  The Bott-Chern cohomology of 6-dimensional nilmanifolds endowed with
invariant complex structure is studied with special attention to the cases when
balanced or strongly Gauduchon Hermitian metrics exist. We consider complex
invariants introduced by Angella and Tomassini and by Schweitzer, which are
related to the $\partial \bar\partial$-lemma condition and defined in terms of
the Bott-Chern cohomology, and show that the vanishing of some of these
invariants is not a closed property under holomorphic deformations. In the
balanced case, we determine the spaces that parametrize deformations in type
IIB supergravity described by Tseng and Yau in terms of the Bott-Chern
cohomology group of bidegree (2,2).
