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Matsumura's extension problem for pluricanonical forms

Let f ⁣:X→Δf\colon X\to \Delta be a proper Kähler family over a disk, with central fiber X0X_0 a simple-normal-crossing fiber, and suppose that the relative canonical bundle KX/ΔK_{X/\Delta} is relatively nef. For every integer m≥1m\ge 1, is the restriction map H0 ⁣(X,mKX/Δ)→H0 ⁣(X0,mKX0)H^0\!\left(X,mK_{X/\Delta}\right)\to H^0\!\left(X_0,mK_{X_0}\right) surjective? Equivalently, does every pluricanonical section on the central fiber extend to a pluricanonical section on the total space?

References

Primary source

arXiv

Additional references

algebraic-geometrycomplex-geometrykhler-geometry
Posted · edited

Progress summary

Refreshed
Claimed progress

A new paper settles important smooth and big cases, but the full extension question remains open.

The problem asks whether pluricanonical sections on a central simple-normal-crossing fibre of a proper Kähler family extend to the total space. The full statement allows a relatively nef canonical bundle and remains unresolved.

Known results

  • Smooth projective families: affirmative extension and invariance of plurigenera (Siu, 1998 and 2002).
  • Smooth central fibre with semi-ample canonical restriction: affirmative (Lev, 1983).
  • Components of a simple-normal-crossing central fibre: extension of pluricanonical sections (Takayama, 1997).
  • Under suitable singular metrics with zero Lelong numbers: affirmative for the corresponding multiplier-ideal subspace.

September 2026 claimed advance

Chen, Rao, and Wang report affirmative extension results for smooth Kähler families and for an essentially Moishezon case, including a bigness condition on a central-fibre component, with the corresponding invariance consequence. This is a partial result, not a solution of the full problem, and the retrieved sources provide no independent mathematical assessment.

Current status (as of October 2026): Classical smooth projective and several metric or positivity cases are settled, while the full Kähler-family problem with relatively nef canonical bundle and simple-normal-crossing central fibre remains open; the September 2026 advance is an unverified partial result.

Sources

Solutions 0

No solutions have been posted yet.