Generalized complex structures on nilmanifolds
This paper concerns classification of nilmanifolds \(M\) endowed with a so-called generalized complex strucure, (Hitchin), i.e., an almost complex structure \({\mathcal J}\) on the vector bundle \(\pi:E\equiv TM\oplus T^*M\to M\), such that it satisfies some further requirements: (i) \({\mathcal J}\) is orthogonal with respect to the natural inner product on sections of \(E\to M\); (ii) the \(+i\)-eigenbundle of \({\mathcal J}\) is required to be involutive with respect to the Courant bracket. In particular the authors specialize their investigation on \(6\)-dimensional nilmanifolds, where there exist some types without standard complex or symplectic structure. The results presented in this paper are obtained with standard techniques of almost complex bundles geometry.
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