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Introduction to generalized complex geometry. Paper from the 26th Brazilian Mathematics Colloquium – Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 29–August 3, 2007. (English) Zbl 1144.53090

Publicações Matemáticas do IMPA. Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada (IMPA) (ISBN 978-85-244-0254-8/pbk). ii, 91 p., open access (2007).
Generalized complex structures (GCS) were introduced by N. Hitchin [Q. J. Math. 54, No. 3, 281–308 (2003; Zbl 1076.32019)] and developed by Gualtieri [D. Ph. Thesis, Oxford (2003)]. This notion is an extension of Dirac structures and Lie algebroids to incorporate complex geometry. Actually, symplectic and complex geometries can be viewed as extremal cases of the geometry of generalized complex structures. GCS also play a leading role in physical string theory, as supersymmetric flux compactifications, which relate 10-dimensional physics to 4-dimensional worlds. Moreover, it is shown how the concepts of B-field, the extended deformation space of Kontsevich, the coisotropic D-branes of Kapustin, as well as the target space of the bi-Hermitian geometry introduced by Gates, Hull and Rocek can be interpreted in terms of GCS.
This paper consists of four chapters. The first chapter gives some algebraic developments of the theory constructed later. It starts with a section dealing with the GCS defined on vector spaces; it was proved that there exists a decomposition of forms similar to the \((p,q)\)-decomposition of forms in a complex manifold. The Courant bracket is defined in the second section. It allows finding the integrability conditions for a GCS on a manifold. In fact, this is a compatibility condition between the pointwise defined GCS and differential structure. Consequently, the pointwise decomposition of forms induced by GCS gives rise to a decomposition of the exterior derivative \(d=\partial + \overline{\partial}\). It is described how the integrability condition unifies the complex and symplectic geometries. In the 4-th section the basic result of Gualtieri’s deformation theorem of GCS is stated. Two important classes of submanifolds of GCS are also studied. The last section contains interesting examples of GCS.
The author presents in the former section of the second chapter (following closely Gualtieri’s and Witt’s expositions), the concept of generalized metric on a vector space and investigate the consequences of the compatibility of this metric with a GCS. By using a metric compatible with a GCS on \(V\oplus V^{\ast}\) a \(\mathbb{Z}^{2}\)-grading on forms is obtained. In the next section the author generalizes these concepts for manifolds. Given a GCS on an exact Courant algebroid \(\mathcal{E}\) one can find a metric compatible with it and, consequently, a Hermitian structure exists on the algebroid; this corresponds to a reduction of the structure group of the Courant algebroid from \(U(n,n)\) to its maximal compact subgroup \(U(n)\times U(n)\). The existence of such a metric implies that every generalized almost complex manifold has an GCS; this is not integrable in general.
Generalized Kähler structures are defined, too. A bi-Hermitian characterization of a generalized Kähler structure is used for finding nontrivial examples of generalized Kähler manifolds. Then, the Hodge theory on a generalized Kähler manifold is studied. The obtained results are used to generalize the theorem of formality of a Kähler manifold. A nontrivial obstruction for a given GCS to be a part of a generalized Kähler structure is given. As an application of this result one can prove that no GCS on a nilpotent Lie algebra is part of a generalized Kähler pair. The generalized complex version of Kähler manifolds correspond to the bi-Hermitian structures of Gates, Hull and Rocek obtained from the study of general (2,2) supersymmetric sigma models.
The next chapter starts giving a description of the group of symmetries of an exact Courant algebroid. Then, the Courant algebras and extended actions are defined. For a given extended action on an exact Courant algebroid over a manifold, the author explaines how to reduce the manifold and the Courant algebroid. In the next section it it shown how to transport Dirac structures from the original Courant algebroid to the reduced Courant algebroid. These results are used to reduce GCSs .
In Chapter 4, the author presents a mathematical version of \(T\)-duality introduced by Bouwknegt, Evslin and Mathai for principal circle bundles with nonzero twisting 3-forms. It is proved that if two principal torus bundles over a common base are T-dual to each other then they can be obtained as reduced spaces from a common space by two torus actions.

MSC:

53C56 Other complex differential geometry
53D20 Momentum maps; symplectic reduction

Citations:

Zbl 1076.32019
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