arXiv math/04124462004-12math.CV

一个推广的Poincaré-Lelong公式

A generalized Poincaré-Lelong formula

Mats Andersson

摘要：

我们证明了经典的Poincaré-Lelong公式的一个推广。给定Hermite向量丛$E\to X$的一个具有零集$Z$的全纯截面$f$，令$S$是$f$张成的$X\setminus Z$上的线丛，令$Q=E/S$ .则陈式$c(D_Q)$在$X$中是局部可积的和闭的，并且存在一个流$W$使得$dd^cW=c(D_E)-c(D_Q)-M,$，其中$M$是一个在$Z$上有支集的流。特别地，顶部Bott-Chern类由支持$Z$的电流表示。我们讨论了这些电流的正性，并揭示了它们与主值和Cauchy-Fantappiè-Leray型剩余电流的密切关系。

英文摘要：

We prove a generalization of the classical Poincaré-Lelong formula. Given a holomorphic section $f$, with zero set $Z$, of a Hermitian vector bundle $E\to X$, let $S$ be the line bundle over $X\setminus Z$ spanned by $f$ and let $Q=E/S$. Then the Chern form $c(D_Q)$ is locally integrable and closed in $X$ and there is a current $W$ such that $dd^cW=c(D_E)-c(D_Q)-M,$ where $M$ is a current with support on $Z$. In particular, the top Bott-Chern class is represented by a current with support on $Z$. We discuss positivity of these currents, and we also reveal a close relation with principal value and residue currents of Cauchy-Fantappiè-Leray type.

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