5.34.2 Lelong level setsE911F3E2C8E9

Proof. In a coordinate chart put $\kappa =\mathrm{dd}^{\mathrm{c}}|z|^2$ and $\mu =T\wedge \kappa ^{n-j}$. By the Lelong monotonicity formula in bidimension $(n-j,n-j)$, the function $r\mapsto r^{-2(n-j)}\mu (B(x,r))$ is nondecreasing; see [Chapter III, formulas (5.5)–(5.8), pp. 158–159, DemCADG]. Fix a relatively compact inner chart and a radius $r_0>0$ such that all balls of radius $r_0$ centered in it lie in a fixed compact subset of the chart. Their masses are bounded by the mass of that compact set. Hence

uniformly for centers in the inner chart. A Borel set in the inner chart with zero Hausdorff $2(n-j)$-measure can be covered by balls $B(x_ i,r_ i)$ with $\sum _ ir_ i^{2(n-j)}$ arbitrarily small, so it has zero $\mu $-mass. Since $T$ is positive, the total variation of each of its coefficient measures is bounded by a constant multiple of $\mu $. A countable cover by charts proves the assertion. $\square$

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. $\square$