Proposition 10.13.393492323D1A9

Proof. Let $\theta +\mathrm{dd}^{\mathrm{c}}\Phi \geq \delta \omega $ be a current as in theorem 10.13.2, and suppose that $x$ is an isolated point of its polar set. Choose holomorphic coordinates $z=(z_1,\ldots ,z_ n)$ on a ball centered at $x$ that contains no other pole. For a sufficiently large constant $A>0$ and any real constant $C$, the local function $q(z)=A|z|^2-C$ satisfies $\theta +\mathrm{dd}^{\mathrm{c}}q\geq \delta '\omega $ on a smaller ball, for some $\delta '>0$.

Since $\Phi $ is smooth on a closed annular collar near the boundary of this smaller ball, we can choose $C$ so that $q<\Phi $ on the collar. Since $\Phi \to -\infty $ at $x$, the function $\max (\Phi ,q)$ agrees with $q$ near $x$ and with $\Phi $ on the collar. By lemma 5.24.2, it glues with $\Phi $ outside the ball to the potential of a Kähler current in $\alpha $ that is smooth near $x$. Its Lelong number at $x$ is zero, contradicting $x\in E_{\mathrm{nK}}(\alpha )$. $\square$