The dimension of harmonic currents on foliated complex surfaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929755739127808 |
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| author | Deroin, Bertrand Dupont, Christophe Kleptsyn, Victor |
| author_facet | Deroin, Bertrand Dupont, Christophe Kleptsyn, Victor |
| contents | Let $\mathcal{F}$ be a singular holomorphic foliation on an algebraic complex surface $S$, with hyperbolic singularities and no foliated cycle. We prove a formula for the transverse Hausdorff dimension of the unique harmonic current, involving the Furstenberg entropy and the Lyapunov exponent. In particular, we extend Brunella's inequality to every holomorphic foliation $\mathcal{F}$ on $\mathbb P^2$: if $\mathcal{F}$ has degree $d \geq 2$, then the Hausdorff dimension of its harmonic current is smaller than or equal to ${d-1 \over d+2}$, in particular the harmonic current is singular with respect to the Lebesgue measure. We also show that the Hausdorff dimension of the harmonic current of the Jouanolou foliation of degree $2$ is equal to $1/4$, and that the same property holds for topologically conjugate foliations on $\mathbb P^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09152 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The dimension of harmonic currents on foliated complex surfaces Deroin, Bertrand Dupont, Christophe Kleptsyn, Victor Differential Geometry Dynamical Systems Let $\mathcal{F}$ be a singular holomorphic foliation on an algebraic complex surface $S$, with hyperbolic singularities and no foliated cycle. We prove a formula for the transverse Hausdorff dimension of the unique harmonic current, involving the Furstenberg entropy and the Lyapunov exponent. In particular, we extend Brunella's inequality to every holomorphic foliation $\mathcal{F}$ on $\mathbb P^2$: if $\mathcal{F}$ has degree $d \geq 2$, then the Hausdorff dimension of its harmonic current is smaller than or equal to ${d-1 \over d+2}$, in particular the harmonic current is singular with respect to the Lebesgue measure. We also show that the Hausdorff dimension of the harmonic current of the Jouanolou foliation of degree $2$ is equal to $1/4$, and that the same property holds for topologically conjugate foliations on $\mathbb P^2$. |
| title | The dimension of harmonic currents on foliated complex surfaces |
| topic | Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2503.09152 |