PomeProposition 5.33.1.EFC5ED339FC3 The relation $\preceq $ is a preorder, and $\sim $ is an equivalence relation. If $u$ is quasi-plurisubharmonic and $f$ is a smooth function on $X$, then $u+f\sim u$. In particular, all smooth functions on $X$ have the same singularity type.
5.33 Singularity type and multiplier ideals9A10411F1BCD
9A10411F1BCD
There are two natural ways to compare the singularities of quasi-plurisubharmonic functions on $X$. The first asks whether one function is bounded above by the other up to an additive constant; it leads to the ordinary singularity type. The second records only which holomorphic functions are locally square integrable against the weights $\mathrm{e}^{-ku}$, $k>0$; it leads to multiplier ideals. Domination implies the corresponding inclusion of multiplier ideals. A logarithmic pole on a disk shows that a single weight loses information, which is why the second comparison uses all positive weights.
Thus $u\preceq v$ means that $u$ is at least as singular as $v$. For instance, a quasi-plurisubharmonic function with a logarithmic pole lies below every bounded potential in this order, but not conversely.
Proof. Reflexivity follows with $C=0$. If $u\leq v+C$ and $v\leq w+D$, then $u\leq w+C+D$, which proves transitivity. As for every preorder, comparison in both directions defines an equivalence relation. Since $X$ is compact, a smooth function $f$ satisfies $|f|\leq A$ for some constant $A$, and then $u-A\leq u+f\leq u+A$. Taking $u=0$ shows that every smooth function has the singularity type of the constants. $\square$
Some later results require the singularities of a potential to be concentrated on a small closed set. A closed subset $A\subseteq X$ is complete pluripolar if every point of $X$ has a neighborhood $U$ with a plurisubharmonic function $p$ on $U$ such that $A\cap U=\{ p=-\infty \} $. A quasi-plurisubharmonic function has small unbounded locus if it is locally bounded outside a closed complete pluripolar set.
Changing the volume form does not change local integrability: on relatively compact coordinate neighborhoods, the ratio of two smooth positive volume forms is bounded above and below by positive constants. In particular, the definition does not involve a choice of Hermitian metric, and it applies verbatim to quasi-plurisubharmonic functions on any complex manifold.
Proposition 5.33.2.32843D040902 If $u\preceq v$, then $\mathcal I(ku)\subseteq \mathcal I(kv)$ for every real $k>0$. If $u\sim v$, then these ideals agree for every $k>0$.
Proof. Let $g$ be a local holomorphic function on $X$. If $u\leq v+C$, then $\mathrm{e}^{-kv}\leq \mathrm{e}^{kC}\mathrm{e}^{-ku}$. Multiplying by $|g|^2$ shows that local integrability of $|g|^2\mathrm{e}^{-ku}$ implies local integrability of $|g|^2\mathrm{e}^{-kv}$. If $u\sim v$, the same argument applies in both directions. $\square$
Example 5.33.3.36A8480CE7DF On a disk in $\mathbb C$ centered at the origin, with coordinate $z$, let $\lambda \geq 0$ and put $u(z)=\lambda \log |z|^2$ for $\lambda >0$, while $u=0$ for $\lambda =0$. Then
\[ \mathcal I(u)_0=(z^{\lfloor \lambda \rfloor }). \]
Indeed, if a nonzero holomorphic germ has order $m$ at the origin, its squared modulus is comparable to $|z|^{2m}$ near the origin. In polar coordinates, its weighted integral over a disk of sufficiently small radius $\epsilon >0$ is finite precisely when
\[ \int _0^\epsilon r^{2m-2\lambda +1}\, \mathrm{d}r<\infty , \qquad \text{equivalently}\qquad m>\lambda -1. \]
The smallest nonnegative integer satisfying this strict inequality is $\lfloor \lambda \rfloor $. When $\lambda $ is an integer, the borderline exponent $-1$ occurs for $m=\lambda -1$ and gives a divergent integral; this is where the strictness of the inequality matters.
The example also shows that a single multiplier ideal does not determine the singularity: different coefficients $\lambda $ can have the same integer part. For a real weight $k>0$, the same calculation applied to $ku$ gives $\mathcal I(ku)_0=(z^{\lfloor k\lambda \rfloor })$. The ideals at all weights $k>0$ recover $\lambda $: if $0\leq \lambda <\mu $, the integer parts of $k\lambda $ and $k\mu $ differ as soon as $k(\mu -\lambda )>1$.
We therefore compare multiplier ideals at all positive weights. For quasi-plurisubharmonic functions $u,v$, we write $u\preceq _{\mathcal I}v$ if $\mathcal I(ku)\subseteq \mathcal I(kv)$ for every real $k>0$, and $u\sim _{\mathcal I}v$ if $\mathcal I(ku)=\mathcal I(kv)$ for every real $k>0$. We call $\preceq _{\mathcal I}$ the ideal order; if $u\sim _{\mathcal I}v$, then $u$ and $v$ have the same multiplier-ideal type, or ideal type. By proposition 5.33.2, $u\preceq v$ implies $u\preceq _{\mathcal I}v$, and $u\sim v$ implies $u\sim _{\mathcal I}v$. These relations make sense on any complex manifold. Their relation to envelopes and to global masses requires further arguments; see section 18.2.
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.