Hessian boundary measures

We will study certain boundary measures related to [Formula: see text]-subharmonic functions on [Formula: see text]-hyperconvex domains. These measures generalize the boundary measures studied by Wan and Wang (see [Complex Hessian operator and Lelong number for unbounded [Formula: see text]-subharmonic functions, Potential. Anal. 44(1) (2016) 53–69]). For the case of plurisubharmonic functions ([Formula: see text]) the boundary measure has been studied by Cegrell and Kemppe (see [Monge–Ampère boundary measures, Ann. Polon. Math. 96 (2009) 175–196]).

A Property of upper level sets of Lelong numbers of currents on ℙ2

Let [Formula: see text] be a positive closed current of bidimension [Formula: see text] with unit mass on the complex projective space [Formula: see text]. For [Formula: see text] and [Formula: see text] we show that if [Formula: see text] has four points with Lelong number at least [Formula: see text], the upper level set [Formula: see text] of points of [Formula: see text] with Lelong number strictly larger than [Formula: see text] is contained within a conic with the exception of at most one point.

Directed harmonic currents near hyperbolic singularities

Let $\mathscr{F}$ be a holomorphic foliation by curves defined in a neighborhood of $0$ in $\mathbb{C}^{2}$ having $0$ as a hyperbolic singularity. Let $T$ be a harmonic current directed by $\mathscr{F}$ which does not give mass to any of the two separatrices. We show that the Lelong number of $T$ at $0$ vanishes. Then 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.