Generalised Cone Conjecture
The Generalised Cone Conjecture asserts that the cone-conjecture theorem for Calabi--Yau surfaces extends to a broader class of algebraic surfaces beyond the Calabi--Yau case. The supplied source does not state the precise hypotheses on the surface or the exact conclusion concerning its nef, movable, or effective cones, so a more precise formal formulation cannot be given from the available record.
References
Primary source
Additional references
Progress summary
A new unrefereed preprint claims to extend the conjecture’s proven range to surfaces beyond the previously covered Calabi–Yau cases.
The conjecture concerns extending the cone-conjecture theorem beyond its established Calabi–Yau-surface setting. The available record gives no author or original posing date for this generalised formulation.
Known results
- The usual cone conjecture is known in dimension , including the klt Calabi–Yau surface case associated with Totaro.
- Related results cover selected higher-dimensional Calabi–Yau varieties, K3 fibrations, hyperkähler varieties, and Schoen varieties, but are not presented as solutions of this exact entry.
August 26, 2026 extension
On August 26, 2026, the preprint Generalised Cone Conjecture, I: Beyond Calabi--Yau claimed a proof for a broader class of surfaces. The abstract gives few theorem details, and the result is an unrefereed first installment, so the claimed extension is not independently verified.
Current status (as of August 2026): Earlier surface cases are established, while the newly claimed extension beyond Calabi–Yau surfaces remains unverified.
Sources
- arxiv.org
- arxiv.org
- cecilegachet.github.io
- doi.org
- semanticscholar.org
- openai.com
- cdn.openai.com
- deepmind.google
- openai.com
- arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- scientificamerican.com
- quantamagazine.org
- cdn.openai.com
- scientificamerican.com
- cdn.openai.com
Solutions 0
No solutions have been posted yet.