Lemma 18.47.1.511A1D146037 For every $v\in \mathrm{PSH}(X,\theta )$,
\[ v\preceq _{\mathcal I}\psi \quad \Longleftrightarrow \quad v\leq \psi +C_ v \quad \hbox{for some finite }C_ v, \]
where $\preceq _{\mathcal I}$ is the multiplier-ideal order of section 5.33. Consequently, for every obstacle, the envelopes defined with these two conditions agree. At either pole, for every real $t>0$,
\[ \mathcal I(t\psi )=\mathfrak m_{p_\pm }^{\max \{ \lfloor t\rfloor -1,0\} }. \]
Away from the poles, $\mathcal I(t\psi )$ is the full structure sheaf.
Proof.
Near $p_+$ use the coordinates $\zeta _1=z_1$, $\zeta _2=1/z_2$ and the frame $T_1Z_2$. In this frame the two pencil sections $T_1T_2$ and $Z_1Z_2$ have coefficients $\zeta _2$ and $\zeta _1$, so
\[ \psi =\log \max (|\zeta _1|^2,|\zeta _2|^2) -\log (1+|\zeta _1|^2)-\log (1+|\zeta _2|^2) =\log |\zeta |^2+O(1). \]
The same calculation applies at $p_-$. The formula for the ideals is lemma 5.37.1 with normal dimension two and coefficient one, including the case of equality in the integrability condition.
Now suppose that $v\preceq _{\mathcal I}\psi $. By theorem 5.37.3, the Lelong numbers satisfy $\nu (v,p_\pm )\geq 1$; here $\nu (v,p)$ is the Lelong number of section 5.34, not the measure $\nu $. Let $\phi $ be the plurisubharmonic local potential of $v$ in a frame near $p_\pm $, and consider its radial maximum
\[ M(a)=\sup _{|\zeta |\leq \mathrm e^{a/2}}\phi (\zeta ) \]
inside a fixed coordinate ball. This function is finite and convex: by the three-circles theorem, the corresponding maximum on each complex line through the pole is convex, and $M$ is the supremum of these functions; lines on which $\phi $ is identically $-\infty $ do not affect the supremum. The limiting slope of $M$ as $a\to -\infty $ is the Lelong number. For $a<a_0$, convexity therefore gives
\[ M(a)\leq M(a_0)+\nu (v,p_\pm )(a-a_0)\leq a+C. \]
Consequently $v\leq \psi +C$ near both poles. After $C$ is increased, the same bound holds on the compact complement, because $v$ is bounded above and $\psi $ is finite and continuous there. Conversely, domination $v\leq \psi +C$ implies the inclusions of multiplier ideals at all weights, by the definition through integrability. Thus the two competitor classes coincide before any obstacle is imposed.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.