Positive-Entropy Automorphisms of Strict Calabi-Yau Threefolds

Definitions and mathematical statement

Let $X$ be a smooth projective variety over ${\mathbb{C}}$ with $\dim X=3$, and let $f\in\operatorname{Aut}(X)$ be a biregular automorphism. The supplied excerpt does not include the promised definitions of "strict Calabi--Yau" and "primitive." Consequently, the following explicit interpretation is provisional and requires comparison with the complete source.

Under the simply connected convention, a strict Calabi--Yau threefold satisfies \[ K_X\simeq\mathcal O_X, \qquad \pi_1(X^{\mathrm{an}})=\{1\}. \] Here $K_X$ denotes the canonical line bundle and $X^{\mathrm{an}}$ the associated complex analytic manifold. The intended convention for "strict" is unresolved in the supplied excerpt.

Using the equivariance framework stated in the excerpt, call $f$ primitive if there do not exist a normal projective variety $B$ over ${\mathbb{C}}$, a dominant rational map $\pi\colon X\dashrightarrow B$, and a birational self-map $f_B\in\operatorname{Bir}(B)$ such that \[ 0<\dim B<3, \qquad \pi\circ f=f_B\circ\pi \] as rational maps. Whether the source additionally requires connected general fibers in this definition must be checked.

For a very ample Cartier divisor $H$ on $X$, define, for $0\leq p\leq3$, \[ d_p(f)=\lim_{n\to\infty} \left(\left((f^n)^*(H^p)\cdot H^{3-p}\right)_X\right)^{1/n}. \] The excerpt states that these limits exist and are independent of $H$. For the smooth projective, biregular setting, its entropy characterization is \[ h_{\mathrm{top}}(f)=\max_{0\leq p\leq3}\log d_p(f), \qquad h_{\mathrm{top}}(f)>0\ \Longleftrightarrow\ d_1(f)>1. \] Here $h_{\mathrm{top}}$ is the topological entropy on $X^{\mathrm{an}}$.

Construction problem, conditional on the terminology above. Construct further pairs $(X,f)$ consisting of a smooth strict Calabi--Yau threefold over ${\mathbb{C}}$ and a primitive biregular automorphism satisfying $d_1(f)>1$. The original request is to find "many examples." Neither a required number of examples nor a criterion for distinctness or novelty is supplied. Thus the source gives an open-ended construction request, rather than a specified assertion about infinitely many isomorphism classes or a classification. The report describes its discussion as speculative rather than a collection of definite results; its broader standing framework allows normal projective varieties of dimension $d$ and birational self-maps, whereas this target concerns smooth threefolds and biregular automorphisms.

Short English statement

Construct additional smooth strict Calabi--Yau threefolds over ${\mathbb{C}}$ with positive-entropy automorphisms that admit no equivariant rational map to a positive-dimensional lower-dimensional variety. The precise conventions for strictness and primitivity, and the meaning of "many" or "further," require review. The supplied curation labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30005961$, exact source problem code catalogue entry; title: Positive-Entropy Automorphisms of Strict Calabi--Yau Threefolds.
• Supplied report citation: Algebraic Geometry: Wall Crossing and Moduli Spaces, Varieties and Derived Categories (2025), Oberwolfach Reports, pp. 1787--1854, DOI 10.4171/owr/2024/32. Original OWR report: https://doi.org/10.4171/owr/2024/32. The registry separately records proposed year 2024; the year discrepancy is unresolved here.
• The excerpt identifies the contribution as Keiji Oguiso's Automorphisms of Calabi--Yau threefolds from algebraic dynamics and the second Chern class. It presents Question 1 as a question after Dinh, Sibony, and D.-Q. Zhang, citing source references [6], [7], and [8] for background, variants, and known results. Those references are not bibliographically identified in the supplied record.
• Supplied literature source, labeled "Oguiso--Truong primary paper": https://arxiv.org/abs/1306.1590. No further bibliographic details are supplied.
• The excerpt attributes well-definedness of dynamical degrees to Dinh--Sibony [3], relative dynamical degrees to Dinh--Nguyên [2], with an additional reference [13], and the entropy formula to Gromov--Yomdin. Full entries for the numbered references are absent from the supplied record.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. Its status is "partially solved" and its assessment states that primitive positive-entropy automorphisms have been explicitly constructed on some strict Calabi--Yau threefolds, while the request for many examples is not a classification theorem. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance assertions, not verification performed for this draft.