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The generalized Lelong numbers and intersection theory

Viet-Anh Nguyen

TL;DR

This work advances the theory of generalized Lelong numbers along a smooth base $V$ within a complex manifold by defining $ u(T,B,oldsymbol ext{ω}^{(j)},r, au,h)$ for currents $T$, and proving their existence, intrinsic nature, and relation to tangent currents. It introduces the horizontal dimension $oldsymbol ext{ħ}$ and Siu-type semicontinuity results, linking these numbers to the Dinh–Sibony cohomology classes ${f c}^{DS}_j(T,V)$, thereby unifying intrinsic Lelong data with tangent-current cohomology. The paper then provides effective criteria for the intersection of multiple positive currents in the DS framework, including wedgeability and continuity results, and culminates with a continuity theory via super-potentials and blow-up models. Collectively, these results extend classical Lelong theory to higher bidegrees, enable robust intersection theory on Kähler manifolds, and offer practical criteria for stability and continuity of tangent/intersection currents with potential applications in complex geometry and dynamics.

Abstract

Let $X$ be a complex manifold of dimension $k,$ and $(V,ω)$ be a Kähler submanifold of dimension $l$ in $X,$ and $B\Subset V$ be a domain with $\mathcal{C}^2$-smooth boundary. Let $T$ be a positive plurisubharmonic current on $X$ such that $T$ satisfies a reasonable approximation condition on $X$ and near $\partial B.$ In our previous work we introduce the concept of the generalized Lelong numbers $ν_j(T,B)\in\mathbb{R}$ of $T$ along $B$ for $0\leq j\leq l.$ When $l=0,$ $V=B$ is a single point $x,$ $ν_0(T,B)$ is none other than the classical Lelong number of $T$ at $x.$ This article has five purposes: Firstly, we formulate the notion of the generalized Lelong number of $T$ associated to every closed smooth $(j,j)$-form on $V.$ This concept extends the previous notion of the generalized Lelong numbers. We also establish their basic properties. Secondly, we define the horizontal dimension $\hbar$ of such a current $T$ along $B.$ Next, we characterize $\hbar$ in terms of the generalized Lelong numbers. We also establish a Siu's upper-semicontinuity type theorem for the generalized Lelong numbers. In their above-mentioned context, Dinh and Sibony introduced some cohomology classes which may be regarded as their analogues of the classical Lelong numbers. Our third objective is to generalize their notion to the broader context where $T$ is (merely) positive pluriharmonic. Moreover, we also establish a formula relating Dinh-Sibony classes and the generalized Lelong numbers. Fourthly, we obtain an effective sufficient condition for defining the intersection of $m$ positive closed currents in the sense of Dinh-Sibony's theory of tangent currents on a compact Kähler manifold. Finally, we establish an effective sufficient condition for the continuity of the above intersection.

The generalized Lelong numbers and intersection theory

TL;DR

This work advances the theory of generalized Lelong numbers along a smooth base within a complex manifold by defining for currents , and proving their existence, intrinsic nature, and relation to tangent currents. It introduces the horizontal dimension and Siu-type semicontinuity results, linking these numbers to the Dinh–Sibony cohomology classes , thereby unifying intrinsic Lelong data with tangent-current cohomology. The paper then provides effective criteria for the intersection of multiple positive currents in the DS framework, including wedgeability and continuity results, and culminates with a continuity theory via super-potentials and blow-up models. Collectively, these results extend classical Lelong theory to higher bidegrees, enable robust intersection theory on Kähler manifolds, and offer practical criteria for stability and continuity of tangent/intersection currents with potential applications in complex geometry and dynamics.

Abstract

Let be a complex manifold of dimension and be a Kähler submanifold of dimension in and be a domain with -smooth boundary. Let be a positive plurisubharmonic current on such that satisfies a reasonable approximation condition on and near In our previous work we introduce the concept of the generalized Lelong numbers of along for When is a single point is none other than the classical Lelong number of at This article has five purposes: Firstly, we formulate the notion of the generalized Lelong number of associated to every closed smooth -form on This concept extends the previous notion of the generalized Lelong numbers. We also establish their basic properties. Secondly, we define the horizontal dimension of such a current along Next, we characterize in terms of the generalized Lelong numbers. We also establish a Siu's upper-semicontinuity type theorem for the generalized Lelong numbers. In their above-mentioned context, Dinh and Sibony introduced some cohomology classes which may be regarded as their analogues of the classical Lelong numbers. Our third objective is to generalize their notion to the broader context where is (merely) positive pluriharmonic. Moreover, we also establish a formula relating Dinh-Sibony classes and the generalized Lelong numbers. Fourthly, we obtain an effective sufficient condition for defining the intersection of positive closed currents in the sense of Dinh-Sibony's theory of tangent currents on a compact Kähler manifold. Finally, we establish an effective sufficient condition for the continuity of the above intersection.
Paper Structure (51 sections, 94 theorems, 321 equations, 1 figure)

This paper contains 51 sections, 94 theorems, 321 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main purpose of this work
  3. Classical Lelong numbers
  4. Tangent currents for $l > 0$ and Dinh-Sibony theory
  5. Alessandrini-Bassanelli theory
  6. Organization of the article and acknowledgments
  7. The generalize Lelong numbers and statement of main results
  8. The generalized Lelong numbers
  9. First main result: existence of the generalized Lelong numbers
  10. Second main results: horizontal dimension and a Siu's upper-semicontinuity type theorem
  11. Third main result: Dinh-Sibony classes versus the generalized Lelong numbers
  12. Fourth main results: intersection theory and effective criteria
  13. Fifth main results: continuity of Dinh-Sibony intersection
  14. Background
  15. Currents and positive currents
  16. ...and 36 more sections

Key Result

Theorem 1.3

(Dinh-Sibony DinhSibony18) Let $X,$$V,$$\mathbb{E},$$\overline \mathbb{E},$$A_\lambda$ and $\tau$ be as above. Let $T$ be a positive closed $(p,p)$-current on $X.$ Assume in addition that $X$ is Kähler and ${\rm supp}(T)\cap V$ is compact. Then:

Figures (1)

  • Figure 1: Illustrations of a Tube ${\rm Tube}(B,r)$ with base $B$ and radius $r,$ its horizontal boundary $\partial_{\rm hor} {\rm Tube}(B,r)$ and its vertical boundary $\partial_{\rm ver} {\rm Tube}(B,r).$ The boundary $\partial B$ of $B$ is represented by $[$ and $]$ on the base $\overline B.$

Theorems & Definitions (131)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 121 more