摘要翻译：
本文的目的是将具有孤立或紧支集奇点的多重调和函数的指数$e^{-2phi}$的可积性与monge-amp\ere质量$(dd^c\phi)^n$的先验界联系起来。该不等式在$\bc^n$中的任意开放子集$\omega$上局部或全局有效。我们证明$\int_\omega(dd\phi)^n<n^n$对于$\omega$中的每个紧致子集$k$意味着$\int_ke^{-2\phi}<+\infty$,而形式为$\phi(z)=n\logz-z_0$,$z_0\in\omega$的函数是极限情况。这一结果是由纯局部代数的一个不等式导出的，它是a.Corti在维数$n=2$上证明的，后来由L.Ein，t.de Fernex和M.Musta\c{t}\v{a}推广到任意维数。
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英文标题：
《Estimates on Monge-Amp\`ere operators derived from a local algebra
  inequality》
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作者：
Jean-Pierre Demailly (Universit\'e de Grenoble I)
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Complex Variables        复变数
分类描述：Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
全纯函数，自守群作用与形式，伪凸性，复几何，解析空间，解析束
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  The goal of this short note is to relate the integrability property of the exponential $e^{-2\phi}$ of a plurisubharmonic function $\phi$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Amp\`ere mass of $(dd^c\phi)^n$. The inequality is valid locally or globally on an arbitrary open subset $\Omega$ in $\bC^n$. We show that $\int_\Omega(dd\phi)^n<n^n$ implies $\int_Ke^{-2\phi}<+\infty$ for every compact subset $K$ in $\Omega$, while functions of the form $\phi(z)=n\log|z-z_0|$, $z_0\in\Omega$, appear as limit cases. The result is derived from an inequality of pure local algebra, which turns out a posteriori to be equivalent to it, proved by A.Corti in dimension $n=2$, and later extended by L.Ein, T.De Fernex and M.Musta\c{t}\v{a} to arbitrary dimensions.
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PDF链接：
https://arxiv.org/pdf/0709.3524