Counterexamples to Shafarevich holomorphic convexity at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

A
D
V
E
R
T

I
S
E
M
E
N
T

Counterexamples to Shafarevich holomorphic convexity

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

thinking... about 3 hours remaining

If this game doesn't work on your computer, we can't help you. No Lean version yet. Some unformalized games could have issues!

expertly designed by an internal OpenAI model

Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	shapes made of equations	Levels:	2
Category:	Algebraic and complex geometry	Lean version:	not yet
Rate this game! 4.2 out of 5 (7,963 votes)

>>> How to Play <<<

Shafarevich counterexamples in dimension two and with large fundamental group. 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.

>>> Level Select <<<

LEVEL 1 A projective fourfold with large fundamental group and non-Stein universal cover

released 2026-10-05 | 4 theorems · 17 lemmas · 22 proofs · 23,849 words | PLAY LEVEL 1 » (pdf)

We 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.

LEVEL 2 A surface counterexample to Shafarevich holomorphic convexity

released 2026-09-23 | 1 theorem · 14 lemmas · 19 proofs · 11,971 words | PLAY LEVEL 2 » (pdf)

We 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.

More Algebraic and complex geometry Games!
Numerical semiampleness and generalized minimal models	The sharp ordinary-double-point volume gap	Fujita's freeness conjecture HOT!	Nagata's conjecture and maximal Seshadri constants HOT!
Bloch's conjecture for complex surfaces HOT!	Hyperkähler SYZ and projective-space bases	Oka classification for K3 surfaces and other compact complex surfaces	$P=W$ for fixed-determinant moduli spaces

COOLMAITH GAMES SPONSORS: (not real)
Big Al's USED GPUs	Epsilon & Delta LLP attorneys at limit	QED Bail Bonds	Hilbert's Hotel always a vacancy!

Cool Links: openai/math Lean Mathlib arXiv the real Coolmath Games

You are visitor number 000719

Thanks for visiting CoolmAIth Games
© 1997-2026 CoolmAIth Games. A parody. Not affiliated with Coolmath.com LLC or OpenAI.
Every proof here was written by an internal OpenAI model and lives at github.com/openai/math.
Best viewed in Netscape Navigator 4.0 at 800x600