Boucksom’s local analytic Bertini conjecture

Boucksom asked whether the local analytic Bertini restriction theorem holds outside a pluripolar exceptional set. The 2026 local-case paper states this in full generality for plurisubharmonic weights and product families.

Known results

• Fubini-type arguments, together with work of Fujino–Matsumura and Meng–Zhou, gave the restriction equality outside a Lebesgue-null exceptional set.
• The projective analogue was proved earlier: outside a pluripolar set of hyperplanes, multiplier ideals restrict correctly; this answered Boucksom’s projective question.

2026 local-case proof

Theorem 1.1 of Analytic Bertini theorem II — The local case asserts that there is a pluripolar set $P$ such that, for every parameter $\eta\notin P$, $\mathcal{I}_{X}(\Phi)\cdot\mathcal{O}_{X_{\eta}}=\mathcal{I}_{X_{\eta}}(\Phi|_{X_{\eta}})$. The proof upgrades the earlier Lebesgue-null exceptional set using fine pluripotential theory, jet bundles, and fiberwise weighted Bergman kernels.

Current status (as of July 2026): The conjecture is settled by the 2026 arXiv preprint in the stated local generality, while independent verification and peer review remain pending.