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Lemma 5.37.4ACC2529A680B

Lemma 5.37.4.ACC2529A680B Let $u,u',v,v'$ be quasi-plurisubharmonic functions on $X$. For any quasi-plurisubharmonic $h$, one has

\[ u+h\sim _{\mathcal I}v+h \quad \Longleftrightarrow \quad u\sim _{\mathcal I}v. \]

If $u\sim _{\mathcal I}v$ and $u'\sim _{\mathcal I}v'$, then

\[ \max (u,u')\sim _{\mathcal I}\max (v,v'),\qquad u+u'\sim _{\mathcal I}v+v',\qquad a u\sim _{\mathcal I}a v\quad (a>0). \]

Proof. Fix a coordinate neighborhood $U\subset X$ and a modification $\pi :Y\to U$ with smooth source. At every $y\in Y$, the elementary rules for the Lelong numbers of maxima, sums and positive multiples give

\[ \begin{aligned} \nu (\max (u,u')\circ \pi ,y) & =\min \{ \nu (u\circ \pi ,y),\nu (u'\circ \pi ,y)\} ,\\ \nu ((u+u')\circ \pi ,y) & =\nu (u\circ \pi ,y)+\nu (u'\circ \pi ,y),\\ \nu (a u\circ \pi ,y)& =a\nu (u\circ \pi ,y). \end{aligned} \]

These rules are unaffected by smooth local corrections. Since $u\sim _{\mathcal I}v$ and $u'\sim _{\mathcal I}v'$, the first item of theorem 5.37.3 shows that the right-hand sides do not change when $u,u'$ are replaced by $v,v'$. The converse direction of the same item, applied on each coordinate neighborhood, gives the three asserted equivalences.

For the cancellation rule, suppose that $u+h\sim _{\mathcal I}v+h$. By the first item of theorem 5.37.3, the pullbacks $(u+h)\circ \pi $ and $(v+h)\circ \pi $ have the same Lelong numbers. The common summand $h\circ \pi $ is quasi-plurisubharmonic, so its Lelong numbers are finite, and subtracting them shows that $u\circ \pi $ and $v\circ \pi $ have the same Lelong numbers. Hence $u\sim _{\mathcal I}v$ by the same item. The reverse implication is the addition rule proved above. $\square$

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