BNU mini-course on several complex variables and complex geometry
Speaker: Dr. Zhangchi Chen (AMSS)
Schedule: 13:30-16:15, Oct 26, -Nov 2,9,16, 23, 2022
Location: Jiaoba 411, BNU
Contact: Zhiwei Wang (zhiwei@bnu.edu.cn)
Title: Ergodicity and density theory of harmonic currents directed by singular holomorphic foliations (5 talks)
Abstract: Holomorphic foliations can be viewed as a continuous-time version of complex dynamical systems, while directed harmonic (closed resp.) currents are analogous to orbits (finite periods resp.) of iterations. By analysing directed harmonic currents, one can prove dynamical results like unique ergodicity of foliations. In this series of talks, I will introduce the related works of Dinh, Fornæss, Nguyên, Sibony, and myself.
Talk 1: Introduction of currents, foliations and ergodicity.
In this talk I will present the analogue of ergodicity between iterations of holomorphic maps and holomorphic foliations. Some basic concepts like directed harmonic currents, leave-wise Poincare measures, Nevanlinna current will be introduced.
Talk 2: Fornæss -Sibony's geometric intersection and unique ergodicity of holomorphic foliations in CP2.
In 2010 Fornæss -Sibony proved that given a holomorphic foliation in CP2 with only hyperbolic singularities and admits no invariant algebraic curves, there is a unique directed harmonic current of mass 1. The proof used the geometric self-intersection of such current.
Talk 3: Dinh-Sibony's tangent currents and ergodicity theorems for foliations in CP2 with an invariant curve.
In 2018, Dinh-Sibony characterised harmonic currents directed by holomorphic foliations in CP2 with an invariant algebraic curve. They proved the existence of tangent currents, and established ergodicity results.
Talk 4: Dinh- Nguyên -Sibony's ergodicity theorem on compact Kähler surfaces.
In 2019, Dinh-Nguyên-Sibony generalized ergodicity theorems to compact Kähler surfaces. The new idea is the density theory of two currents, which is defined by the tangent current of their tensor products.
Talk 5: Lelong number of directed harmonic currents.
The Lelong number is a power tool to study local density of harmonic currents. In 2014 Nguyên proved that the Lelong number of directed harmonic currents near hyperbolic singularities vanish, assuming that such currents give no mass on separatrices. In 2022 Chen analysed the Lelong number in the non-hyperbolic case and obtained both vanishing and non-vanishing results. Geometrically, the Lelong number reveals the visibility of one regular point from another point.
References
[1] Chen, Z.C.: Directed harmonic currents near non-hyperbolic linearizable singularities, Ergodic Theory Dynam. Systems (2022), First View, pp. 1-30, DOI: 10.1017/etds.2022.46
[2] Dinh, T.C.; Sibony, N.: Unique ergodicity for foliations in P2 with an invariant curve, Invent. Math. 211 (2018), no. 1, 1-38.
[3] Dinh, T.C.; Sibony, N.: Density of positive closed currents, a theory of non-generic intersections, J. Algebraic Geom. 27 (2018), no.3, 497-551.
[4] Dinh, T.C.; Nguyên, V.A.; Sibony, N.: Unique Ergodicity for foliations on compact Kähler surfaces, arxiv:1811.07450v2
[5] Fornæss, J.E.; Sibony, N.: Unique ergodicity of harmonic currents on singular foliations of P2, Geom. Funct. Anal. 19 (2010), no. 5,1334-1377.
[6] Nguyên, V.A.: Directed harmonic currents near hyperbolic singularities, Ergodic Theory Dynam. Systems 38 (2018), no. 8, 3170-3187.
[7] Nguyên, V.A.: Ergodic theorems for laminations and foliations: recent results and perspectives, Acta Math. Vietnam. 46 (2021)
