TITLE: Example of a non-Kähler manifold with varying plurigenera
QUESTION [7 upvotes]: Let $X \stackrel{\pi}{\to} \mathbb{D}$ be a proper holomorphic family with fibres $X_t = \pi^{-1}(t)$. Siu proved, when the $X_t$'s are projective, that the plurigenera $h^0(X_t, mK_{X_t})$ are constant. It is conjectured that the same will hold true when the fibres $X_t$ are Kähler. 
Are there examples known where this fails when the fibres are not Kähler?
My usual example of a compact non-Kähler manifold is the Hopf surface, but if the fibres of such holomorphic family are all Hopf surfaces, then the plurigenera are all zero (in particular, they are constant).

REPLY [6 votes]: The first example of this phenomenon was discovered by Iku Nakamura in his 1975 paper Complex parallelisable manifolds and their small deformations, J. Differential Geom. 10 (1975), 85-112.
The manifold $X$ is a $3$-dimensional solvmanifold, and he writes down its small deformations explicitly, see his Theorem 2 and the discussion on pages 96-99.
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