Lemma 13.24.7487BF774470F

Proof. The Poincaré–Lelong formula gives $\mathrm{dd}^{\mathrm{c}}\ell _ j=[H_ j]-\kappa $. Unitary invariance reduces its integral to the distribution of $x=|Z_0|^2/\| Z\| ^2$, whose density on $[0,1]$ is $2(1-x)$. Thus

The decreasing series has finite $L^1$ norm and a common lower curvature bound $-A\kappa $. Its canonical limit is quasi-plurisubharmonic. Distributional convergence gives the current identities and strict positivity.

Fix $\pi $ and $E$, and choose a Gauduchon form $\gamma $ on the compact surface $Z$. For the tail $r_ N=\sum _{j>N}a_ j\ell _ j$ of total coefficient $A_ N=\sum _{j>N}a_ j$, the closed positive current

has trace mass $A_ N\int _ Z\pi ^*\kappa \wedge \gamma $, by distributional Stokes. The pullback of $r_ N$ is quasi-plurisubharmonic, since $\pi $ is an isomorphism on a dense open set where $r_ N$ is finite almost everywhere. Divisorial removal gives

The denominator is positive and the ratio is finite. Finite additivity of Lelong numbers now identifies the partial profile with $\nu (u\circ \pi ,E)$ up to an error tending to zero. This proves convergence and the asserted equality. $\square$