格林函数、塞格雷数和金公式
Green functions, Segre numbers, and King's formula
摘要：
设$\mathcal J$是复流形$X$上的一个凝聚理想层，其零集为$Z$，且$G$是一个多重次调和函数，使得$G=\log|f|+\mathcal O(1)$局部地位于$Z$上，其中$f$是定义了$\mathcal J$的全纯函数元组.我们给出了一个意义的Monge-Ampère产品$(dd^c G)^k$的$k=0,1,2,...$，并证明了Lelong数的电流$M_k^{\mathcal J}:=\mathbf 1_Z(dd^c G)^k$在$x$与所谓的Segre数的$\mathcal J$在$x$，独立介绍的Tworzewski，加夫尼-Gassler，和Ephilles-Manaresi。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。
英文摘要：
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-Ampère 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.
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