Example 5.34.4697E97CED780

Proof. For a linear form $\ell $ with coefficient vector of norm one, the function $v_\ell =\log (|\ell (Z)|^2/\| Z\| ^2)$ is nonpositive and satisfies $\mathrm{dd}^{\mathrm{c}}v_\ell =[\ell =0]-\sigma $. Its $L^1(X,\sigma ^2)$ norm is finite and independent of $\ell $, because the unitary group $U(3)$ acts on homogeneous coordinates, preserves $\sigma $, and acts transitively on such forms. Since $\sum _ j\lambda _ j=1/2$, the partial sums of the series defining $u$ decrease to an $L^1$ function. They have the common curvature lower bound $-\frac32\sigma $, so their limit is quasi-plurisubharmonic and

The series of positive closed currents converges, since every line has $\sigma $-trace mass one and the coefficients are summable.

We compute the Lelong numbers from the local mass formula. In affine coordinates $w=(w_1,w_2)$ centered at a point $q$, put $\kappa =\mathrm{dd}^{\mathrm{c}}|w|^2$. An affine complex line $H$ at distance $d$ from $q$ satisfies

Consequently $r^{-2}\int _{H\cap B(q,r)}\kappa $ is at most one and tends to $\mathbf1_ H(q)$ as $r\downarrow 0$; a line that does not meet the chart contributes zero. Tonelli's theorem and dominated convergence for the summable coefficients therefore give

Here we used that a smooth local correction does not change the Lelong number. The computation also covers points at which infinitely many lines accumulate.

The lines $L_ j$ meet $D$ at the distinct points $p_ j$ and meet one another only at $p_\infty =[0:1:0]\notin D$. Thus $\nu (u,p_ j)=1+\lambda _ j$, whereas $\nu (u,q)=1$ for $q\in D\setminus \{ p_ j:j\geq 1\} $. Outside $D$, the Lelong number is at most $\sum _ j\lambda _ j=1/2$. This proves the two level-set identities. Finally, $p_ j\to p_*=[0:0:1]$ and $\nu (u,p_*)=1$, so the strict level set is not closed and hence not analytic. $\square$