Several Complex Variables Problem List — Problem 67

Definitions and mathematical statement

Let ${\mathbb{P}}^2({\mathbb{C}})$ denote the complex projective plane, whose points are complex lines in ${\mathbb{C}}^3$. Interpret the supplied phrase "a holomorphic map on $\mathbb{P}^2$ of degree $d$" as a holomorphic self-map \[ f\colon {\mathbb{P}}^2({\mathbb{C}})\longrightarrow{\mathbb{P}}^2({\mathbb{C}}) \] of algebraic degree $d$. Under this convention, $f$ has a representation \[ f([z_0:z_1:z_2])=[F_0(z):F_1(z):F_2(z)], \] where the $F_i$ are homogeneous complex polynomials of common degree $d$ with no common zero in ${\mathbb{C}}^3\setminus\{0\}$. The complex ground field, self-map interpretation, and algebraic degree convention are implicit in the supplied wording rather than explicitly specified there.

A point $p\in{\mathbb{P}}^2({\mathbb{C}})$ is a repelling fixed point of $f$ if $f(p)=p$ and every eigenvalue $\lambda$ of the complex derivative \[ D f_p\colon T_p{\mathbb{P}}^2({\mathbb{C}})\longrightarrow T_p{\mathbb{P}}^2({\mathbb{C}}) \] satisfies $|\lambda|>1$.

The supplied question, with these conventions, is: for every integer $d\geq2$ and every holomorphic self-map $f$ of algebraic degree $d$, must there exist $p\in{\mathbb{P}}^2({\mathbb{C}})$ such that \[ f(p)=p \quad\text{and}\quad \operatorname{Spec}(D f_p)\subset\{\lambda\in{\mathbb{C}}:|\lambda|>1\}? \]

Short English statement

Does every holomorphic self-map of the complex projective plane of algebraic degree at least two have a fixed point at which both derivative eigenvalues have modulus strictly greater than one? The complex self-map and degree conventions make explicit the interpretation used in this draft.

Sources

• Buzzard, Fornaess, Krantz, The Several Complex Variables Problem List (1995), Problem 67. Supplied source URL: https://arxiv.org/abs/math/9509202. The source record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance statements are supplied metadata, not independently verified access.
• Local UnsolvedMath registry, permanent ID $800268$, source problem code catalogue entry, title "Several Complex Variables Problem List --- Problem 67," in the AMR Open Problem Lists collection. The registry attributes the problem to Buzzard, Fornaess, Krantz, with proposed year 1995, and labels it open; this is a supplied status label, not an independently verified current mathematical assessment.