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Let $M$ be a compact complex manifold, $S\subset M$ a submanifold of codimension $2$, let $\omega$ be a k\"ahler metric on $M\setminus S$. Then we know by Reese Harvey's paper "Removable singularities for positive currents" that $\omega$ extends to a unique closed positive current $\tilde{\omega}$ with $L^1_{loc}(M)$ coefficients.

I want to know the behavior of $\omega$ near $S$. The extension $\tilde{\omega}$ may not be smooth.For example, if $A=\{0\}\subset \mathbb{C}^2$ and $\omega=i\partial\bar{\partial}|z|^2+i\partial\bar{\partial}\log|z|$ on $\mathbb{C}^2-\{0\}$. Obviously $\omega$ is Kähler on $\mathbb{C}^2-\{0\}$, but $\tilde{\omega}$ is not smooth at the origin.

Especially I'm interested in the order of infinity $\omega$ can possess near $S$. I feel that the speed of $\omega$ going to infinity near $S$ might be controlled by $\partial\bar{\partial}\log (dist(x,S))$ as $x\in M$ tends to $S$. But I don't know how to prove this statement is right or not.

How to set out to study $\omega$? Can you give me some advice? Thanks in advance!

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  • $\begingroup$ In fact Harvey proved the following statement: Let $A$ be a closed subset of an n-dimensional complex manifold $X$ such that the Hausdorff $(2n−3)$-measure of $A$ vanishes; then each positive holomorphic line bundle on $X−A$ extends to $X$ . See Sibony, Nessim: Quelques problemes de prolongement de courants en analyse complexe. Duke Math. J. 52, 157–197 (1985). projecteuclid.org/euclid.dmj/1077304283 $\endgroup$
    – user21574
    Nov 25, 2017 at 23:09
  • $\begingroup$ See the Proposition 1 of arxiv.org/pdf/1507.06195.pdf $\endgroup$
    – user21574
    Nov 25, 2017 at 23:21

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