Deformation of rational singularities and Hodge structure

Abstract

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.