The decomposition of forms and cohomology of generalized complex manifolds

Gil R. Cavalcanti

doi:10.1016/j.geomphys.2006.02.006

The decomposition of forms and cohomology of generalized complex manifolds

Gil R. Cavalcanti^*

^*Corresponding author for this work

Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, United Kingdom

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We study the decomposition of forms induced by a generalized complex structure giving a complete description of the bundles involved and, around regular points, of the operators ∂ and over(∂, -) associated to the generalized complex structure. We prove that if the generalized ∂ over(∂, -)-lemma holds then the decomposition of forms gives rise to a decomposition of the cohomology of the manifold, H^• (M) = ⊕_{- n}ⁿ G H^k (M), and the canonical spectral sequence degenerates at E₁. We also show that if the generalized ∂ over(∂, -)-lemma holds, any generalized complex submanifold can be associated to a Poincaré dual cohomology class in the middle cohomology space G H⁰ (M).

Original language	English
Pages (from-to)	121-132
Number of pages	12
Journal	Journal of Geometry and Physics
Volume	57
Issue number	1
DOIs	https://doi.org/10.1016/j.geomphys.2006.02.006
Publication status	Published - 31 Dec 2006
Externally published	Yes

Bibliographical note

Funding Information:
I would like to thank Marco Gualtieri for many useful conversations and Nigel Hitchin for his guidance and so many suggestions which improved this paper greatly. This research was partially supported by CAPES, grant 1326/99-6, and EPSRC, grant EP/C525124/1.

Funding

I would like to thank Marco Gualtieri for many useful conversations and Nigel Hitchin for his guidance and so many suggestions which improved this paper greatly. This research was partially supported by CAPES, grant 1326/99-6, and EPSRC, grant EP/C525124/1.

Keywords

Algebraic and differential topology
Cohomology decomposition
Generalized complex geometry

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10.1016/j.geomphys.2006.02.006

Cite this

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title = "The decomposition of forms and cohomology of generalized complex manifolds",

abstract = "We study the decomposition of forms induced by a generalized complex structure giving a complete description of the bundles involved and, around regular points, of the operators ∂ and over(∂, -) associated to the generalized complex structure. We prove that if the generalized ∂ over(∂, -)-lemma holds then the decomposition of forms gives rise to a decomposition of the cohomology of the manifold, H• (M) = ⊕- nn G Hk (M), and the canonical spectral sequence degenerates at E1. We also show that if the generalized ∂ over(∂, -)-lemma holds, any generalized complex submanifold can be associated to a Poincar{\'e} dual cohomology class in the middle cohomology space G H0 (M).",

keywords = "Algebraic and differential topology, Cohomology decomposition, Generalized complex geometry",

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note = "Funding Information: I would like to thank Marco Gualtieri for many useful conversations and Nigel Hitchin for his guidance and so many suggestions which improved this paper greatly. This research was partially supported by CAPES, grant 1326/99-6, and EPSRC, grant EP/C525124/1.",

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N2 - We study the decomposition of forms induced by a generalized complex structure giving a complete description of the bundles involved and, around regular points, of the operators ∂ and over(∂, -) associated to the generalized complex structure. We prove that if the generalized ∂ over(∂, -)-lemma holds then the decomposition of forms gives rise to a decomposition of the cohomology of the manifold, H• (M) = ⊕- nn G Hk (M), and the canonical spectral sequence degenerates at E1. We also show that if the generalized ∂ over(∂, -)-lemma holds, any generalized complex submanifold can be associated to a Poincaré dual cohomology class in the middle cohomology space G H0 (M).

AB - We study the decomposition of forms induced by a generalized complex structure giving a complete description of the bundles involved and, around regular points, of the operators ∂ and over(∂, -) associated to the generalized complex structure. We prove that if the generalized ∂ over(∂, -)-lemma holds then the decomposition of forms gives rise to a decomposition of the cohomology of the manifold, H• (M) = ⊕- nn G Hk (M), and the canonical spectral sequence degenerates at E1. We also show that if the generalized ∂ over(∂, -)-lemma holds, any generalized complex submanifold can be associated to a Poincaré dual cohomology class in the middle cohomology space G H0 (M).

KW - Algebraic and differential topology

KW - Cohomology decomposition

KW - Generalized complex geometry

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DO - 10.1016/j.geomphys.2006.02.006

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EP - 132

JO - Journal of Geometry and Physics

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ER -

The decomposition of forms and cohomology of generalized complex manifolds

Abstract

Bibliographical note

Funding

Keywords

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