Constancy of plurigenera under a Kähler-fiber assumption on only the central fiber

Prove, using Kodaira–Spencer deformation theory or a power-series method, that for a smooth proper surjective holomorphic map with connected fibers, if the central fiber is Kähler and has nef canonical bundle, then the m-genus of the fibers is constant near the central parameter for each positive integer m, possibly with the neighborhood depending on m.

Background

The paper proves constancy of plurigenera for a smooth proper surjective Kähler morphism when the canonical bundles of the fibers are nef. The authors explain that the total space may fail to be Kähler even when the central fiber is Kähler, so their L²-extension argument does not directly apply under the weaker hypothesis.

The unresolved question asks whether Kodaira–Spencer deformation theory, for example through a power-series construction, can establish local constancy of the plurigenera without assuming that the total space is Kähler or that the family is a Kähler morphism. The question explicitly allows the neighborhood of the central parameter to depend on the plurigenus index m.

References

A natural question arises: Let f:X\to\Delta be a smooth proper surjective holomorphic map of complex manifolds with connected fibers, and write X_t:=f{-1}(t). Assume that X_0 is Kähler and that K_{X_0} is nef. Can one prove, using Kodaira--Spencer deformation theory (for instance, via a power-series method), that for each integer m\geq 1, the m-genus P_m(X_t) of X_t is constant in a neighborhood of 0 (possibly depending on m)?

— Matsumura's extension problem for pluricanonical forms in Kähler families I: the smooth and essentially Moishezon cases  (2609.40040 - Chen et al., 30 Sep 2026) in Question in Section 1, subsection “The smooth family case”