Pome691315D9AA1CLemma 20.4.14.691315D9AA1C Let $w$ be a quasi-plurisubharmonic function on $Y$, and take the ambient manifold in (20.3.3) to be $Y$ itself. For every prime divisor $E$ on a smooth compact modification $\pi :Z\to Y$, the resulting self-trace satisfies
In particular, for the potential $w$ of the closed positive current $T$, one has $N_ Z(T)=S_ Z(w;Y)$.
Proof. Since $w$ is not identically $-\infty $, its Lelong number vanishes at almost every point; hence $\nu (w,Y)=0$, and the self-trace is defined. Fix a divisor $E$ on a compact modification and apply lemma 5.38.9 on a coordinate neighborhood of a point of its image in $Y$. With $\nu _ E=\nu (w\circ \pi ,E)$ and $A_ E$ the log discrepancy of $E$, it gives
The limit along the integers defining $\tau _{Y,w}(E)$ is therefore $\nu _ E$. Since all coefficients agree, the two divisorial currents are equal. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.