On generalized Gauduchon nilmanifolds

We construct invariant generalized Gauduchon metrics on the product of two complex nilmanifolds that do not necessarily admit this kind of metrics. In particular, we prove that the product of a locally conformal Kähler nilmanifold and a balanced nilmanifold admits a generalized Gauduchon metric. In complex dimension 4, generalized Gauduchon nilmanifolds with (the highest possible) nilpotency step s=5 are given, as well as 3-step and 4-step examples for which the center of their underlying Lie algebras does not contain any non-trivial J-invariant ideal. These examples show strong differences between the SKT and the generalized Gauduchon geometries of nilmanifolds.