Brunella non-Steinness conjecture

Let $C\subset \mathbb{P}^{2}$ be a smooth cubic curve, let $p_{1},\ldots,p_{9}\in C$ be distinct points, and let $\pi\colon X=\operatorname{Bl}_{p_{1},\ldots,p_{9}}\mathbb{P}^{2}\to\mathbb{P}^{2}$ be the blow-up. If $\widetilde{C}$ denotes the strict transform of $C$ , then the conjecture asserts that the quasi-projective surface $X\setminus\widetilde{C}$ is not Stein.

References

Primary source

arXiv

Additional references

Deformation families of open Calabi-Yau manifolds and Steinness — arXiv

algebraic-geometry complex-geometry

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

New deformation examples shed light on the conjecture but do not settle whether the relevant spaces are Stein.

The conjecture concerns Steinness after nine-point blowups. No proposer or date is identified in the retrieved material.

September 2026 deformation families

A September 2026 report links a preprint presenting compactifiable deformation families with holomorphic involutions and examples of both claimed non-Stein and Stein fibers. This adds deformation information but explicitly does not claim a complete resolution.

Current status (as of September 2026): The conjecture remains unresolved; new deformation families provide claimed examples of both Stein and non-Stein fibers, but the reported advance is unverified and does not settle the conjecture.

Sources

No solutions have been posted yet.