Stein neighborhood basis problem for complex–totally real unions

The problem asks when unions involving complex and totally real pieces admit Stein neighborhood bases. The newly reported work separates a tractable low-dimensional case from the unrestricted question, rather than classifying all unions.

Known results

• Two totally real planes in $\mathbb{C}^2$ admit strongly pseudoconvex Stein neighborhoods under a smallness condition on a real $2\times2$ matrix $A$ (Starčič, 2016).
• Certain unions of two maximal totally real subspaces in $\mathbb{C}^n$ admit regular Stein neighborhoods under spectral hypotheses on $A$ (Starčič, 2016).
• Compact real surfaces in complex surfaces have such bases under finitely many flat hyperbolic complex points (Slapar, 2003).

September 2026 claimed counterexample

An arXiv preprint claims a positive result for the $\mathbb{C}^2$ configuration and a negative example for unrestricted unions. This would refute unrestricted existence, but the claim has not been independently verified and does not settle the full classification.

Current status (as of September 2026): Special configurations, including the reported $\mathbb{C}^2$ case, have positive results, while unrestricted existence is claimed false; the general classification remains open and the new counterexample is unverified.