Uniqueness of tangent currents for positive closed currents

2025/02/10 by Nguyen, Viet-Anh, Truong, Tuyen Trung · 1 citation
#14C30 #32J25 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2502.06532

Abstract

Let X be a complex manifold X of dimension k, and let V⊂ X be a Kähler submanifold of dimension l, and let B⊂ V be a piecewise C²-smooth domain. Let T be a positive closed currents of bidegree (p,p) in X such that T satisfies a mild reasonable assumption in a neighborhood of ∂ B in X and that the j-th average mean ν_j(T,B,r) for every j with max(0,l-p)≤ j≤min(l,k-p) converges sufficiently fast to the j-th generalized Lelong number ν_j(T,B) as r tends to 0 so that r^-1(ν_j(T, B,r)-ν_j( T,B)) is locally integrable near r=0. Then we show that T admits a unique tangent current along B. A local version where we replace the condition of T near B by the conditions on a finite cover of B by piecewise C²-smooth domains in V is also given. When T is a current of integration over a complex analytic set, we show that ν_j(T,B,r)-ν_j(T,B)=O(r^ρ) for some ρ>0, and hence this condition is satisfied. Our result may be viewed as a natural generalization of Blel-Demailly-Mouzali's criterion from the case l=0 to the case l>0. The result has applications in the intersection theory of positive closed currents.

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