5.16 Lelong numbers and multiplier ideal sheavesEB678D619AD9

Proof. For every $c>0$, the analytic Lelong-level-set theorem proves that $\{x:\nu(\varphi,x)\geq c\}$ is an analytic subset. Since Lelong numbers are nonnegative, for $c\leq0$ this upper level set is all of $X$. These are exactly the upper level sets needed for upper semicontinuity in the analytic Zariski topology. The normalization is the same in both statements: $\mathrm{dd}^{\mathrm c}=\mathrm i\partial\bar\partial/(2\pi)$ and $\nu(\log|z|^2,0)=1$.

Proof. On each coordinate ball, the multiplier-ideal coherence theorem proves that the germs $f$ for which $|f|^2\exp(-\varphi)$ is locally integrable form a coherent analytic ideal sheaf. This is exactly the definition of $\mathcal I(\varphi)$ used here. These local sheaves agree on overlaps, since local integrability is unchanged by a holomorphic coordinate change with a nonvanishing Jacobian. Coherence is local, so $\mathcal I(\varphi)$ is coherent on $X$. In particular the argument uses the weight $\exp(-\varphi)$, not $\exp(-2\varphi)$.

Proof. On $X\times X$, with projections $p_1,p_2$, consider $\Phi=\varphi\circ p_1+\psi\circ p_2$. The external-product identity in the proved restriction and product theorem gives \[\mathcal I(\Phi)=p_1^*\mathcal I(\varphi)\,p_2^*\mathcal I(\psi).\] Restrict to the diagonal $\iota:X\to X\times X$. The restriction part of that theorem gives $\mathcal I(\iota^*\Phi)\subseteq\mathcal I(\Phi)|_X^{\mathrm{im}}$. Here $\iota^*\Phi=\varphi+\psi$, and the image ideal on the right is $\mathcal I(\varphi)\mathcal I(\psi)$. This proves the inclusion stalkwise and hence as sheaves. The earlier theorem includes the weighted extension proof needed for the restriction step.

Proof. By the divisorial multiplier-ideal estimate, for every positive integer $k$, \[k\nu(\varphi,E)-A_X(E)<\operatorname{ord}_E\mathcal I(k\varphi)\leq k\nu(\varphi,E).\] The log discrepancy $A_X(E)$ is a fixed finite number on a fixed smooth model carrying $E$. Divide by $k$ and let $k\to\infty$. Both bounds tend to $\nu(\varphi,E)$, giving the formula. The order is that of the pulled-back ideal on the model; it is independent of its chosen generators and of a further model, as proved in the same estimate.

Proof. For a coherent ideal $\mathfrak a$ near $x$, the order of $\mathfrak a\mathcal O_Y$ along the exceptional divisor of the blowup at $x$ is its ordinary vanishing order at $x$. Indeed, in a blowup chart substitute $z=(t,tw_2,\ldots,tw_n)$ into each Taylor series. Its lowest nonzero homogeneous term factors as $t^d$ times a polynomial in $w$ which is not identically zero on the exceptional divisor, so its generic order there is $d$; taking the minimum over generators proves the ideal assertion.

Apply this to $\mathfrak a=\mathcal I(k\varphi)$. The pointwise order estimate gives $k^{-1}\operatorname{ord}_x\mathcal I(k\varphi)\to\nu(\varphi,x)$, while its divisorial counterpart gives $k^{-1}\operatorname{ord}_E\mathcal I(k\varphi)\to\nu(\varphi,E)$. The two sequences are identical, proving the equality. In dimension one the blowup of the point ideal is an isomorphism and the assertion reduces to the point itself.

Qualification and proof. For the general germ statement, the precise formulation is the local one: divisors range over modifications of a sufficiently small neighborhood on which $f$ is represented. The proved uniform valuative criterion gives exactly \[f\in\mathcal I(\varphi)_x\quad\Longleftrightarrow\quad\exists\varepsilon>0:\ \operatorname{ord}_Ef+A_X(E)\geq(1+\varepsilon)\nu(\varphi,E)\] for all such divisors whose centers contain $x$. The same $\varepsilon$ must work for every divisor.

In the analytic-singularity clause, one must test all components of the total pole divisor on the log resolution whose centers contain $x$, including nonexceptional components. Restricting the test to exceptional divisors is false: on $\mathbb C$ take $\varphi=\log|z|^2$ and the identity resolution. There are no exceptional divisors, but $1\notin\mathcal I(\varphi)_0$ since $|z|^{-2}$ is not locally integrable.

Here is the finite test with this correction. On a log resolution write $\varphi\circ\pi=\sum_i a_i\log|s_i|^2+O(1)$ and let $b_i$ be the order of the holomorphic Jacobian along $D_i=\{s_i=0\}$. If $c_i=\operatorname{ord}_{D_i}(f\circ\pi)$, change of variables makes the integrand, up to a bounded factor and a holomorphic residual factor, a product of $|s_i|^{2(c_i+b_i-a_i)}$. At a generic point of each divisor this residual factor is nonzero. The one-variable integral $\int_0^\delta r^{2(c_i+b_i-a_i)+1}\,\mathrm dr$ is finite exactly when $c_i+b_i+1>a_i$. Since the divisor is simple normal crossings, Fubini proves sufficiency for their simultaneous inequalities and the generic-point calculation proves necessity. Thus the test is $\operatorname{ord}_{D_i}f+A_X(D_i)>\nu(\varphi,D_i)$ for all the indicated total pole components. This is also the log-resolution integrability criterion. Resolution itself remains the permitted Hironaka input.

Proof. The multiplier ideals $\mathcal I(\psi_j)_x$ form an increasing sequence, since $\psi_j\leq\psi_{j+1}$. By monotone multiplier-ideal stability, their union is $\mathcal I(\varphi)_x$. Choose finitely many generators $f_1,\ldots,f_r$ of the latter ideal; finite generation follows from coherence. Each $f_l$ belongs to some $\mathcal I(\psi_{j_l})_x$. For $j\geq\max_l j_l$, all generators belong, so $\mathcal I(\varphi)_x\subseteq\mathcal I(\psi_j)_x$. The reverse inclusion follows from $\psi_j\leq\varphi$ almost everywhere. Thus equality holds for every sufficiently large $j$, with the index allowed to depend on $x$.

Proof. This is local on $Y$. In coordinates in which $Y$ is a coordinate subspace, take a germ $f\in\mathcal I(\varphi|_Y)$. The weighted restriction theorem extends $f$ to a holomorphic germ $F$ in the ambient coordinates with $|F|^2\exp(-\varphi)$ integrable. Thus $F\in\mathcal I(\varphi)$ and its restriction is $f$. This is precisely the inclusion in the image ideal $\operatorname{Res}_Y\mathcal I(\varphi)$. The hypothesis that the restricted weight is not identically $-\infty$ ensures that its multiplier ideal has the intended nontrivial local definition; the earlier theorem also covers the zero-ideal case.