The Calabi–Yau type conjecture for varieties with polarized endomorphisms

At least 9 years old · documented by 2016

Let $X$ be a normal complex projective variety admitting a non-invertible polarized endomorphism.

Calabi–Yau type conjecture. Then $X$ is of Calabi–Yau type.

This is presented as another important conjecture concerning polarized endomorphisms. The supplied status evidence resolves it under the additional assumptions that $X$ is $\mathbb{Q}$ -Gorenstein and the endomorphism is étale in codimension one; the general statement remains open.

References

Primary source

Shou Yoshikawa, “Singularities of non-Q-Gorenstein varieties admitting a polarized endomorphism”, arXiv:1811.01795 (2021).

Additional references

2 papers in this index state this conjecture (2016–2018). The statement above is taken from the most recent of them; the others are arXiv:1607.01382.

Posted 1 month ago · edited 3 hours ago

Progress summary

Refreshed 3 hours ago

Claimed progress

A recent unrefereed theorem advances the conjecture for a stronger class of self-maps, but the full statement remains open.

The conjecture asserts that every normal complex projective variety with a non-invertible polarized endomorphism is of Calabi–Yau type. The general claim is unresolved; known results cover several important special cases.

Known results

Broustet and Gongyo (2016): the conjecture holds for surfaces.
Yoshikawa (2021): if the endomorphism is étale in codimension one, then $K_X$ is $\mathbb{Q}$ -linearly trivial and $X$ is of Calabi–Yau type, without requiring $X$ to be $\mathbb{Q}$ -Gorenstein.
The conjecture holds for smooth projective threefolds; a specific projective-bundle case remains unresolved.

October 2026 int-amplified advance

Shou Yoshikawa’s preprint Construction of log Calabi--Yau boundaries from int-amplified endomorphisms claims the Calabi–Yau-type conclusion under the stronger int-amplified-endomorphism hypothesis. This is claimed progress, not a resolution, and the preprint is unrefereed.

Current status (as of October 2026): The conjecture is proved in several special settings, including the étale-in-codimension-one case and smooth threefolds, while the general polarized-endomorphism statement remains open; the int-amplified extension is unverified.

Sources

No solutions have been posted yet.