1 · The essentials
These two manuscripts describe examples that challenge a proposed rule about the geometry of complex shapes. The rule concerns what happens when a shape is replaced by a covering space that removes its loops. It predicts that this covering space has a particular kind of convexity, defined using complex functions rather than straight lines. One manuscript claims a counterexample in complex dimension two. The main manuscript claims another in complex dimension four, even with an additional restriction on the shape’s loops. In that example, the failure occurs despite the absence of compact complex pieces of positive dimension inside the covering space. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine unrolling a patterned object into a much larger sheet and asking whether the sheet always has a certain organizing property. The four-dimensional example claims that this property can fail even when the unrolled sheet contains none of the closed-off pieces one might suspect of causing trouble. This image does not capture complex geometry or its notion of convexity.
Why it matters
The manuscripts claim two distinct limits on Shafarevich holomorphic convexity: failure already in complex dimension two, and failure even under the large-fundamental-group hypothesis. The fourfold example also separates the absence of positive-dimensional compact analytic subvarieties from the stronger properties of being Stein or holomorphically convex.
2 · Going further
The problem concerns the universal cover of a smooth projective complex variety. In the formulation addressed by these manuscripts, Shafarevich’s holomorphic-convexity conjecture predicts that this cover is holomorphically convex. This is a condition expressed through holomorphic functions, not ordinary straight-line convexity.
The main manuscript claims a smooth projective complex fourfold with large fundamental group whose universal cover is not Stein. It also states that the cover contains no positive-dimensional compact complex-analytic subvariety and presents the construction as a counterexample to holomorphic convexity even under the large-fundamental-group hypothesis. The repository summary explicitly describes the cover as neither Stein nor holomorphically convex.
The companion manuscript claims a smooth connected projective complex surface whose universal cover is not holomorphically convex. Its role is to place the failure in complex dimension two; the supplied abstract does not assert a large fundamental group or the absence of compact analytic subvarieties for this surface. Both abstracts describe their approach as constructing examples, but neither supplies construction details or the intermediate argument, so no more specific mechanism can be extracted from the supplied text.
Useful notions
- Complex dimension
- Smooth projective complex varieties
- Fundamental groups and universal covers
- Holomorphic functions and holomorphic convexity
- Stein spaces and analytic subvarieties
What was known before / what is new
Before
The source identifies Shafarevich’s holomorphic-convexity conjecture as the claim being challenged. It also distinguishes the version considered under a large-fundamental-group hypothesis. No earlier partial results or historical development are supplied.
New
The main manuscript claims a fourfold counterexample with large fundamental group and no positive-dimensional compact analytic subvariety in its universal cover. The cover is nevertheless described as neither Stein nor holomorphically convex. The companion manuscript separately claims a surface counterexample to unrestricted holomorphic convexity.
Limits and assumptions
The fourfold claim concerns a smooth projective complex variety of complex dimension four with large fundamental group; the surface claim concerns a smooth connected projective complex variety of complex dimension two. The source does not define the large-fundamental-group hypothesis or give the constructions and arguments needed to assess the claims in detail. The main result is not formalized in Lean and has not been formally verified. No remaining open cases are specified, and the additional properties asserted for the fourfold should not be attributed to the surface.
Mini glossary
- Universal cover —
- A simply connected covering space of the original space, which removes its noncontractible loops.
- Fundamental group —
- The group recording loops in a space up to continuous deformation.
- Holomorphic convexity —
- A condition requiring the hull of every compact set, defined by bounds on holomorphic functions, to remain compact.
- Stein —
- A complex-analytic property that includes holomorphic convexity and sufficient holomorphic functions to distinguish points.
- Compact complex-analytic subvariety —
- A compact subset locally described by complex-analytic equations.
- Surface and fourfold —
- In this setting, these mean complex varieties of complex dimension two and four, respectively.
Keywords : Shafarevich conjecture · Universal cover · Holomorphic convexity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
A projective fourfold with large fundamental group and non-Stein universal cover
05/10/2026We construct a smooth projective complex fourfold with large fundamental group whose universal cover is not Stein. The universal cover contains no positive-dimensional compact complex-analytic subvariety, so this disproves Shafarevich's holomorphic-convexity conjecture even under the large-fundamental-group hypothesis.
A surface counterexample to Shafarevich holomorphic convexity
23/09/2026We construct a smooth connected projective complex surface whose universal cover is not holomorphically convex. This disproves the Shafarevich conjecture on holomorphic convexity in complex dimension two.