Holomorphic Foliation Theory — Problem 14

Definitions and mathematical statement

A biholomorphism of ${\mathbb{C}}^n$ is a bijective holomorphic map $f:{\mathbb{C}}^n\to{\mathbb{C}}^n$ whose inverse is holomorphic. Write $\operatorname{Aut}_{\mathrm{hol}}({\mathbb{C}}^n)$ for the set of these maps.

The supplied Problem 14 asks to investigate "the above properties" for the following classes:

1. biholomorphisms of ${\mathbb{C}}^n$;
2. volume-preserving automorphisms of ${\mathbb{C}}^n$;
3. symplectic automorphisms of ${\mathbb{C}}^n$, with $n$ even.

Here preservation of a specified differential form $\eta$ means $f^*\eta=\eta$. This explains the preservation terminology conditionally: the supplied record does not identify the intended volume form or symplectic form, or specify whether these structures are real or holomorphic.

Unresolved formulation. The phrase "the above properties" refers to preceding material absent from the supplied record. Consequently, the properties to investigate, their hypotheses, and their quantifiers cannot be recovered from this record. The source also supplies no dimension restriction beyond evenness of $n$ in the symplectic case. This draft therefore preserves the research request but cannot state a definite conjecture or a fully specified mathematical target without the missing context.

Short English statement

Problem 14 asks for an investigation of previously described properties for biholomorphisms and volume-preserving automorphisms of ${\mathbb{C}}^n$, and for symplectic automorphisms when $n$ is even. The referenced properties and the preserved structures are unspecified in the supplied record, so a precise conjecture cannot yet be formulated.

Sources

• Local UnsolvedMath registry: permanent ID $800687$, source problem code catalogue entry, title "Holomorphic Foliation Theory --- Problem 14," in the AMR Open Problem Lists collection. The registry attributes the problem to Dinh and Sibony, with proposed year 2018, and labels it open. This is a supplied status label, not an independently verified mathematical or literature assessment.
• Dinh, Sibony, Some Open Problems on Holomorphic Foliation Theory (2018), Problem 14. Supplied source URL: https://arxiv.org/abs/1807.05640. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; neither access nor the source context has been independently verified for this draft.