  For a one-parameter degeneration of reduced compact complex analytic spaces
of dimension $n$, we prove the invariance of the frontier Hodge numbers
$h^{p,q}$ (that is, with $pq(n{-}p)(n{-}q)=0$) for the intersection cohomology
of the fibers and also for the cohomology of their desingularizations, assuming
that the central fiber is reduced, projective, and has only rational
singularities. This can be shown to be equivalent to the invariance of the
dimension of the cohomology of structure sheaf (which is known in the
algebraizable case), since we can prove the Hodge symmetry for all the Hodge
numbers $h^{p,q}$ together with $E_1$-degeneration of the Hodge-to-de Rham
spectral sequence for nearby fibers, assuming only the projectivity of the
central fiber.
  For the proof of the main theorem, we calculate the graded pieces of the
induced $V$-filtration for the first non-zero member of the Hodge filtration on
the intersection complex Hodge module of the total space, which coincides with
the direct image of the dualizing sheaf of a desingularization (related to
Koll\'ar's conjecture on the direct images of dualizing sheaves of smooth
varieties). This calculation implies also that the order of nilpotence of the
local monodromy is smaller than the general singularity case by 2 in the
situation of the main theorem assuming further smoothness of general fibers. We
can prove a partial converse of the main theorem under some hypothesis.
