Several Complex Variables Problem List — Problem 74

Definitions and mathematical statement

Let ${\mathbb{P}}^2({\mathbb{C}})$ denote the complex projective plane. Let $f:{\mathbb{P}}^2({\mathbb{C}})\to{\mathbb{P}}^2({\mathbb{C}})$ be a holomorphic map of degree $d\geq 2$. The supplied source refers to an ergodic measure $\mu=T\wedge T$, but does not define $T$ or its normalization. Here $T\wedge T$ denotes the wedge product of the intended current with itself; its definition and the hypotheses ensuring that it is an $f$-invariant ergodic probability measure require clarification.

For the intended measure $\mu$, let $\lambda_1\leq\lambda_2$ be the two complex Lyapunov exponents of the derivative cocycle of $f$. Thus, when these exponents exist, if $s_1(Df_x^n)\leq s_2(Df_x^n)$ are the singular values with respect to a smooth Hermitian metric, then for $\mu$-almost every $x$, \[ \lambda_i=\lim_{n\to\infty}\frac{1}{n}\log s_i(Df_x^n),\qquad i=1,2. \] The source question is whether, for every such $f$ and its intended measure $\mu=T\wedge T$, \[ \lambda_1>0. \] This formulation remains conditional on identifying the intended current $T$ and the meaning of degree. The supplied excerpt does not specify whether degree means algebraic degree or topological degree; these conventions differ for endomorphisms of the complex projective plane.

Short English statement

For a holomorphic self-map of the complex projective plane of degree at least two, must the smaller Lyapunov exponent of the intended ergodic measure $T\wedge T$ be strictly positive? The supplied excerpt leaves $T$ and the degree convention unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $800275$, exact source code catalogue entry; title: Several Complex Variables Problem List --- Problem 74. The registry attributes the problem to Buzzard, Fornaess, and Krantz, with proposed year 1995.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 74. Supplied source URL: https://arxiv.org/abs/math/9509202. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independently verified access.
• The supplied registry labels this problem open. This is a source status label, not an independently verified assessment of the literature.