[Paper Review] Convergence in capacity on compact Kähler manifolds

Sławomir Dinew, Phạm Hoàng Hiệp|ArXiv.org|Apr 27, 2009

Geometry and complex manifolds7 references3 citations

TL;DR

This paper establishes sharp stability and convergence results for sequences of functions in the Cegrell class $υ(X,\omega)$ on compact Kähler manifolds under convergence in capacity. By leveraging new estimates from pluripotential theory and uniform absolute continuity with respect to the $C_{X,\omega}$-capacity, it proves that $L^1$-convergence and uniform control of Monge-Ampère measures imply convergence in capacity, removing prior assumptions such as boundedness or minorant conditions.

ABSTRACT

The aim of this note is to study the convergence in capacity for functions in the class $\mathcal E(X,ω)$. We obtain several stability theorems. Some of these are (optimal) generalizations of results of Xing, while others are new.

Motivation & Objective

• To establish optimal stability theorems for convergence in capacity within the Cegrell class $υ(X,\omega)$ on compact Kähler manifolds.
• To generalize and sharpen prior results by Xing and Kołodziej by removing technical assumptions such as uniform lower bounds or boundedness.
• To prove that $L^1$-convergence and uniform control of Monge-Ampère measures imply convergence in capacity, even without boundedness.
• To demonstrate that uniform absolute continuity of measures with respect to $C_{X,\omega}$-capacity is sufficient for stability under convergence in capacity.

Proposed method

• Utilizes the $C_{X,\omega}$-capacity introduced by Kołodziej and studied by Guedj-Zeriahi, defined via suprema of Monge-Ampère masses over bounded $ω$-psh functions.
• Applies the Cegrell class $υ(X,\omega)$, consisting of $ω$-psh functions where the Monge-Ampère operator is well-defined and total mass is preserved.
• Employs truncation techniques: define $u_{jt} = \max(u_j, -t)$, $u_t = \max(u, -t)$, and use their convergence in capacity to control singularities.
• Uses the comparison principle for Monge-Ampère measures: $\int_{\{u \leq v\}} \omega_{\max(u,v)}^k \wedge T = \int_{\{u \leq v\}} \omega_u^k \wedge T$ for $T$ a product of $\omega$-psh currents.
• Applies uniform absolute continuity of $\omega_{v_j}^n$ with respect to $C_{X,\omega}$, ensuring that small capacity sets have small measure mass.
• Combines estimates on sets where $u_j < u - \delta$, using decomposition into $\{u_j \leq -t\}$, $\{u \leq -t\}$, $\{v_j \leq -t\}$, and $\{u_{jt} > u_t + \epsilon\}$, with capacity decay and measure control.

Experimental results

Research questions

• RQ1 Can convergence in capacity be established for sequences in $\u03c5(X,\omega)$ without assuming boundedness or a uniform minorant?
• RQ2 Under what conditions does $L^1$-convergence and uniform control of Monge-Ampère measures imply convergence in capacity?
• RQ3 How does the uniform absolute continuity of Monge-Ampère measures with respect to $C_{X,\omega}$-capacity affect stability theorems?
• RQ4 To what extent can the stability results of Xing and Kołodziej be generalized to the full $\u03c5(X,\omega)$ class?
• RQ5 What role does the truncation $\max(u_j, -t)$ play in controlling singularities and proving convergence in capacity?

Key findings

• The paper proves that if $u_j \to u$ in $L^1(X)$ and $\omega_{u_j}^n \leq A \omega_{v_j}^n$ for some $A > 1$, and $v_j \to v$ in $C_{X,\omega}$, then $u_j \to u$ in $C_{X,\omega}$, even without boundedness or minorant assumptions.
• The key estimate shows $\varlimsup_{j\to\infty} \int_{\{u_j < u - \delta\}} \omega_{u_j}^n \leq \sup_{j \geq 1} \int_{\{u_j \leq -t\} \cup \{u \leq -t\} \cup \{v_j \leq -t\}} \omega_{v_j}^n + \frac{2\epsilon}{\delta}$, which tends to zero as $t \to \infty$ and $\epsilon \to 0$.
• Uniform absolute continuity of $\omega_{v_j}^n$ with respect to $C_{X,\omega}$ ensures that $\int_{\{u_{jt} > u_t + \epsilon\}} \omega_{v_{jt}}^n \to 0$ as $j \to \infty$ for fixed $t, \epsilon > 0$.
• The convergence $\int_X (u_t - u_{jt}) \omega_{v_t}^n \to 0$ is established via [Ce3], relying on $L^1$-convergence and truncation control.
• The capacity $C_{X,\omega}(\{u_{jt} > u_t + \epsilon\}) \to 0$ as $j \to \infty$ by Hartogs’ lemma and truncation stability.
• The result generalizes Theorem 5 from Xing (2005) and Theorem 3.4 from Kołodziej (2005), removing the need for a fixed minorant function $v_0 \in \u03c5(X,\omega)$.