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Generalized complex geometry. (English) Zbl 1235.32020
As mentioned in the introduction, the article is largely based upon the author’s doctoral thesis [Generalized complex geometry. PhD thesis. Oxford University (2004), arXiv:math/0401221]. It is structured into the following sections:
1. Linear geometry of \(V\oplus V^*\). 1.1 Maximal isotropics and pure spinors.
2. The Courant bracket (2.1. Symmetries of the Courant bracket; 2.2 Dirac structures; 2.3 Tensor product of Dirac structures);
3. Generalized complex structures (3.1 Type and canonical line bundle; 3.2 Courant integrability; 3.3 Hamiltonian symmetries; 3.4 The Poisson structure and its modular class; 3.5 Interpolation);
4. Local structure: the generalized Darboux theorem (4.1 Type jumping);
5. Deformation theory (5.1 Lie bialgebroids and the deformation complex; 5.2 The deformation theorem; 5.3 Examples of deformed structures);
6. Generalized complex branes;
7. Appendix.
There are some references to other works. For example, the notion of the tensor product of Dirac structures was obtained independently by A. Alekseev, H. Bursztyn and E. Meinrenken in [Astérisque 327, 131–199 (2009; Zbl 1251.53052)]. In the case when the author deformes a complex structure, some cohomology groups are related to [S. Barannikov and M. Kontsevich, Int. Math. Res. Not. 1998, No. 4, 201–215 (1998; Zbl 0914.58004)]. Finally, the complex branes in Chapter 6 were considered also by A. Kapustin and D. Orlov in [J. Geom. Phys. 48, No. 1, 84–99 (2003; Zbl 1029.81058)].

MSC:
32Qxx Complex manifolds
53Dxx Symplectic geometry, contact geometry
53C27 Spin and Spin\({}^c\) geometry
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53C80 Applications of global differential geometry to the sciences
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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