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Pré-Publication, Document De Travail Année : 2023

On slice measures of Green currents on CP(2)

Résumé

Let $f$ be a holomorphic map of $\mathbb{C}\mathbb{P}^2$ of degree $d\geq 2$, let $T$ be its Green current and $\mu=T\wedge T$ be its equilibrium measure. We give a new proof of a theorem due to Dujardin asserting that $\mu \ll T\wedge\omega_{\mathbb{P}^2}$ implies $\lambda_2=\frac{1}{2} \log\ d$, where $\lambda_1 \geq \lambda_2$ are the Lyapunov exponents of $\mu$. Then, assuming $\mu\ll T\wedge\omega_{\mathbb{P}^2}$, we study slice measures $\nu :=T\wedge dd^c|W|^2$, where $W$ is a holomorphic local submersion. We give sufficient conditions on the Radon-Nikodym derivative of $\mu$ with respect to the trace measure $T\wedge\omega_{\mathbb{P}^2}$ ensuring $\mu=\nu$. The involved submersion $W$ comes from normal coordinates for the inverse branches of the iterates of $f$.
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Dates et versions

hal-04256303 , version 1 (24-10-2023)
hal-04256303 , version 2 (23-02-2024)

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Christophe Dupont, Virgile Tapiero. On slice measures of Green currents on CP(2). 2023. ⟨hal-04256303v1⟩
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