10.1112/plms.12093

Cited by 24 publications

(84 citation statements)

References 38 publications

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“…We will show that, in analogy to Eq. (7), the closure of Graph ω is the direct sum of ker ω and of the graph of Π restricted to the annihilator of ker ω.…”

Section: 4mentioning

confidence: 99%

Removable Presymplectic Singularities and the Local Splitting of Dirac Structures

Blohmann

2016

Int Math Res Notices

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We call a singularity of a presymplectic form ω removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the removability of singularities is given in terms of regularizing functions for pure spinors. All removable singularities are poles in the sense that the norm of ω is not locally bounded. The points at which removable singularities occur are the non-regular points of the Dirac structure for which we prove a general splitting theorem: Locally, every Dirac structure is the gauge transform of the product of a tangent bundle and the graph of a Poisson structure. This implies that in a neighborhood of a removable singularity ω can be split into a non-singular presymplectic form and a singular presymplectic form which is the partial inverse of a Poisson bivector that vanishes at the singularity. An interesting class of examples is given by log-Dirac structures which generalize logsymplectic structures. The analogous notion of removable singularities of Poisson structures is also studied.

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“…We will show that, in analogy to Eq. (7), the closure of Graph ω is the direct sum of ker ω and of the graph of Π restricted to the annihilator of ker ω.…”

Section: 4mentioning

confidence: 99%

Removable Presymplectic Singularities and the Local Splitting of Dirac Structures

Blohmann

2016

Int Math Res Notices

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“…These simple obstructions tell us for example that there are no type one generalized Calabi-Yau structures on products of spheres of dimension bigger than 1. Notice however that the manifolds S 1 × S 3 and S 1 × S 5 do admit a type one generalized complex structure with topologically trivial canonical bundle [4] but, by the results above, these are not generalized Calabi-Yau.…”

mentioning

confidence: 95%

“…The type of a generalized complex structure on M is an integer-valued lower semi-continuous function with locally constant parity. In the generic even case, generalized complex structures may be viewed as symplectic structures with singularities along loci where the type jumps from 0 to 2 or higher; when these singular loci are required to satisfy a transversality condition, we have stable generalized complex structures, which were studied by Cavalcanti and Gualtieri in [4]. If, instead, we are in the case of odd type, a generic generalized complex structure would be of type 1 almost everywhere; very little is currently known about type 1 structures.…”

Section: Introductionmentioning

confidence: 99%

Type one generalized Calabi–Yaus

Bailey

Cavalcanti

Gualtieri

2017

Journal of Geometry and Physics

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We study type one generalized complex and generalized Calabi-Yau manifolds. We introduce a cohomology class that obstructs the existence of a globally defined, closed 2-form which agrees with the symplectic form on the leaves of the generalized complex structure, the twisting class. We prove that in a compact, type one, 4n-dimensional generalized complex manifold the Euler characteristic must be even and equal to the signature modulo four. The generalized Calabi-Yau condition places much stronger constrains: a compact type one generalized Calabi-Yau fibers over the 2-torus and if the structure has one compact leaf, then this fibration can be chosen to be the fibration by the symplectic leaves of the generalized complex structure. If the twisting class vanishes, one can always deform the structure so that it has a compact leaf. Finally we prove that every symplectic fibration over the 2-torus admits a type one generalized Calabi-Yau structure. * Utrecht University; m.a.bailey@uu.nl the symplectic leaves are the fibers of the fibration. As a special case, we obtain a correspondence in four dimensions: a compact four-manifold admits a type 1 generalized Calabi-Yau structure if and only if it is an oriented fibration over T 2 . These results may be viewed as the generalized complex analogues of the results obtained by Guillemin, Miranda and Pires for codimension one Poisson structures [6].

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“…Among all type-changing generalized complex structures, one kind seems to deserve special attention: stable generalized complex structures. These are the structures whose canonical section of the anticanonical bundle vanishes transversally along a codimension-two submanifold, D, endowing it with the structure of an elliptic divisor in the language of [8]. Consequently, the type of such a structure is 0 on X \ D, while on D it is equal to two.…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, the type of such a structure is 0 on X \ D, while on D it is equal to two. Many examples of stable generalized complex structures were produced in dimension four [7,10,15,16] and a careful study was carried out in [8]. One of the outcomes of that study was that it related stable generalized complex structures to symplectic structures on a certain Lie algebroid.…”

Section: Introductionmentioning

confidence: 99%

Classification of Boundary Lefschetz Fibrations over the Disc

Behrens¹,

Cavalcanti²,

Klaasse³

2018

Geometry and Physics: Volume II

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We show that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to S 1 × S 3 #nCP 2 , #mCP 2 #nCP 2 or #m(S 2 × S 2 ). Given the relation between boundary Lefschetz fibrations and stable generalized complex structures, we conclude that the manifolds S 1 × S 3 #nCP 2 , #(2m + 1)CP 2 #nCP 2 and #(2m+1)S 2 ×S 2 admit stable structures whose type change locus has a single component and are the only four-manifolds whose stable structure arise from boundary Lefschetz fibrations over the disc. arXiv:1706.09207v1 [math.DG] 28 Jun 2017 2 STEFAN BEHRENS GIL R. CAVALCANTI RALPH L. KLAASSE Theorem ([8, Theorem 3.7]). Let D be a co-orientable elliptic divisor on X. Then there is a correspondence between gauge equivalence classes of stable generalized complex structures on X which induce the divisor D, and zero-residue symplectic structures on (X, D).This results paves the way for the use of symplectic techniques to study stable structures. One result that exemplifies that use is the following.

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Stable generalized complex structures

Cited by 24 publications

References 38 publications

Removable Presymplectic Singularities and the Local Splitting of Dirac Structures

Removable Presymplectic Singularities and the Local Splitting of Dirac Structures

Type one generalized Calabi–Yaus

Classification of Boundary Lefschetz Fibrations over the Disc

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