A family of complex nilmanifolds with infinitely many real homotopy types

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We find a one-parameter family of non-isomorphic nilpotent Lie algebras gₐ, with a ∈ [0,∞), of real dimension eight with (strongly non-nilpotent) complex structures. By restricting a to take rational values, we arrive at the existence of infinitely many real homotopy types of 8-dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.