Continuity on arbitrary bounded B-regular domains

Determine whether the pluricomplex Green function with a reduced hypersurface divisor is necessarily continuous in the extended-real sense on every bounded B-regular domain, without assuming convexifiability or a sectorial inward deformation.

Background

The paper proves extended-real continuity of the pluricomplex Green function G_{A,\Omega} when the domain is B-regular and convexifiable, and more generally when the pair (\Omega,A) admits a sectorial inward deformation (SID). The SID hypothesis supplies a global holomorphic motion into the domain with quantitative boundary depth, enabling analytic-disc insertions that recover the logarithmic mass lost by bounded truncation.

The unresolved issue is whether B-regularity alone suffices for continuity when A=(f)|_\Omega is a reduced divisor extending holomorphically across the boundary. The authors indicate that a proof in this generality would need an alternative to the global holomorphic inward motion used in their approximation argument.

References

Let $\Omega\Subsetn$ be an arbitrary bounded B-regular domain and let $A=(f)|\Omega$ be a reduced divisor extending across the boundary. Is $G{A,\Omega}$ necessarily continuous in the extended-real sense?

— 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 1

What remains true when the pole set is an analytic set of codimension at least two, or more generally is defined by a coherent ideal sheaf?

— 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 3

Which geometric conditions imply the SID property? Example~\ref{ex:KN-sectorial} shows that convexifiability is not necessary.

— 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 4