Generalized complex geometry
As mentioned in the introduction, the article is largely based upon the author's doctoral thesis [Generalized complex geometry. PhD thesis. Oxford University (2004), \url{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 \textit{A. Alekseev}, \textit{H. Bursztyn} and \textit{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 [\textit{S. Barannikov} and \textit{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 \textit{A. Kapustin} and \textit{D. Orlov} in [J. Geom. Phys. 48, No. 1, 84--99 (2003; Zbl 1029.81058)].
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