Pome

Corollary 18.7.25AE1D9158EE3

Corollary 18.7.2.5AE1D9158EE3 Let $\epsilon _ j\geq 0$ tend to zero, and let $a_ j\in \mathrm{PSH}(X,\theta +\epsilon _ jh)$ satisfy $u\preceq _{\mathcal I}a_ j$. Suppose that, for every integer $m\geq 1$ and every $\delta >0$,

\[ \mathcal I((1+\delta )m a_ j)\subseteq \mathcal I(mu) \quad \text{for all sufficiently large }j. \]

Then the conclusions of proposition 18.7.1 hold.

Proof. Fix a local divisor $E$ over a coordinate neighborhood $U$, and write $A=A_ U(E)$ for its log discrepancy. By lemma 5.38.9, the assumed inclusion implies

\[ (1+\delta )m\nu _ E(a_ j) \geq \operatorname {ord}_ E\mathcal I((1+\delta )m a_ j) \geq \operatorname {ord}_ E\mathcal I(mu) \geq m\nu _ E(u)-A \]

for all large $j$. Taking the lower limit in $j$ first, and then letting $m\to \infty $ and $\delta \downarrow 0$, we obtain $\liminf _ j\nu _ E(a_ j)\geq \nu _ E(u)$. The opposite inequality $\nu _ E(a_ j)\leq \nu _ E(u)$ holds for every $j$ because $u\preceq _{\mathcal I}a_ j$. Hence proposition 18.7.1 applies. $\square$

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub