Singular holomorphic foliations by curves. III: Zero Lelong numbers

2020/09/14 by Nguyen, Viet-Anh
#37A30 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Primary: 37F75 #Secondary: 57R30

paper · doi:10.48550/arxiv.2009.06566

Abstract

Let F be a holomorphic foliation by curves defined in a neighborhood of 0 in ℂⁿ (n≥ 2) having 0 as a weakly hyperbolic singularity. Let T be a positive harmonic current directed by F which does not give mass to any of the n coordinate invariant hyperplanes \z_j=0\ for 1≤ j≤ n. Then we show that the Lelong number of T at 0 vanishes. Moreover, an application of this local result in the global context is given. We discuss also the relation between several basic notions such as directed positive harmonic currents, directed positive ddc-closed currents, Lelong numbers etc. in the framework of singular holomorphic foliations.

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