Directed harmonic currents near hyperbolic singularities

2014/11/24 by Nguyen, Viet-Anh
#37A30 #Complex Variables (math.CV) #FOS: Mathematics #Primary: 37F75 #Secondary: 57R30

paper · doi:10.48550/arxiv.1411.6421

Abstract

Let \Fc be a holomorphic foliation by curves defined in a neighborhood of 0 in \C² having 0 as a hyperbolic singularity. Let T be a harmonic current directed by \Fc which does not give mass to any of the two separatrices. Then we show that the Lelong number of T at 0 vanishes. Next, we apply this local result to investigate the global mass-distribution for directed harmonic currents on singular holomorphic foliations living on compact complex surfaces. Finally, we apply this global result to study the recurrence phenomenon of a generic leaf.

Related

• Harmonic currents directed by foliations by Riemann surfaces
• Singular holomorphic foliations by curves. III: Zero Lelong numbers
• Unique Ergodicity of Harmonic Currents on Singular Foliations of P2
• The dimension of harmonic currents on foliated complex surfaces
• Singular holomorphic foliations by curves II: Negative Lyapunov exponent