The invariance of plurigenera is one most central problems in deformation theory. I mainly concern about the (singular) surface setting in this post since Iitaka had proved that the deformation invariance of smooth surfaces.
Wilson had considered the singular setting, he obtained the lower semi-continuity of the plurigenera of projective surfaces under degeneration of surfaces having only isolated Gorenstein singularities in 1978. Afterwards, in 1980 Kawamata proved the deformation invariance of the arithmetic genera for the dualizing sheaf the reduced algebraic surfaces with at most simple elliptic or simple quasi-elliptic singularities.
Now, if one considers (almost) the widest assumptions, namely, given a flat proper family $\pi:\mathcal X\rightarrow\Delta$, suppose that all fibers are in the Fujiki class $\mathcal C$ and with at most log canonical singularities. Do we also have the invariance of plurigenera?
ps. For an arbitrary compact complex variety, we define its $m$-genus as the $m$-genus of its arbitrary non-singular model since in the smooth category the $m$-genus is a bimeromorphic invariant.
I would greatly appreciate it if some examples/counterexamples or recent developments could be provided.
