Pome

Lemma 5.37.2EBB1890494B1

Lemma 5.37.2.EBB1890494B1 Let $\omega $ be a Hermitian form on $X$, and fix distinct points $p_1,\ldots ,p_ N$ of $X$. There are constants $C_ j>0$ such that, whenever $a_ j>0$ and $\sum _ jC_ ja_ j<1$, there is $u\in \mathrm{PSH}(X,\omega )$, smooth off these points, with

\[ u=a_ j\log |z|^2+O(1)\quad \hbox{near }p_ j,\qquad \omega +\mathrm{dd}^{\mathrm{c}}u\geq \left(1-\sum _ jC_ ja_ j\right)\omega . \]

Here $z$ denotes local coordinates centered at $p_ j$. The function can be chosen to vanish outside arbitrarily small disjoint coordinate neighborhoods, with the constants depending on those neighborhoods.

Proof. Choose disjoint coordinate balls centered at the points $p_ j$ and smooth cutoffs $\chi _ j$ supported in them and equal to one near their centers. The function $\chi _ j\log |z|^2$, extended by zero, is plurisubharmonic near its pole and smooth elsewhere, so its complex Hessian is bounded below by $-C_ j\omega $. Set $u=\sum _ ja_ j\chi _ j\log |z|^2$. These lower bounds give the asserted current inequality, and the remaining properties follow from the disjointness of the supports. $\square$

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