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Uniqueness of tangent currents for positive closed currents

Viet-Anh Nguyen, Tuyen Trung Truong

TL;DR

This work extends the uniqueness criteria for tangent currents from the classical $l=0$ setting to higher-dimensional centers $V$ by leveraging the Dinh–Sibony theory and generalized Lelong numbers. It defines intrinsic, higher-dimensional Lelong numbers $ u_j(T,B, au,h)$ and develops the CL classes ${ m CL}^{p;m,m'}$ to handle currents with boundary control, proving that under suitable integrability conditions on the Lelong-number discrepancies, tangent currents along $B$ are unique. The paper provides a local version of the criteria applicable to finite covers of $B$, and it specializes to currents of integration over analytic sets, where it yields a quantified rate of convergence to the tangent current and a construction of cylindrical cones in the normal bundle. The results have applications to intersection theory of positive closed currents and extend the applicability of tangent-current techniques to positive-dimensional centers in complex geometry.

Abstract

Let $X$ be a complex manifold $X$ of dimension $k,$ and let $V\subset X$ be a Kähler submanifold of dimension $l,$ and let $B\subset V$ be a piecewise $\mathcal{C}^2$-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 $\partial B$ in $X$ and that the $j$-th average mean $ν_j(T,B,r)$ for every $j$ with $\max(0,l-p)\leq j\leq\min(l,k-p)$ 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 $\mathcal{C}^2$-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.

Uniqueness of tangent currents for positive closed currents

TL;DR

This work extends the uniqueness criteria for tangent currents from the classical setting to higher-dimensional centers by leveraging the Dinh–Sibony theory and generalized Lelong numbers. It defines intrinsic, higher-dimensional Lelong numbers and develops the CL classes to handle currents with boundary control, proving that under suitable integrability conditions on the Lelong-number discrepancies, tangent currents along are unique. The paper provides a local version of the criteria applicable to finite covers of , and it specializes to currents of integration over analytic sets, where it yields a quantified rate of convergence to the tangent current and a construction of cylindrical cones in the normal bundle. The results have applications to intersection theory of positive closed currents and extend the applicability of tangent-current techniques to positive-dimensional centers in complex geometry.

Abstract

Let be a complex manifold of dimension and let be a Kähler submanifold of dimension and let be a piecewise -smooth domain. Let be a positive closed currents of bidegree in such that satisfies a mild reasonable assumption in a neighborhood of in and that the -th average mean for every with converges sufficiently fast to the -th generalized Lelong number as tends to so that is locally integrable near Then we show that admits a unique tangent current along A local version where we replace the condition of near by the conditions on a finite cover of by piecewise -smooth domains in is also given. When is a current of integration over a complex analytic set, we show that for some and hence this condition is satisfied. Our result may be viewed as a natural generalization of Blel-Demailly-Mouzali's criterion from the case to the case The result has applications in the intersection theory of positive closed currents.

Paper Structure

This paper contains 25 sections, 32 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Tangent currents for $l=0$
  3. Tangent currents for $l > 0$ and Dinh-Sibony theory, and Alessandrini-Bassanelli's Lelong number
  4. The generalized Lelong numbers
  5. Classes ${\rm CL}^{p;m,m'}$ ($0\leq m'\leq m$) of positive closed currents and a tangent theorem
  6. Statement of the main results
  7. Preparatory results
  8. Positive currents
  9. (Strongly) admissible maps
  10. Lelong-Jensen formula for closed currents
  11. Technical tools
  12. Forms $\alpha_{\rm ver}$ and $\beta_{\rm ver}$
  13. Analysis in local coordinates
  14. Strong Standing Hypothesis
  15. Representative currents
  16. ...and 10 more sections

Key Result

Theorem 1.1

(Blel-Demailly-Mouzali BlelDemaillyMouzali) Let $T$ be a positive closed $(p,p)$ current on $X$ and $x$ a point in $X.$ Suppose that one of the following conditions is satisfied for some $r_0>0.$ Then $T$ admits a unique tangent current.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Proposition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 36 more