Continuity and truncation convergence on strongly pseudoconvex domains

Establish whether the pluricomplex Green function G_{A,\Omega} is necessarily continuous in the extended-real sense for a strongly pseudoconvex domain with C^2 boundary and a reduced divisor extending across the boundary, without assuming convexifiability or a sectorial inward deformation; determine additionally whether the bounded envelopes P_\Omega\min\{-\log|f|,j\} increase to the desingularized Green function \widetilde G_A on \Omega\setminus A.

Background

The paper’s main approximation theorem establishes convergence of the bounded envelopes P_\Omega\min{-\log|f|,j} to \widetilde G_A under the SID condition, and its continuity theorem then applies on B-regular domains. Strongly pseudoconvex domains with sufficiently smooth boundary form an important class of B-regular domains, but they need not satisfy the global holomorphic deformation property required by the paper.

The authors explicitly leave open both continuity and the stronger truncation-convergence statement for arbitrary strongly pseudoconvex domains, even when the boundary is smooth. The stated obstacle is retaining quantitative control of divisor intersections near the boundary without a global sectorial deformation.

References

In particular, let $\Omega\Subsetn$, $n\ge2$, be a strongly pseudoconvex domain with $C2$ boundary, and let $f$ be holomorphic and define a reduced divisor on a neighborhood of $\overline\Omega$. For $A=(f)|\Omega$, is $G{A,\Omega}$ necessarily continuous in the extended-real sense, without assuming convexifiability or an SID? A stronger approximation question is whether

P_\Omega\min{-\log|f|,j}\nearrow\widetilde G_A \quad\text{on }\Omega\setminus A.

The arguments of this paper do not settle these questions for an arbitrary strongly pseudoconvex domain, even when its boundary is $C\infty$.

— Continuity of pluricomplex Green functions with hypersurface poles on certain B-regular domains  (2610.00909 - Dao et al., 1 Oct 2026) in Section “Open questions,” item 2