An arXiv preprint reports that the Cœuré–Loeb fibration, whose original total space is not Stein, can nevertheless be realized as an open subset of a Stein space. More precisely, the paper says the example is biholomorphic to the complement of an analytic subset in a Stein space. In ordinary language, “biholomorphic” means the two descriptions are identified without changing the complex-analytic structure being studied.
That addresses the smooth open-immersion problem: whether the Cœuré–Loeb fibration is isomorphic to an open subset of a Stein space. The author interprets the positive result as showing that this fibration is not a counterexample to the problem.
A general construction behind the example
The result rests on a more general theorem. Under its stated assumptions, the specified principal holomorphic additive bundle is biholomorphic to an open subset of a Stein space. A central condition is that the bundle’s torsor class—the cohomological data used to describe it—extends from a modification of the base.
The paper uses a proof-based complex-analytic approach, with the main proof divided into steps. It first shows that the torsor class is killed by a sufficiently high power of the maximal ideal at the puncture. It then constructs finitely many functions on the bundle from base functions and weighted fiber coordinates to define a biholomorphic map.
The constructed ambient space is a closed analytic subset equipped with its reduced complex-space structure. It is Stein because it is a closed analytic subspace of the Stein product of the base with affine space.
The proof identifies the original bundle with the punctured part of that constructed space, namely the complement of the fiber over the puncture. The result is therefore an open-subset realization, not a claim that the original bundle itself is Stein.
How the Cœuré–Loeb case fits
For the named example, the paper represents the Cœuré–Loeb construction by a quotient whose relevant projection is a principal holomorphic additive bundle. That places the example within the framework of the general theorem, subject to its extension condition.
The application verifies that condition through a partial compactification of the base. The compactification adds a finite rational normal-crossing cycle with a negative-definite intersection matrix, which contracts to a Stein surface. The paper then identifies the torsor class of the Cœuré–Loeb bundle on the punctured base as the restriction of the class of the extended bundle on the compactified base.
With that geometric input in place, the paper’s second theorem states that the Cœuré–Loeb example is biholomorphic to the complement of an analytic subset in a Stein space. The construction supplies the open-subset realization at the centre of the paper’s research question.
What remains unresolved
The result does not show that the original Cœuré–Loeb total space is Stein. Nor does it resolve the smooth open-immersion problem in full generality; the author’s interpretation is specifically that this example is not a counterexample.
The application also depends on the torsor class extending from a modification of the base. The paper’s conclusions therefore rest on the stated extension condition for the general theorem and on its verification for the Cœuré–Loeb construction.
Preprint status
The supplied document is labeled arXiv version 1 and dated 26 Aug 2026.
Paper data and sources
Original title: The Cœuré-Loeb example as an open subset of a Stein space
Authors: Ovidiu Preda
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text