On slice measures of Green currents on CP(2)

Abstract

Let $f$ be a holomorphic map of $\mathbb{C}\mathbb{P}^2$ of degree $d\geq 2$, let $T$ be its Green current and $μ=T\wedge T$ be its equilibrium measure. We give a new proof of a theorem due to Dujardin asserting that $μ\ll T\wedgeω_{\mathbb{P}^2}$ implies $λ_2=\frac{1}{2} \log\ d$, where $λ_1 \geq λ_2$ are the Lyapunov exponents of $μ$. Then, assuming $μ\ll T\wedgeω_{\mathbb{P}^2}$, we study slice measures $ν:=T\wedge dd^c|W|^2$, where $W$ is a holomorphic local submersion. We give sufficient conditions on the Radon-Nikodym derivative of $μ$ with respect to the trace measure $T\wedgeω_{\mathbb{P}^2}$ ensuring $μ=ν$. The involved submersion $W$ comes from normal coordinates for the inverse branches of the iterates of $f$.