Lemma 10.8.157EB0767C032B

Proof. We compute on a log resolution adapted to $\Phi $ and $Y$; by lemma 10.8.10, only the strict transform of the exceptional divisor of the blowup of $Y$ contributes to the limit.

A resolution adapted to $\Phi $ and $Y$. Let $\operatorname {Bl}_ YX\to X$ be the blowup of $Y$, with exceptional divisor $E$; for $q=1$ this is the identity and $E=Y$. By proposition 9.6.3, its Jacobian determinant is $\pm t^{q-1}$ in the charts of (9.6.4), where $E=\{ t=0\} $, so its relative canonical divisor is $(q-1)E$. Let $\mathfrak a$ be the ideal in the type of $\Phi $. We apply theorem 9.6.7 on the compact manifold $\operatorname {Bl}_ YX$ to the pullback of $\mathfrak a$, which is nonzero, and to the divisor $E$. This gives a finite sequence of blowups with smooth centers, each having normal crossings with the accumulated divisor of theorem 9.6.7, which starts with $E$; this accumulated divisor is the boundary in the sense of lemma 10.8.10. On the final model the boundary has simple normal crossings, and the pullback of $\mathfrak a$ is the ideal of an effective divisor supported in the boundary. Denote the composite map to $X$ by $\pi $. It is a composite of blowups along closed submanifolds that principalizes $\mathfrak a$, that is, a log resolution of $\mathfrak a$ in the sense of section 6.14. Since definition 6.14.12 requires a smooth remainder on every log resolution, $\Phi \circ \pi $ has a smooth remainder.

Divisor orders. In a normal-crossing chart on the final model write

with $v$ smooth, $r$ smooth and strictly positive, and $a_ D,b_ D\geq 0$. The Jacobian factor of the pulled-back volume is $\prod _ D|s_ D|^{2(A_ D-1)}$ times a smooth positive function, where $A_ D=A_ X(D)$ is the log discrepancy. Let $F_ i$ be one of the holomorphic extensions used in $G$, and suppose that the chart lies over the open set $U_ i$ of lemma 10.8.11. Then $F_ i\circ \pi $ vanishes to some integral order $e_{i,D}\geq 0$ along each coordinate divisor $D$ of the chart. By (10.8.12), the integral of $|F_ i|_{h_0}^2\mathrm e^{-(1+\kappa )\Phi }R_ Y^{-q}(-\log R_ Y)^{-2}$ over $U_ i$ is finite, and this forces

Indeed, fix the other coordinates away from their divisors and integrate radially in $s_ D$: the integrand behaves like $|s_ D|^{2(e_{i,D}+A_ D-(1+\kappa )a_ D-qb_ D)-2}(-\log |s_ D|^2)^{-2}$ when $b_ D>0$, and the squared logarithm in the denominator makes the exponent $-1$ of the radial variable integrable but cannot compensate a smaller exponent. If $b_ D=0$, the logarithmic factor is bounded above and below near generic points of $D$, and the inequality must be strict. This is the computation of theorem 6.12.8 with one additional logarithmic factor.

Shrinking the chart, we may assume that for each $i$ either the chart lies over $U_ i$ or $\eta _ i\circ \pi $ vanishes on it, because $\operatorname{Supp}\eta _ i$ is a compact subset of the open set $U_ i$. Then, on a smaller chart, the pullback of $G=\sum _ i\eta _ iF_ i$ is the monomial $\prod _ Ds_ D^{e_ D}$, with $e_ D=\min _ ie_{i,D}$ over the indices $i$ of the first kind, times a smooth bounded function. The density of $|G|_ h^2R_ Y^{-q}\, \mathrm dV$ is therefore bounded by a constant times $\prod _ D|s_ D|^{2(\gamma _ D-1)}$, where

If $a_ D>0$, (10.8.17) gives $\gamma _ D\geq \kappa a_ D>0$, and if $a_ D=b_ D=0$, it gives $\gamma _ D>0$. If $a_ D=0$ and $b_ D>0$, the formula for the relative canonical divisor of the first blowup gives $A_ D=A_{\operatorname {Bl}_ YX}(D)+(q-1)\operatorname {ord}_ D(E)$, and $b_ D=\operatorname {ord}_ D(E)$ because $\mathcal I_ Y$ pulls back to the ideal of $E$; hence

By lemma 10.8.10, applied with $Z=\operatorname {Bl}_ YX$, this is positive unless $D$ is the strict transform of $E$. For that divisor $b_ E=1$ and $A_ E=q$, and $a_ E=0$: since the restriction of $h$ to $Y$ is defined, generic points of $Y$ lie outside the pole set of $\Phi $. Hence $\gamma _ E=e_ E\geq 0$, and the strict transform of $E$ is the only divisor along which $\gamma _ D$ can vanish.

The annular limit. By a finite partition of unity on the compact inverse image of $Y$, it suffices to consider the integral over a small chart. If no coordinate divisor of the chart is the strict transform of $E$ with $\gamma _ E=0$, the density is integrable, the shells $\{ \varepsilon /2\leq R_ Y\leq \varepsilon \} $ shrink to the null set $\pi ^{-1}(Y)$, and dominated convergence shows that the integral over the shell tends to zero. Otherwise let $t$ be the coordinate defining the strict transform of $E$. The density is $|t|^{-2}$ times a smooth bounded coefficient and the factors $|s_ D|^{2(\gamma _ D-1)}$, $D\ne E$, which are integrable since $\gamma _ D>0$. For fixed values of the remaining coordinates $s$, the shell is $\varepsilon /2\leq |t|^2m(s)r(t,s)\leq \varepsilon $, with $m(s)=\prod _{D\ne E}|s_ D|^{2b_ D}$. Since $r$ is bounded above and below, the shell lies in a range of $\log |t|$ of bounded length, so the integral of $|t|^{-2}$ over it is bounded independently of $\varepsilon $ and $s$; this gives an integrable dominating function. As $\varepsilon \to 0$, the shell shrinks to $t=0$, the smooth coefficient tends to its value at $t=0$, and the integral of $|t|^{-2}$ over the shell tends to $\pi \log 2$. By dominated convergence, only the restriction to $t=0$ of the coefficient of $|t|^{-2}$ in the density contributes to the limit.

Identification of the limit. The argument just given, applied after multiplying the density by a continuous function with compact support, shows that the integrals over the shells converge to $\pi \log 2$ times the total mass of a measure $\mu $ on the strict transform of $E$; in a chart as above, $\mu $ has the density obtained by restricting to $t=0$ the smooth coefficient of $|t|^{-2}$, multiplied by the factors $|s_ D|^{2(\gamma _ D-1)}$, $D\ne E$. Since $\mu $ is determined by the limits of integrals over the shells $\{ \varepsilon /2\leq R_ Y\leq \varepsilon \} $, it does not depend on the coordinates in which it is computed, and since its density in every chart is integrable, it gives no mass to proper analytic subsets. We may therefore compute $\mu $ on the dense open subset of the strict transform of $E$ lying over the complement of the pole set of $\Phi $ in $Y$ and over which the further blowups are isomorphisms, using coordinates of the first blowup $\operatorname {Bl}_ YX$.

Let $(z',w)$ be coordinates on $X$ in which $Y=\{ w=0\} $, and write $\mathrm dV=J(z',w)\, \mathrm dV(z')\, \mathrm dV(w)$ with $J$ smooth and positive. By lemma 10.8.8, $R_ Y=Q_{z'}(w)+O(|w|^3)$ locally uniformly, where $Q_{z'}$ is a positive definite Hermitian form on $\mathbb C^ q$ depending continuously on $z'$. Over this chart, $\operatorname {Bl}_ YX$ is covered by charts with coordinates $(z',t,\xi )$, $\xi \in \mathbb C^{q-1}$, in which, after renumbering the coordinates $w$, we have $w=t(1,\xi )$ and $E=\{ t=0\} $ (for $q=1$ there is no $\xi $, and $t=w$). There $\mathrm dV(w)=|t|^{2(q-1)}\, \mathrm dV(t)\, \mathrm dV(\xi )$ and $R_ Y=|t|^2\bigl (Q_{z'}(1,\xi )+O(|t|)\bigr )$, so away from the poles of $\Phi $ the density of $|G|_ h^2R_ Y^{-q}\, \mathrm dV$ is $|t|^{-2}$ times the smooth function

whose restriction to $t=0$ equals $|f(z')|_ h^2Q_{z'}(1,\xi )^{-q}J(z',0)$, because $G|_ Y=f$. The points of the projective normal fibers not covered by such a chart form a hyperplane in each fiber and are negligible. Integrating over $\xi $, we find that the image of $\mu $ under the projection to $Y$ has the density $c(z')|f(z')|_ h^2$ with respect to $\mathrm dV(z')$, where

Since $Q_{z'}(1,\xi )\geq c_0(1+|\xi |^2)$ with $c_0>0$ locally uniformly in $z'$, and $(1+|\xi |^2)^{-q}$ is integrable on $\mathbb C^{q-1}$, the function $c$ is locally bounded. It depends only on $\omega $, $R_ Y$ and the chart. Covering the compact manifold $Y$ by finitely many such charts and comparing $\mathrm dV(z')$ with $\mathrm dV_ Y$, we obtain (10.8.16). $\square$