Several Complex Variables Problem List — Problem 1

Definitions and mathematical statement

Let $V$ be the germ at $0$ of an irreducible complex variety in ${\mathbb{C}}^n$. For a sufficiently small $\epsilon>0$, choose a representative of this germ on $B_\epsilon:=\{z\in{\mathbb{C}}^n:|z|<\epsilon\}$, and put \[ E_\epsilon:=V\cap{\mathbb{R}}^n\cap B_\epsilon. \] Here $|\cdot|$ denotes the Euclidean norm, and $\Im z\in{\mathbb{R}}^n$ is the vector of imaginary parts of the coordinates of $z$. The supplied text defines \[ U_{E_\epsilon}(z):=\sup\left\{u(z):u\leq0\text{ on }E_\epsilon,\quad 0\leq u\leq1\text{ on }B_\epsilon\right\},\qquad z\in B_\epsilon. \] The source does not specify the function class over which this supremum is taken. If its displayed conditions are read literally, the admissible functions are all maps $u:B_\epsilon\to{\mathbb{R}}$ satisfying those conditions; no continuity, plurisubharmonicity, or other regularity condition is stated.

For a fixed sufficiently small $\epsilon$, the displayed classification target is to characterize those germs $V$ for which \[ \exists A>0\quad\forall z\in B_{\epsilon/2},\qquad U_{E_\epsilon}(z)\leq A|\Im z|.\tag{$\alpha$} \] The displayed domain of this estimate is the ambient ball $B_{\epsilon/2}$. The source does not state whether $A$ may depend on $\epsilon$, or whether a single $A$ must work for every sufficiently small $\epsilon$. Consequently, the intended classification remains incompletely specified until the admissible function class and the quantification of $\epsilon$ and $A$ are clarified. The source also refers to Section 7.2, whose contents are not supplied.

Short English statement

Classify irreducible complex variety germs whose real locus has the stated extremal-function bound by the distance to ${\mathbb{R}}^n$. The supplied formulation omits the admissible function class and leaves the dependence of the bound on $\epsilon$ unresolved.

Sources

• Local UnsolvedMath registry: permanent ID $800205$, source code catalogue entry, title Several Complex Variables Problem List --- Problem 1. The supplied record attributes the problem to Buzzard, Fornaess, and Krantz, with proposed year 1995, and labels it open; this status is supplied metadata, not an independently verified assessment.
• Buzzard, Fornaess, Krantz, The Several Complex Variables Problem List (1995), Problem 1. Supplied source URL: https://arxiv.org/abs/math/9509202. The excerpt marks the item "New" and refers to Section 7.2.
• Supplied provenance records access on 2026-10-16 and extraction method arxiv_latex. This is source-reported provenance; no independent source visit or verification is claimed.