Păun’s metric-extension question

Let $\pi\colon X\to S$ be a smooth projective family, let $s_0\in S$ and write $X_{s_0}=\pi^{-1}(s_0)$ . Given an effective line bundle $L$ on $X$ and a prescribed singular Hermitian metric $h_0$ on $L|_{X_{s_0}}$ with semipositive curvature, does there exist a neighbourhood $U$ of $X_{s_0}$ in $X$ and a singular Hermitian metric $h$ on $L|_U$ with semipositive curvature such that $h|_{X_{s_0}}=h_0$ ? A stronger version asks whether one can require $h|_{X_{s_0}}$ to have the same singularity type as $h_0$ .

References

Primary source

arXiv

Additional references

Failure of semipositive metric extension in a smooth projective family — arXiv — Xiangsen Qin

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 preprint claims a counterexample that would disprove Păun’s metric-extension question, but the claim has not been independently verified.

Păun’s metric-extension question asks whether semipositive metrics can be extended under the prescribed conditions. It is identified as Question 38 in a standard problem list.

September 2026 counterexample

Xiangsen Qin’s preprint claims a negative answer, with rational surfaces exhibiting degenerating collinearity; the obstruction allegedly persists even when extensions must have the same singularity type. The work is unrefereed.

Current status (as of September 2026): the question is claimed solved negatively by an unrefereed counterexample, but that claim has not been independently verified.

Sources

No solutions have been posted yet.