Green functions, Segre numbers, and King's formula
Green functions, Segre numbers, and King's formula
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格林函数、Segre 数和 King 公式
DOI：
10.5802/aif.2922
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Elizabeth Wulcan
中科院分区：
文献类型：
--
作者：
M. Andersson;Elizabeth Wulcan
设$\mathcal J$是复流形$X$上具有零点集$Z$的凝聚理想层，$G$是多重次调和函数，使得$G=\log| F| +\mathcal O（1）$局部在$Z$，其中$f$是定义$\mathcal J$的全纯函数的元组。本文对k= 0，1，2，.给出了Monge-齐次积（dd^c G）^k的意义，并证明了电流$M_k^{\mathcal J}：=\mathbf 1_Z（dd^c G）^k$在$x$处的Lelong数与由Tworzewski，Gaffney-Gassler，and Spilles-Manaresi独立引入的$\mathcal J$在$x$处的所谓塞格雷数一致。更一般地，我们证明了$M_k^{\mathcal J}$满足经典King公式的某种推广。
Let $\mathcal J$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=\log|f|+\mathcal O(1)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $\mathcal J$. We give a meaning to the Monge-Amp\`{e}re products $(dd^c G)^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{\mathcal J}:=\mathbf 1_Z(dd^c G)^k$ at $x$ coincide with the so-called Segre numbers of $\mathcal J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{\mathcal J}$ satisfy a certain generalization of the classical King formula.