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Abstract. In this paper, we investigate the notion of q-pseudoconvexity to discuss and describe some geometric characterizations of q-pseudoconvex domains Ω ⊂ C n . In particular, we establish that Ω is q-pseudoconvex, if and only if, for every boundary point, the Levi form of the boundary is semipositive on the intersection of the holomorphic tangent space to the boundary with any (n−q +1)-dimensional subspace E ⊂ C n . Furthermore, we prove that the Kiselman's minimum principal holds true for all q-pseudoconvex domains in C p ×C n such that each slice is a convex tube in C n .

We establish that a sequence (Xk)k∈N of analytic subsets of a domain Ω in Cn, purely dimensioned, can be released as the family of upper-level sets for the Lelong numbers of some positive closed current. This holds whenever the sequence (Xk)k∈N satisfies, for any compact subset L of Ω, the growth condition Σ k∈N Ck mes(Xk ∩ L) < ∞. More precisely, we built a positive closed current Θ of bidimension (p, p) on Ω, such that the generic Lelong number mXk of Θ along each Xk satisfies mXk = Ck. In particular, we prove the existence of a plurisubharmonic function v on Ω such that, each Xk is contained in the upper-level set ECk (ddcv)

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