5.38 Lelong numbers and divisorial estimates2C2417A75DF4

Proof. Work on a coordinate ball and choose a smooth function $\rho $ with $\mathrm{dd}^{\mathrm{c}}\rho \geq \beta $. The current

is closed and positive. A smooth form, viewed through its coefficient measures, assigns zero mass to every proper analytic set. Thus $T$ and $\beta +\mathrm{dd}^{\mathrm{c}}w$ have the same restriction to $\bigcup _ jE_ j$, and the consequence of Siu's theorem stated above identifies this restriction with $\sum _ j\nu (w,E_ j)[E_ j]$. Subtracting it from the positive current $\beta +\mathrm{dd}^{\mathrm{c}}w$ leaves the restriction of $\beta +\mathrm{dd}^{\mathrm{c}}w$ to the complement of $\bigcup _ jE_ j$, which is positive. This proves the inequality locally, hence globally. Note that $\beta +\mathrm{dd}^{\mathrm{c}}w$ itself need not be closed. $\square$

Proof. The current $S=\mathrm{dd}^{\mathrm{c}}(v-u)$ is real, closed, of order zero and supported on $Z$. By the support theorem in [Chapter III, Corollaries 2.11 and 2.14, pp. 141–143, DemCADG],

where $Z_ j$ are the prime divisorial components of $Z$. Indeed, the support theorem first gives real constants $c_ j$ such that $S-\sum _ jc_ j[Z_ j]$ is supported in codimension at least two; this closed current of order zero then vanishes by the same theorem. Near a generic smooth point of $Z_ j$, with local equation $z_1=0$, the function $v-u-c_ j\log |z_1|^2$ has zero complex Hessian and is therefore smooth and pluriharmonic. This identifies $c_ j$ with the difference of the generic Lelong numbers, and $c_ j\leq 0$ because $u\leq v$.

Thus $\mathrm{dd}^{\mathrm{c}}(u-v)=-S\geq 0$. Hence the locally integrable distribution $u-v$ has a global plurisubharmonic representative, which is constant by the maximum principle on the compact connected manifold $X$. Since quasi-plurisubharmonic functions that agree almost everywhere agree everywhere, $v=u+C$ for a constant $C$. $\square$

Proof. The local $\mathrm{dd}^{\mathrm{c}}$-Poincaré lemma gives real distributional potentials $v_ i$ on the sets of a cover of $Z$ by coordinate neighborhoods, with $T=\mathrm{dd}^{\mathrm{c}}v_ i$; for positive $T$ these potentials are plurisubharmonic. See [Chapter III, Section 1.18 and Proposition 1.19, DemCADG]. On overlaps $\mathrm{dd}^{\mathrm{c}}(v_ i-v_ j)=0$. Taking the Euclidean trace shows that $v_ i-v_ j$ is a harmonic distribution, so it is smooth by elliptic regularity.

Choose a smooth locally finite partition of unity $(\chi _ i)$ subordinate to the cover and put $v=\sum _ i\chi _ i v_ i$, each term being extended by zero as a distribution. On the $j$th neighborhood,

is smooth. Thus $\beta =T-\mathrm{dd}^{\mathrm{c}}v=-\mathrm{dd}^{\mathrm{c}}(v-v_ j)$ is smooth on each neighborhood, hence on $Z$, and it is real and closed. For positive $T$, the same identity expresses $v$ locally as a plurisubharmonic function plus a smooth function. Finally, if $T=\beta +\mathrm{dd}^{\mathrm{c}}v=\beta '+\mathrm{dd}^{\mathrm{c}}v'$ are two such decompositions, then $\mathrm{dd}^{\mathrm{c}}(v-v')=\beta '-\beta $ is smooth; taking a smooth Hermitian trace and applying elliptic regularity shows that $v-v'$ is smooth. $\square$

Proof. We first note that $\nu (v,D_ j)=c_ j$. Indeed, the restriction of $S$ to $D_ j$ is $c_ j[D_ j]$: distinct divisors intersect in codimension at least two, and such sets carry no mass of any divisor current, also after countable summation. Smooth forms give every divisor zero mass. The local form of Siu's decomposition, applied after a smooth correction, therefore gives $\nu (v,D_ j)=c_ j$.

Ordinary domination implies ideal domination by proposition 5.33.2, and ideal domination implies the asserted inequalities for generic Lelong numbers by theorem 5.37.3. It remains to prove that these inequalities imply ordinary domination.

Choose a Hermitian form $\Lambda $ with $a\in \mathrm{PSH}(Y,\Lambda )$. By lemma 5.38.1 and the inequalities $\nu (a,D_ j)\geq c_ j$, each finite partial sum of $S$ can be subtracted from $\Lambda +\mathrm{dd}^{\mathrm{c}}a$ without losing positivity. Letting the partial sums increase to $S$ gives

Consequently $\mathrm{dd}^{\mathrm{c}}(a-v)\geq \beta -\Lambda $. Hence the integrable function $a-v$ has a quasi-plurisubharmonic representative, which is bounded above on the compact manifold $Y$. This yields $a\leq v+C$ almost everywhere, and lemma 5.24.3 extends this inequality everywhere. Thus $a\preceq v$. $\square$

Proof. Lemma 5.38.4, applied to $T=S$, gives the decomposition, with $v$ quasi-plurisubharmonic since $S\geq 0$, and shows that any two such potentials differ by a smooth function. By proposition 5.33.1, they have the same ordinary singularity type. $\square$

Proof. On a coordinate ball about $x$, add one smooth function to all the potentials to make them plurisubharmonic, and write $p_ j,p$ for the resulting functions. Let $n$ be the dimension and $\kappa $ the Euclidean form of the chart. Distributional convergence gives weak convergence of the positive trace measures $\mathrm{dd}^{\mathrm{c}}p_ j\wedge \kappa ^{n-1}$ to $\mathrm{dd}^{\mathrm{c}}p\wedge \kappa ^{n-1}$. For every sufficiently small radius $r$ such that the sphere of radius $r$ about $x$ has zero mass for the limit measure, the masses of the balls converge. The Lelong monotonicity formula bounds each $\nu (p_ j,x)$ by the normalized trace mass of $p_ j$ on such a ball. Taking the upper limit in $j$, and then letting these radii decrease to zero, gives $\limsup _ j\nu (p_ j,x)\leq \nu (p,x)$. Since smooth additions do not change Lelong numbers, this proves the lemma. $\square$

Proof. Since adding a constant does not change Lelong numbers, we may assume that $\sup _ Zv=0$. By theorem 5.25.2, these normalized potentials have a common $L^1$ bound. Cover $Z$ by finitely many triples of concentric coordinate balls, a smaller, an intermediate and a larger one, such that the smaller balls cover $Z$. On each larger ball choose a fixed smooth quadratic function $r$ with $\mathrm{dd}^{\mathrm{c}}r\geq \beta $, so that $p=v+r$ is plurisubharmonic there. The functions $p$ obtained in this way have uniformly bounded local $L^1$ norms.

Write $d=\dim Z$ and let $\kappa $ be the Euclidean form in the chart. Choose a nonnegative smooth cutoff $\chi $ supported in the larger ball and equal to one on an intermediate ball. Positivity and distributional integration by parts give

Every point of the smaller ball is the center of a Euclidean ball of a fixed radius contained in the intermediate ball. The local Lelong monotonicity formula, recalled in Proposition 6.7.1, bounds the Lelong number at this center by a fixed multiple of the trace mass of the intermediate ball. The finite cover gives one bound on $Z$. Since smooth additions and additive constants do not change Lelong numbers, this proves the assertion for the original potentials. $\square$

Proof. The upper estimate follows from the local Bergman lower bound in proposition 5.35.1: the logarithm of the sum of squared moduli of local generators of $\mathcal I(mu)$, divided by $m$, is at least $u$ up to a locally bounded term. Pulling back and taking generic Lelong numbers along $E$ reverses this inequality.

For the lower estimate, take a local generator $f$ of the ideal and set $k=\operatorname {ord}_ E(f\circ \pi )$. Near a generic point of $E$, choose a local coordinate $z_1$ defining $E$. Divisorial removal (lemma 5.38.1), after a smooth correction, gives

Shrink around a point where the factors of $f\circ \pi $ and of the Jacobian complementary to the powers of $z_1$ do not vanish. The change-of-variables formula and local integrability of $|f|^2\mathrm{e}^{-mu}$ then require integrability in $z_1$ of

This implies $k+A_ U(E)>m\nu _ E(u)$, and in particular the weaker non-strict lower bound in the statement. Taking the minimum over a finite set of local generators and using that orders are nonnegative proves the lower estimate. Since $\nu _ E(a_ m)=m^{-1}\operatorname {ord}_ E\mathcal I(mu)$, the last assertion follows by letting $m\to \infty $. $\square$

Proof. If $n\geq 2$, let $\rho :\widetilde Y\to Y$ be the blowup of $y$ and let $E$ be its exceptional divisor. For $n=1$, take $\rho $ to be the identity and $E=\{ y\} $. Set $F=\pi \circ \rho $ and write $J_ F$ for its local holomorphic Jacobian determinant. Generic orders along $E$ give

On a standard blowup chart the Jacobian of $\rho $ has order $n-1$ along $E$. The chain rule therefore gives the log discrepancy over $U$ as

Since $g\in \mathcal I(mu)$, the order of $g\circ F$ along $E$ is at least the order of this ideal. The lower estimate in lemma 5.38.9, applied to $F$, therefore proves the inequality. $\square$

Proof. Principalize the coherent ideal of an analytic presentation of $a$ by a log resolution $\pi :Y\to X$ with smooth $Y$. In suitable local coordinates $z_1,\ldots ,z_ n$ on $Y$,

where $r\leq n$. The finitely many inequalities in the statement give $\nu (v\circ \pi ,\{ z_ i=0\} )\geq b_ i$; if $v\preceq _{\mathcal I}a$, these inequalities follow from theorem 5.37.3. Add a smooth local function to $v\circ \pi $ to make it plurisubharmonic. Divisorial removal for this closed positive complex Hessian shows that

has a quasi-plurisubharmonic representative; here the difference is first interpreted off the divisor, or as a locally integrable distribution. Such a representative is locally bounded above. Since $B$ is locally bounded, $v\circ \pi \leq a\circ \pi +C$ on each relatively compact chart of $Y$, away from the divisor. Since $\pi $ is proper, finitely many such charts cover the inverse image of a smaller coordinate neighborhood in $X$, which makes $C$ uniform there. Push the inequality down off the analytic exceptional set; as quasi-plurisubharmonic functions are determined by their values almost everywhere, it then holds everywhere. Finitely many neighborhoods cover the compact manifold, so $v\leq a+C$ globally.

The only input from resolution of singularities is the principalization of a coherent ideal. Divisorial removal is applied to the closed positive complex Hessian obtained after a local smooth correction, so no closed background on $X$ is needed. $\square$

Proof. If $s$ is bounded, upper bounds for $U_ m$ and $V_ m$ give the assertion with $\delta _ m=1/m$ for $m>1$. Otherwise principalize the ideal of $s$ by a log resolution $\pi :\widetilde X\to X$ with smooth $\widetilde X$. Let $E_ i$ be the components of the resulting divisor, with positive coefficients $c_ i$ in $s\circ \pi $; in local coordinates $z_ i$ defining these components, the singular part of $s\circ \pi $ is $\sum _ i c_ i\log |z_ i|^2$. Set $A_ i=1+\operatorname {ord}_{E_ i}(\operatorname {Jac}\pi )$. The hypothesis and lemma 5.38.9 give

Choose a positive integer $N\geq \max _ i2A_ i/c_ i$ and put $\delta _ m=N/m$, which is less than one for large $m$. Then the right-hand side above is at least $(1-\delta _ m)c_ i$. After a smooth local correction, divisorial removal from the positive Hessian of $(U_ m+V_ m)\circ \pi $ shows that

has a quasi-plurisubharmonic representative, which is locally bounded above. A finite cover of $\widetilde X$ by charts and descent along the proper map $\pi $, as in the proof of lemma 5.38.11, therefore give $U_ m+V_ m\leq (1-\delta _ m)s+O(1)$. The bounded constant may depend on $m$. $\square$