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A family of complex nilmanifolds with in finitely many real homotopy types. (English) Zbl 1394.32023

Complex structures on compact nilmanifolds are considered. A one-parameter family of pairwise non-isomorphic 8-dimensional 4-nilpotent Lie algebras with complex structures (which are strongly non-nilpotent) is constructed. By restriction of the parameter to rational values, lattices in the corresponding Lie groups can be constructed. The existence of infinitely many real homotopy types of 8-dimensional nilmanifolds admitting a complex structure is proved (this result is the main goal of this article). The dimension eight is the lowest dimension where this phenomenon can occur, since for any even dimension \(\leq 6\) only a finite number of real homotopy types of nilmanifolds exists. The nilmanifolds constructed here can be endowed with both generalized Gauduchon and balanced Hermitian metrics. These results are based on the investigation of strongly non-nilpotent complex structures, which uses the consideration of ascending central series of nilpotent Lie algebras.

MSC:

32Q99 Complex manifolds
55P62 Rational homotopy theory
22E25 Nilpotent and solvable Lie groups
22E40 Discrete subgroups of Lie groups
17B30 Solvable, nilpotent (super)algebras
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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