Harmonic currents directed by foliations by Riemann surfaces

Dinh, Tien-Cuong; Wu, Hao

doi:10.1090/proc/15470

Harmonic currents directed by foliations by Riemann surfaces
HTML articles powered by AMS MathViewer

by Tien-Cuong Dinh and Hao Wu PDF

Proc. Amer. Math. Soc. 149 (2021), 3453-3461 Request permission

Abstract:

We study local positive $dd^{c}$-closed currents directed by a foliation by Riemann surfaces near a hyperbolic singularity which have no mass on the separatrices. A theorem of Nguyên says that the Lelong number of such a current at the singular point vanishes. We prove that this property is sharp: one cannot have any better mass estimate for this current near the singularity.

References

D. V. Anosov, S. Kh. Aranson, V. I. Arnold, I. U. Bronshtein, V. Z. Grines, and Yu. S. Il′yashenko, Ordinary differential equations and smooth dynamical systems, Springer-Verlag, Berlin, 1997. Translated from the 1985 Russian original by E. R. Dawson and D. O’Shea; Third printing of the 1988 translation [Dynamical systems. I, Encyclopaedia Math. Sci., 1, Springer, Berlin, 1988; MR0970793 (89g:58060)]. MR 1633529
Bo Berndtsson and Nessim Sibony, The $\overline \partial$-equation on a positive current, Invent. Math. 147 (2002), no. 2, 371–428. MR 1881924, DOI 10.1007/s002220100178
Tien-Cuong Dinh and Nessim Sibony, Unique ergodicity for foliations in $\Bbb P^2$ with an invariant curve, Invent. Math. 211 (2018), no. 1, 1–38. MR 3742755, DOI 10.1007/s00222-017-0744-2
John Erik Fornæss and Nessim Sibony, Unique ergodicity of harmonic currents on singular foliations of $\Bbb P^2$, Geom. Funct. Anal. 19 (2010), no. 5, 1334–1377. MR 2585577, DOI 10.1007/s00039-009-0043-1
John Erik Fornæss, Nessim Sibony, and Erlend Fornæss Wold, Examples of minimal laminations and associated currents, Math. Z. 269 (2011), no. 1-2, 495–520. MR 2836081, DOI 10.1007/s00209-010-0747-9
Lucy Garnett, Foliations, the ergodic theorem and Brownian motion, J. Functional Analysis 51 (1983), no. 3, 285–311. MR 703080, DOI 10.1016/0022-1236(83)90015-0
Viêt-Anh Nguyên, Directed harmonic currents near hyperbolic singularities, Ergodic Theory Dynam. Systems 38 (2018), no. 8, 3170–3187. MR 3868026, DOI 10.1017/etds.2017.2
Viêt-Anh Nguyên, Singular holomorphic foliations by curves I: integrability of holonomy cocycle in dimension 2, Invent. Math. 212 (2018), no. 2, 531–618. MR 3787833, DOI 10.1007/s00222-017-0772-y
Viêt-Anh Nguyên, Singular holomorphic foliations by curves III: zero Lelong numbers, 2020. arXiv:2009.06566.
Yum Tong Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974), 53–156. MR 352516, DOI 10.1007/BF01389965
Dennis Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255. MR 433464, DOI 10.1007/BF01390011