6.17 Smooth cutoff sums and decreasing approximationB4D51CAC5833

Proof. Step 1: proof of (2). Let $w_ j$, $\chi _ j$, $S$ and $\alpha _ j$ be as in (2), and put $f_ j=\chi _ j^2\mathrm e^{w_ j}$, so that $S=\sum _ jf_ j$. Then $\partial f_ j=\chi _ j\mathrm e^{w_ j}(\chi _ j\partial w_ j+2\partial \chi _ j)=\chi _ j\mathrm e^{w_ j}\alpha _ j$, so that $\partial S=\sum _ j\chi _ j\mathrm e^{w_ j}\alpha _ j$. Since $\chi _ j$ and $w_ j$ are real, $\bar\partial f_ j=\overline{\partial f_ j}=\chi _ j\mathrm e^{w_ j}\bar\alpha _ j$ with $\bar\alpha _ j=\chi _ j\bar\partial w_ j+2\bar\partial \chi _ j$, and

where the last line follows by expanding $\alpha _ j\wedge \bar\alpha _ j=\chi _ j^2\partial w_ j\wedge \bar\partial w_ j+2\chi _ j\partial w_ j\wedge \bar\partial \chi _ j+2\chi _ j\partial \chi _ j\wedge \bar\partial w_ j+4\partial \chi _ j\wedge \bar\partial \chi _ j$. Next, $\partial \bar\partial \log S=\partial \bar\partial S/S-\partial S\wedge \bar\partial S/S^2$, and since $\sum _ l\chi _ l^2\mathrm e^{w_ l}=S$,

In the last sum the terms with $j=l$ vanish, and for $j<l$ the terms with indices $(j,l)$ and $(l,j)$ add up to $\mathrm e^{w_ j+w_ l}(\chi _ j\alpha _ l-\chi _ l\alpha _ j)\wedge \overline{(\chi _ j\alpha _ l-\chi _ l\alpha _ j)}$. Multiplying by $\mathrm i/2\pi $ and using $\mathrm{dd}^{\mathrm{c}}=\frac{\mathrm i}{2\pi }\partial \bar\partial $, we obtain (6.17.2). The first term on the right of (6.17.2) is semipositive, because the coefficients $\mathrm e^{w_ j+w_ l}/(2\pi S^2)$ are positive and, for a $(1,0)$-form $\gamma =\sum _ k\gamma _ k\, \mathrm dz_ k$, the coefficient matrix $(\gamma _ k\bar\gamma _{k'})$ of $\mathrm i\gamma \wedge \bar\gamma $ is positive semidefinite.

We now prove (1) in three steps.

Step 2: a curvature estimate. By lemma 6.14.1, applied on the compact manifold $X$ to the finitely many smooth forms below, there is a constant $A>0$, depending only on the fixed geometry, such that $\chi _ j\mathrm{dd}^{\mathrm{c}}\chi _ j-\frac{\mathrm i}{2\pi }\partial \chi _ j\wedge \bar\partial \chi _ j\geq -A\omega $ for every $j$. Let $W\subset X$ be open, and for the indices $j$ with $W\cap \operatorname{Supp}\chi _ j\neq \varnothing $ let $w_ j$ be smooth real functions on $W$ such that

for all such $i,j$. We claim that $S=\sum _ j\chi _ j^2\mathrm e^{w_ j}$ is positive on $W$ and that

Let $x\in W$ and choose $i$ with $x\in U_ i$. Then $\chi _ i(x)=1$, so $S(x)\geq \mathrm e^{w_ i(x)}>0$, and $\mathrm e^{w_ j}/S\leq \mathrm e^{C_ B}$ at $x$ for every $j$ with $\chi _ j(x)\neq 0$. Since $S>0$ on $W$, the identity (6.17.2) of (2) holds on $W$. If $\chi _ j(x)=0$, the $j$th summand of the last term in (6.17.2) vanishes at $x$, because the nonnegative function $\chi _ j$ attains its minimum at $x$, so that $\partial \chi _ j(x)=0$. Hence the last term is at least $-2JA\mathrm e^{C_ B}\omega $ at $x$. The coefficients $\chi _ j^2\mathrm e^{w_ j}/S$ are nonnegative with sum one, so the middle term is at least $-m\theta $, and the first term is semipositive. This proves the claim.

Step 3: $G_ m$ is quasi-plurisubharmonic and satisfies the curvature bound. Since the kernels $a_{j,m}$ may vanish, we apply Step 2 to smooth approximations of $\log a_{j,m}$ and pass to the limit. Fix $x_0\in X$ and a small coordinate ball $W$ about $x_0$ such that $W\subset U'_ j$ whenever $x_0\in \operatorname{Supp}\chi _ j$, and $W\cap \operatorname{Supp}\chi _ j=\varnothing $ otherwise; then the indices $j$ with $W\cap \operatorname{Supp}\chi _ j\neq \varnothing $ are those with $x_0\in \operatorname{Supp}\chi _ j$. Choose a smooth function $g$ on $W$ with $\mathrm{dd}^{\mathrm{c}}g=\theta $, and for each such $j$ write $h_ j-g=\operatorname{Re}b_ j$ on $W$ with $b_ j$ holomorphic, by lemma 6.14.2. Since $h_ j=g+\operatorname{Re}b_ j$ on $W$, we have $a_{j,m}=\mathrm e^{-mh_ j}K_{j,m}=\mathrm e^{-mg}k_ j$ on $W$, where, by (6.15.2) and since $|\mathrm e^{-mb_ j/2}|^2=\mathrm e^{-m\operatorname{Re}b_ j}$, $k_ j=\mathrm e^{-m\operatorname{Re}b_ j}K_{j,m}=\sum _\ell |\mathrm e^{-mb_ j/2}e_\ell |^2$ for an orthonormal basis $(e_\ell )$ of the space defining $K_{j,m}$.

For $\delta >0$ the function $\log (k_ j+\delta )$ is smooth, by lemma 6.15.3, and plurisubharmonic: it is the increasing limit of the logarithms of the finite partial sums of these squared moduli together with $|\sqrt\delta |^2$, which are plurisubharmonic by proposition 5.10.3, and since the limit is continuous, theorem 5.10.2 applies. Put $w_ j^\delta =\log (k_ j+\delta )-mg$, so that $\mathrm{dd}^{\mathrm{c}}w_ j^\delta \geq -m\theta $. On $W\cap U_ i\cap \operatorname{Supp}\chi _ j$, which lies in $\overline{U'_ j}\cap \overline{U'_ i}$, lemma 6.16.2 and the identities $a_{j,m}=\mathrm e^{-mg}k_ j$, $a_{i,m}=\mathrm e^{-mg}k_ i$ give $k_ j\leq \mathrm e^{C_ B}k_ i$, and since $\mathrm e^{C_ B}\geq 1$ also $k_ j+\delta \leq \mathrm e^{C_ B}(k_ i+\delta )$. Thus the functions $w_ j^\delta $ satisfy the hypotheses of Step 2 on $W$, and the function

is plurisubharmonic on $W$, where $g'$ is a smooth function on $W$ with $\mathrm{dd}^{\mathrm{c}}g'=\theta +(2JA\mathrm e^{C_ B}/m)\omega $, given by lemma 6.14.2. Indeed, by Step 2 this function is smooth with positive semidefinite complex Hessian, so proposition 5.9.2(1) applies.

As $\delta \downarrow 0$, the sums $\sum _ j\chi _ j^2\mathrm e^{w_ j^\delta }=\mathrm e^{-mg}\sum _ j\chi _ j^2(k_ j+\delta )$ decrease to $R_ m$, so these functions decrease to $G_ m+g'$, which is not identically $-\infty $, since $G_ m\geq v_{i,m}$ on $U_ i$. By theorem 5.10.1, $G_ m+g'$ is plurisubharmonic on $W$. Hence $G_ m$ is quasi-plurisubharmonic, and the curvature bound holds with $c'=2JA\mathrm e^{C_ B}$, which depends only on the fixed geometry.

Step 4: the singularities of $G_ m$ are gentle. Each summand of $R_ m$ is smooth by lemma 6.15.3. On $U_ i$ we have $a_{i,m}\leq R_ m\leq J\mathrm e^{C_ B}a_{i,m}$ by lemma 6.16.2, and $a_{i,m}$ is comparable with $\sum _ k|g_ k|^2$ for local generators $g_ k$ of $\mathcal I(m(\varphi +h_ i))=\mathcal I(m\varphi )$. Thus $G_ m$ has analytic singularities of type $(m^{-1},\mathcal I(m\varphi ))$, and $\exp (mG_ m)=R_ m$ is smooth on $X$.

Now let $\pi \colon Y\to X$ be any log resolution of $\mathcal I(m\varphi )$, with $\pi ^*\mathcal I(m\varphi )=\mathcal O(-D)$. Its restriction $\pi ^{-1}(B_ j)\to B_ j$ satisfies $\pi ^*\mathcal I(m(\varphi +h_ j))=\mathcal O(-D)$ on $\pi ^{-1}(B_ j)$, so by lemma 6.15.3, applied with the volume form $\omega ^ n/n!$, the pulled-back kernel $K_{j,m}\circ \pi $ is locally $|s_ D|^2$ times a smooth strictly positive function on $\pi ^{-1}(B_ j)$. The factors $\mathrm e^{-mh_ j}\circ \pi $ and $\chi _ j^2\circ \pi $ are smooth, and the latter vanishes outside the closed set $\pi ^{-1}(\operatorname{Supp}\chi _ j)\subset \pi ^{-1}(B_ j)$; hence $(R_ m\circ \pi )/|s_ D|^2$ is smooth on $Y$, and it is strictly positive because at every point some $\chi _ i\circ \pi $ equals one. Therefore $G_ m\circ \pi $ is locally the sum of $m^{-1}\log |s_ D|^2$ and a smooth function. These are exactly the requirements of definition 6.14.12. $\square$

Proof. We first bound $G_ m$ from below. Let $x\in U_ i$. Then $R_ m(x)\geq a_{i,m}(x)$, and the lower bound in (6.15.5), in the version of theorem 6.15.4 with the measure $\omega ^ n/n!$, gives $m^{-1}\log K_{i,m}(x)\geq \varphi (x)+h_ i(x)-C/m$. Since $h_ i$ cancels, $G_ m\geq v_{i,m}\geq \varphi -C/m$ on $U_ i$, where $C\geq 0$ depends only on the fixed geometry.

For an upper bound, let $r>0$ be smaller than the distance, in the coordinates of $B_ j$, from $\operatorname{Supp}\chi _ j$ to $\partial B_ j$ for every $j$, and for $x\in \operatorname{Supp}\chi _ j$ let $B_ j(x,r)\subset B_ j$ denote the ball of radius $r$ about $x$ in these coordinates. If $\chi _ j(x)\neq 0$, the upper bound in (6.15.5) on $B_ j$, again in the version with the measure $\omega ^ n/n!$, gives

where $C_2$ depends only on the fixed geometry. By the mean value theorem, $\sup _{B_ j(x,r)}h_ j-h_ j(x)\leq C_ hr$, where $C_ h$ bounds the gradients of the finitely many functions $h_ j$ on the compact sets $\overline{B_ j}$ and therefore depends on $\theta $. Since $R_ m(x)$ is a sum of at most $J$ terms, each at most $\max _ ja_{j,m}(x)$ over the indices with $\chi _ j(x)\neq 0$,

Fix $x$. Letting first $m\to \infty $ and then $r\downarrow 0$, and using that the balls $B_ j(x,r)$ shrink to $x$, the upper semicontinuity of $\varphi $ and the lower bound, we obtain $G_ m(x)\to \varphi (x)$, also at points of the pole set. For a fixed $r$, the upper bound shows that $G_ m\leq \sup _ X\varphi +C_ hr+|\log (JC_2r^{-2n})|$ on $X$ for every $m$, where $\sup _ X\varphi <\infty $ because $\varphi $ is upper semicontinuous on the compact space $X$. Since moreover $\varphi -C\leq G_ m$ and $\varphi \in L^1(X)$, dominated convergence gives $G_ m\to \varphi $ in $L^1(X)$.

By lemma 6.17.1(1), $G_ m=m^{-1}\log \sum _ k|g_ k|^2+O(1)$ near $x$ with generators $g_ k$ of $\mathcal I(m\varphi )$. Since $G_ m\geq \varphi -C/m$ and $\varphi $ is locally integrable (lemma 6.14.4(1)), $G_ m$ is not identically $-\infty $ near $x$, so the germ of some $g_ k$ at $x$ is not zero and $\mathcal I(m\varphi )_ x\neq 0$. Hence lemma 6.14.7 gives $\nu (G_ m,x)=m^{-1}\operatorname {ord}_ x\mathcal I(m\varphi )$. Write $\varphi =u+f$ on a connected neighborhood of $x$, with $u$ plurisubharmonic and $f$ smooth; then $\mathcal I(m\varphi )=\mathcal I(mu)$ and $\nu (\varphi ,x)=\nu (u,x)$ near $x$, and the Lelong bounds follow from lemma 6.9.1, applied to the weight $u$.

For the integrability, note that $\mathrm e^{m(G_ m-\varphi )}=R_ m\mathrm e^{-m\varphi }=\sum _ j\chi _ j^2K_{j,m}\mathrm e^{-m(\varphi +h_ j)}$. Cover the compact set $\operatorname{Supp}\chi _ j$ by finitely many open sets on which $K_{j,m}\leq C_ m\sum _ i|g_ i|^2$ with members $g_ i$ of the weighted space on $B_ j$, by lemma 6.15.3. Since these $g_ i$ have finite weighted norm, every summand is integrable.

For the doubling inequality, lemma 6.15.7, applied on each $B_ j$ with $u=\varphi +h_ j$ and the measure $\omega ^ n/n!$, gives $K_{j,2m}\leq C_*K_{j,m}^2$, hence $a_{j,2m}\leq C_*a_{j,m}^2$, with $C_*$ depending only on the fixed geometry. Let $x\in U_ i$. For every $j$ with $\chi _ j(x)\neq 0$, lemma 6.16.2 gives $a_{j,m}(x)\leq \mathrm e^{C_ B}a_{i,m}(x)$, and $R_ m(x)\geq a_{i,m}(x)$. Consequently

at $x$, also when both sides vanish. Taking logarithms and dividing by $2m$ gives $G_{2m}\leq G_ m+D/m$ with $D=\frac12\log \max (1,C_*J\mathrm e^{2C_ B})$. $\square$

Proof. After shrinking the neighborhood we may subtract a constant from $v$ and assume $v\leq 0$; this changes neither the ideals nor the integrability of $\mathrm e^{m(v-u)}$. Put $p=\lambda '/\lambda >1$, and note that $m\geq \lambda p/(p-1)>\lambda $. Let $f\in \mathcal I(\lambda 'v)_ x$, defined on a small neighborhood on which $|f|^2\leq C$ for a constant $C$. All the following inequalities are taken outside the pole sets of $u$ and $v$, which have measure zero. Where $u\geq pv$, we have $\lambda u\geq \lambda 'v$ and hence $|f|^2\mathrm e^{-\lambda u}\leq |f|^2\mathrm e^{-\lambda 'v}$. Where $u<pv$, since $m-\lambda >0$, $v\leq 0$ and $(m-\lambda )p-m=m(p-1)-\lambda p\geq 0$,

that is, $-\lambda u\leq m(v-u)$; hence $|f|^2\mathrm e^{-\lambda u}\leq C\mathrm e^{m(v-u)}$. Both bounds are integrable near $x$, so $f\in \mathcal I(\lambda u)_ x$. $\square$

Proof. Put $\Phi _ j=G_{2^ j}+2D/2^ j$, with $D$ as in lemma 6.17.3. The doubling inequality gives

so the sequence is decreasing, and by lemma 6.17.3 its pointwise and $L^1$ limit is $\varphi $. Adding a constant changes neither the curvature, nor the analytic type, nor the Lelong numbers. Moreover $\exp (2^ j\Phi _ j)=\mathrm e^{2D}R_{2^ j}$ is smooth, and the remainder on a log resolution changes only by a constant. Hence $\Phi _ j$ has gentle analytic singularities of type $(2^{-j},\mathcal I(2^ j\varphi ))$ by lemma 6.17.1(1), and $\theta +\mathrm{dd}^{\mathrm{c}}\Phi _ j\geq -c'2^{-j}\omega $.

It remains to prove the inclusions of multiplier ideals. Let $0<\lambda <\lambda '$. The function $\exp (2^ j(\Phi _ j-\varphi ))=\mathrm e^{2D}\mathrm e^{2^ j(G_{2^ j}-\varphi )}$ is integrable on $X$ by lemma 6.17.3. If $j$ is so large that $2^ j\geq \lambda \lambda '/(\lambda '-\lambda )$, then lemma 6.17.4, with $u=\varphi $, $v=\Phi _ j$ and $m=2^ j$, gives $\mathcal I(\lambda '\Phi _ j)_ x\subseteq \mathcal I(\lambda \varphi )_ x$ at every point $x\in X$. Since this threshold for $j$ depends only on $\lambda $ and $\lambda '$, the inclusion holds as an inclusion of sheaves on $X$ for all such $j$. $\square$