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Singularities of currents of full mass intersection

Shuang Su

TL;DR

This paper addresses how singularities of closed positive currents with full mass intersection can be compared in the mixed setting on a compact Kähler manifold. It develops and uses the theory of envelopes, relative non-pluripolar products, and Demailly's analytic approximation to relate Lelong numbers along analytic centers. The main contributions show that, under full mass intersection, at least one current must share its Lelong number with its competitor along a given center, and, in the self-intersection case, provides a quantitative bound linking the deficit in total mass to the product of Lelong-number gaps. The work extends Vu's results to broader classes (big, $\mathcal{I}$-model currents) and yields a sharp inequality for $m=n$ with potential relaxations of the modeling hypothesis, advancing understanding of singularities in mixed Monge-Ampère-type products on Kähler manifolds.

Abstract

In this paper, we study currents that have full mass intersection with respect to given currents in the mixed setting on a compact Kähler manifold. We compare their singularities by using Lelong numbers. Our main theorems generalize some results of Vu.

Singularities of currents of full mass intersection

TL;DR

This paper addresses how singularities of closed positive currents with full mass intersection can be compared in the mixed setting on a compact Kähler manifold. It develops and uses the theory of envelopes, relative non-pluripolar products, and Demailly's analytic approximation to relate Lelong numbers along analytic centers. The main contributions show that, under full mass intersection, at least one current must share its Lelong number with its competitor along a given center, and, in the self-intersection case, provides a quantitative bound linking the deficit in total mass to the product of Lelong-number gaps. The work extends Vu's results to broader classes (big, -model currents) and yields a sharp inequality for with potential relaxations of the modeling hypothesis, advancing understanding of singularities in mixed Monge-Ampère-type products on Kähler manifolds.

Abstract

In this paper, we study currents that have full mass intersection with respect to given currents in the mixed setting on a compact Kähler manifold. We compare their singularities by using Lelong numbers. Our main theorems generalize some results of Vu.

Paper Structure

This paper contains 8 sections, 15 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singularity type and Envelopes
  4. Relative non-pluripolar products
  5. Demailly's analytic approximation theorem
  6. Proof of the main theorems
  7. Proof of Theorem \ref{['maintheorem']}
  8. Proof of Theorem \ref{['maintheroem2']}

Key Result

Theorem 1.1

Let $\{\theta_{1}\}, \dots ,\{\theta_{m}\}$ be big classes, and let $T'_{j}, T_{j} \in \{\theta_{j}\}$ be closed positive $(1,1)$-currents such that Let $V$ be a proper irreducible analytic subset such that $\dim(V) \geq n-m$. If $T'_1, \dots , T'_m$ are of full mass intersection with respect to $T_{1}, \dots , T_{m}$. Then, there exists $1 \leq j \leq m$ such that $\nu(T'_{j},V)=\nu (T_j,V)$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • proof
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6: Viet-generalized-nonpluri, Theorem 4.4
  • Lemma 2.7
  • ...and 21 more