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).



Example of a non-Kähler manifold with varying plurigenera
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.](https://projecteuclid.org/euclid.jdg/1214432677)


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.


