Ohsawa's Bergman-kernel restriction conjecture

About 4 years old · traced to 2022

Let $M$ be a Stein manifold of complex dimension $n$ , let $S$ be a closed complex submanifold, and let $G(,S)$ be the upper envelope of the polar functions associated with $S$ . Let $b4b4_M$ and $b4b4_{M/S}$ denote the Bergman kernels of $M$ and $S$ defined from complete orthogonal systems of the corresponding $L^2$ holomorphic canonical sections. If $S_{n-k}$ is the $n-k$ -dimensional component of $S$ , then Ohsawa's conjecture.

\frac{\pi^k}{k!}\kappa_M(x)\ge\kappa_{M/S}(x)

for any $x\in S_{n-k}$ . The conjecture was proved by Guan and Zhou, so the inequality is no longer open.

References

Primary source

Qi'an Guan and Zheng Yuan, “Concavity property of minimal L^2 integrals with Lebesgue measurable gain IV: product of open Riemann surfaces”, arXiv:2211.04953 (2022).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

Guan and Zhou announced a proof in 2014, so the conjecture is regarded as settled, although this automated report does not independently verify the proof.

Ohsawa’s conjecture asserts a comparison between the Bergman kernels of a Stein manifold and a complex submanifold, namely $\frac{\pi^k}{k!}\kappa_M(x)\ge\kappa_{M/S}(x)$ for $x\in S_{n-k}$ .

November 2014 proof

On November 28, 2014, Guan and Zhou’s paper Optimal constant in an $L^2$ extension problem and a proof of a conjecture of Ohsawa reported the conjecture as a consequence of their optimal extension theorem; the arXiv record is dated November 29, 2014. The catalogue also records the problem as solved in November 2022. This report treats the resolution as claimed rather than independently verified.

Current status (as of September 2026): Guan and Zhou’s claimed proof settles the conjecture, with no reported counterexample or later challenge, but the resolution remains unverified in this automated summary.

Sources

No solutions have been posted yet.