Several Complex Variables Problem List — Problem 9

Definitions and mathematical statement

For a complex manifold $X$, let $\mathcal O(X)$ denote its holomorphic functions. Recall that $X$ is Stein if global holomorphic functions separate distinct points, provide local holomorphic coordinates at every point, and make $X$ holomorphically convex: for every nonempty compact $K\subseteq X$, the set \[ \widehat K_X=\{x\in X: |f(x)|\leq \sup_{y\in K}|f(y)|\text{ for every }f\in\mathcal O(X)\} \] is compact.

The source does not define "locally Stein." A conventional boundary formulation for an open subset $U\subseteq M$ is \[ \forall p\in\partial_M U\quad \exists\text{ an open neighborhood }V_p\subseteq M\text{ of }p \quad U\cap V_p\text{ is Stein}. \] This interpretation is provisional and requires review against the source's intended convention.

Problem (N. Sibony). Find appropriate conditions $\mathcal C$ on a compact complex manifold $M$ ensuring that \[ \mathcal C(M)\ \Longrightarrow\ \left[\forall U\subseteq M\text{ open},\quad U\text{ locally Stein}\ \Longrightarrow\ U\text{ Stein}\right]. \] This is a request for sufficient conditions, with no specified class of admissible conditions or optimality requirement. The source does not require $U$ to be proper, connected, or to have nonempty boundary. In particular, its quantifier includes $U=M$, for which the displayed boundary condition is vacuous. The intended treatment of this case is unresolved; no additional restriction is imposed here.

Short English statement

Find conditions on a compact complex manifold under which every locally Stein open subset is Stein. The meaning of locally Stein and the treatment of the allowed case $U=M$ require clarification. The supplied registry labels this problem open; that status is not independently verified.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $800213$, exact source code catalogue entry, title "Several Complex Variables Problem List --- Problem 9." Supplied status: open.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 9. Supplied source URL: https://arxiv.org/abs/math/9509202. The record attributes the individual problem to N. Sibony, with contact sibony@anh.matups.fr; its collection-level proposer field names Buzzard, Fornaess, and Krantz.
• Supplied provenance: accessed 2026-10-16; extraction method arxiv_latex; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. These are supplied provenance statements, not claims of independent source access or verification.