Henon Maps: Open Problems — Problem 44

Definitions and mathematical statement

A polynomial automorphism of ${\mathbb{C}}^d$ is a polynomial map $f:{\mathbb{C}}^d\to{\mathbb{C}}^d$ having a polynomial inverse. The supplied record restricts attention to regular polynomial automorphisms, but does not define regularity or the object $S(f)$. Thus the following transcribes the requested conclusions conditionally on those missing definitions; it is not a complete standalone formulation.

For every regular polynomial automorphism $f\in\operatorname{Aut}[{\mathbb{C}}^d]$, with $S(f)$ understood as the intended complex manifold, establish and investigate the following:

1. There exist a compact complex manifold $\bar S(f)$, a holomorphic open embedding $j:S(f)\hookrightarrow\bar S(f)$, and a divisor $D$ on $\bar S(f)$ such that \[ \bar S(f)\setminus j(S(f))=\operatorname{Supp}(D). \] Here an open embedding identifies its domain biholomorphically with an open subset, and $\operatorname{Supp}(D)$ denotes the underlying analytic hypersurface of the divisor. This expresses the source's set-theoretic assertion that the complement is a divisor; multiplicities are not specified.
2. Any two compact complex manifolds satisfying the preceding compactification requirement for the same $f$ are bimeromorphic, meaning that there is a meromorphic map between them with a meromorphic inverse. The source does not specify whether this bimeromorphism must restrict to the identity under the two identifications of $S(f)$.
3. Describe complex objects on $\bar S(f)$, including analytic subvarieties, vector fields, holomorphic foliations, and positive closed currents, and compute its deformation space.

The dimension range, the construction of $S(f)$, and the intended meaning of deformation space must be supplied before these requests become precise. The last request also leaves unspecified whether a particular compactification is intended or whether the description should apply to every compactification satisfying the first requirement.

Short English statement

For each regular polynomial automorphism, the problem asks for a compactification of its associated space $S(f)$ with divisor boundary, uniqueness up to bimeromorphism, and a description of geometric objects and deformations. The supplied record does not define $S(f)$ or regularity, so this is a conditional statement requiring mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $801092$, source problem code catalogue entry; supplied title: Henon Maps: Open Problems --- Problem 44. The registry labels this record open; that label is supplied source data, not an independently verified mathematical assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Problem 44, as attributed in the supplied record. Supplied source URL: https://arxiv.org/abs/2312.03907. The background records access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance statements, not a claim of independent source consultation.