Several Complex Variables Problem List — Problem 52

Definitions and mathematical statement

Let $n\geq 1$, and let $(V,0)$ be the germ at the origin of an irreducible complex analytic variety in ${\mathbb{C}}^n$. Choose a representative on a neighborhood of $0$. For sufficiently small $\epsilon>0$, write \[ B_\epsilon=\{z\in{\mathbb{C}}^n:|z|<\epsilon\},\qquad E_\epsilon=V\cap{\mathbb{R}}^n\cap B_\epsilon, \] where $|\cdot|$ denotes the Euclidean norm and $\Im z=(\Im z_1,\ldots,\Im z_n)$. The source displays the extremal function \[ U_{E_\epsilon}(z):=\sup\left\{u(z):u\leq0\text{ on }E_\epsilon,\quad 0\leq u\leq1\text{ on }B_\epsilon\right\},\qquad z\in B_\epsilon. \] The admissible function class for $u$ is not specified in the supplied record. In particular, no continuity, subharmonicity, or plurisubharmonicity condition is stated. Thus the intended extremal function is not fully specified; adding such a condition would be an additional hypothesis.

The source asks to classify those germs $(V,0)$ for which there is a constant $A>0$ such that \[ (\alpha)\qquad U_{E_\epsilon}(z)\leq A|\Im z|\qquad\text{for every }z\in B_{\epsilon/2}. \] The displayed estimate is stated on the entire ambient ball, without a restriction $z\in V$. The construction is made for any sufficiently small $\epsilon$, but the source does not explicitly state whether $A$ may depend on $\epsilon$ or must be uniform over all sufficiently small radii. These unresolved qualifications must be settled before this becomes a precise analytic classification problem.

Short English statement

Classify irreducible complex analytic germs at $0$ whose stated extremal function associated with their real points satisfies a linear bound in the distance to ${\mathbb{R}}^n$. The supplied formulation omits the admissible function class and leaves the dependence of the bound on the radius unclear.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800253$, source code catalogue entry, title "Several Complex Variables Problem List --- Problem 52." The registry attributes the record to Buzzard, Fornaess, and Krantz, with proposed year $1995$, and labels it open; that label is supplied metadata, not an independently verified mathematical status.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 52, marked "(New)" in the supplied statement. Supplied source link: https://arxiv.org/abs/math/9509202. The provenance records access on $2026\text{-}10\text{-}16$ and extraction by arxiv_latex; these are supplied provenance claims, not access or verification performed for this draft.