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arXiv 2106.10883math.CV
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非多极能量与复Monge-Ampère算子

Non-pluripolar energy and the complex Monge-Ampère operator

Mats Andersson, David Witt Nyström, Elizabeth Wulcan

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38 pages
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摘要：

给定整环$Ω\subset \mathbf C^n$，在$\mathcal G(Ω)$上引入一类多次调和函数$\mathcal G(Ω)$和Monge-Ampère算子$u\mapsto [dd^c u]^p$，$p\leq n$，它们是Bedford-Taylor-Demailly Monge-Ampère算子的推广.此处$[dd^c u]^p$是二度$(p,p)$的闭合正电流，其主导非多极Monge-Ampère电流$\langle dd^c u\rangle^p$。我们证明了$[dd^c u]^p$是$u$的某些自然正则化的Monge-Ampère电流的极限。在紧致Kähler流形$(X, ω)$上，我们引入了非多极能量的概念和相应的有限能量类$\mathcal G(X, ω)\subset \text{PSH}(X, ω)$，它是$\mathcal G(Ω)$的整体形式.从局部构造，我们得到全局Monge-Ampère电流$[dd^c φ+ ω]^p$为$φ\in \mathcal G(X,ω)$，只依赖于当前的$dd^c φ+ ω$。$φ$的某些自然正则化的Monge-Ampère电流的极限可以用$j\leq p$的$[dd^c φ+ ω]^j$表示。我们得到了一个包含电流$[dd^c φ+ω]^p$的质量公式，它描述了非多极Monge-Ampère测度$\langle dd^c φ+ω\rangle^n$的质量损失。类$\mathcal G(X, ω)$包括具有解析奇点的$ω$ -psh函数和有限能量的$ω$ -psh函数类$\mathcal E(X, ω)$以及某些其他凸能量类，尽管它本身不是凸的。

英文摘要：

Given a domain $Ω\subset \mathbf C^n$ we introduce a class of plurisubharmonic (psh) functions $\mathcal G(Ω)$ and Monge-Ampère operators $u\mapsto [dd^c u]^p$, $p\leq n$, on $\mathcal G(Ω)$ that extend the Bedford-Taylor-Demailly Monge-Ampère operators. Here $[dd^c u]^p$ is a closed positive current of bidegree $(p,p)$ that dominates the non-pluripolar Monge-Ampère current $\langle dd^c u\rangle^p$. We prove that $[dd^c u]^p$ is the limit of Monge-Ampère currents of certain natural regularizations of $u$. On a compact Kähler manifold $(X, ω)$ we introduce a notion of non-pluripolar energy and a corresponding finite energy class $\mathcal G(X, ω)\subset \text{PSH}(X, ω)$ that is a global version of $\mathcal G(Ω)$. From the local construction we get global Monge-Ampère currents $[dd^c φ+ ω]^p$ for $φ\in \mathcal G(X,ω)$ that only depend on the current $dd^c φ+ ω$. The limits of Monge-Ampère currents of certain natural regularizations of $φ$ can be expressed in terms of $[dd^c φ+ ω]^j$ for $j\leq p$. We get a mass formula involving the currents $[dd^c φ+ω]^p$ that describes the loss of mass of the non-pluripolar Monge-Ampère measure $\langle dd^c φ+ω\rangle^n$. The class $\mathcal G(X, ω)$ includes $ω$-psh functions with analytic singularities and the class $\mathcal E(X, ω)$ of $ω$-psh functions of finite energy and certain other convex energy classes, although it is not convex itself.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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