0B2C9E3CF86E
From now on $(X,\omega )$ is a compact Kähler manifold of dimension $n$. If $n=0$, then $X$ is a finite set, every quasi-plurisubharmonic function on $X$ is real-valued and hence smooth, and every multiplier ideal is the unit ideal; the constant sequence $\Phi _ j=\varphi $ then has all the approximation properties proved below. We therefore assume $n\geq 1$.
We fix, once and for all, finitely many concentric coordinate balls
\[ U_ j\Subset U'_ j\Subset B_ j\Subset B_ j^+, \qquad j=1,\ldots ,J,\qquad X=\bigcup _ jU_ j, \]
and smooth functions $\chi _ j\colon X\to [0,1]$ with compact support in $U'_ j$ and $\chi _ j=1$ on $U_ j$. We require in addition that whenever $\overline{U'_ j}\cap \overline{U'_ l}\neq \varnothing $, the union $B_ j^+\cup B_ l^+$ is contained in a coordinate ball $\mathcal B_{jl}$ whose closure lies in a coordinate chart. By lemma 6.14.1, applied to the forms $\mathrm{dd}^{\mathrm{c}}\chi _ j$ on $X$, there is a constant $C_\chi \geq 0$ with $\mathrm{dd}^{\mathrm{c}}\chi _ j\geq -C_\chi \omega $ for all $j$.
Such balls and cutoff functions exist. To see this, we first construct a metric $d$ defining the topology of $X$. Choose finitely many charts $\kappa _ k\colon V_ k\to \mathbb C^ n$, $1\leq k\leq N$, and compact sets $L_ k\subset V_ k$ whose interiors cover $X$, and, by proposition 5.2.6, smooth functions $\psi _ k\colon X\to [0,1]$ with compact support in $V_ k$ and $\psi _ k=1$ on $L_ k$. The map $\Psi =(\psi _ k,\psi _ k\kappa _ k)_{1\leq k\leq N}$ from $X$ to $\mathbb R^ N\times \mathbb C^{nN}$, where $\psi _ k\kappa _ k$ is extended by zero outside $V_ k$, is continuous and injective: if $\Psi (x)=\Psi (y)$ and $x\in L_ k$, then $\psi _ k(y)=\psi _ k(x)=1$, so $y\in V_ k$ and $\kappa _ k(y)=\kappa _ k(x)$, whence $y=x$. A continuous injective map from a compact space to a Hausdorff space is a homeomorphism onto its image, so $d(x,y)=|\Psi (x)-\Psi (y)|$ is a metric defining the topology of $X$.
Now cover $X$ by finitely many coordinate balls whose closures lie in coordinate charts, and let $\delta >0$ be a Lebesgue number of this cover for $d$, so that every subset of $X$ of $d$-diameter less than $\delta $ lies in one of these balls. About every point of $X$ choose concentric coordinate balls $U\Subset U'\Subset B\Subset B^+$, with $\overline{B^+}$ contained in a coordinate chart and of $d$-diameter less than $\delta /2$. By compactness finitely many of the balls $U$ cover $X$; these give the $U_ j$, and proposition 5.2.6 gives the cutoff functions $\chi _ j$. If $\overline{U'_ j}\cap \overline{U'_ l}\neq \varnothing $, then $B_ j^+\cup B_ l^+$ is a union of two intersecting sets of diameter less than $\delta /2$, so it has diameter less than $\delta $ and lies in one ball $\mathcal B_{jl}$ of the cover.
All the quantities attached to these choices are now fixed: the bounds for the densities of $\omega ^ n/n!$ with respect to Lebesgue measure in the charts, the coordinate changes, the diameters of the balls, the distances from $\overline{U'_ j}$ to $\partial B_ j$, and the derivatives of the cutoffs. We say that a constant depends only on the fixed geometry if it depends only on these data, hence only on $(X,\omega )$ once the choices have been made.
Lemma 6.16.1.6A23B839899E Let $\theta $ be a smooth closed real $(1,1)$-form on $X$, let $a\geq 0$, and let $v_ j\in \mathrm{PSH}(B_ j,\theta )$ for $j=1,\ldots ,J$ satisfy $v_ j\leq v_ l+a$ on $\overline{U'_ j}\cap \overline{U'_ l}$ for all $j,l$. Then the function
\[ v(x)=\max _{j:\, x\in \overline{U'_ j}}\bigl (v_ j(x)+a\chi _ j(x)\bigr ) \]
is quasi-plurisubharmonic on $X$ and satisfies $\theta +\mathrm{dd}^{\mathrm{c}}v\geq -aC_\chi \omega $ and $v_ j\leq v\leq v_ j+2a$ on $U_ j$.
Proof.
Since the $U_ i$ cover $X$, the maximum is taken over a nonempty set at every point. Fix $x_0\in X$ and choose $i$ with $x_0\in U_ i$. Indices $j$ with $x_0\notin \overline{U'_ j}$ do not occur near $x_0$. If $x_0\in \partial U'_ j$, then $\chi _ j=0$ near $x_0$, because $\chi _ j$ has compact support in $U'_ j$; so near $x_0$ the $j$th term, where it is defined, equals $v_ j\leq v_ i+a=v_ i+a\chi _ i$, since $\chi _ i=1$ on the neighborhood $U_ i$ of $x_0$. Such a term never exceeds the $i$th term and can be omitted. Hence on a small neighborhood of $x_0$, $v$ is the maximum of $v_ j+a\chi _ j$ over the fixed set of indices $j$ with $x_0\in U'_ j$. Each of these functions satisfies $\theta +aC_\chi \omega +\mathrm{dd}^{\mathrm{c}}(v_ j+a\chi _ j)\geq 0$. Adding a local potential of the closed form $\theta +aC_\chi \omega $ on a coordinate ball about $x_0$ (lemma 6.14.2), which makes these functions plurisubharmonic by lemma 6.14.4(2), and applying proposition 5.9.5, we obtain that $v$ is quasi-plurisubharmonic near $x_0$ with $\theta +\mathrm{dd}^{\mathrm{c}}v\geq -aC_\chi \omega $.
On $U_ j$ the $j$th term is $v_ j+a\geq v_ j$, and every term with index $l$ is at most $v_ l+a\leq v_ j+2a$. This proves the two bounds, also at points of the pole sets.
$\square$
Now fix a smooth closed real $(1,1)$-form $\theta $ on $X$. By lemma 6.14.2 we can choose smooth real functions $h_ j$ on $B_ j^+$ with $\mathrm{dd}^{\mathrm{c}}h_ j=\theta $. Let $\varphi \in \mathrm{PSH}(X,\theta )$ and let $m\geq 1$ be an integer. Then $\varphi +h_ j$ is plurisubharmonic on $B_ j$ by lemma 6.14.4(2), and we let $K_{j,m}$ be the Bergman kernel of section 6.15 on the ball $B_ j$ for the weight $m(\varphi +h_ j)$ and the volume form $\omega ^ n/n!$:
\[ K_{j,m}(x)=\sup \Bigl \{ |F(x)|^2:F\in \mathcal O(B_ j),\ \int _{B_ j}|F|^2\mathrm e^{-m(\varphi +h_ j)}\frac{\omega ^ n}{n!}\leq 1\Bigr \} . \]
We put
\[ a_{j,m}=\mathrm e^{-mh_ j}K_{j,m},\qquad v_{j,m}=m^{-1}\log a_{j,m}=m^{-1}\log K_{j,m}-h_ j. \]
The closure of $B_ j$ lies in a chart, so in the coordinates of $B_ j$ the density of $\omega ^ n/n!$ with respect to Lebesgue measure is bounded above and below by positive constants on $B_ j$. Hence theorem 6.15.4, in the version with the measure $\varrho \, \mathrm dV=\omega ^ n/n!$, applies on $B_ j$: $m^{-1}\log K_{j,m}$ is plurisubharmonic, and $v_{j,m}\in \mathrm{PSH}(B_ j,\theta )$.
Lemma 6.16.2.82376665E5F7 There is a constant $C_ B\geq 0$, depending only on the fixed geometry and in particular independent of $\theta $, $\varphi $ and $m$, such that
\[ a_{j,m}\leq \mathrm e^{C_ B}a_{l,m}\qquad \text{on }\overline{U'_ j}\cap \overline{U'_ l} \]
for all $j,l$. In particular $a_{j,m}$ and $a_{l,m}$ have the same zeros on $\overline{U'_ j}\cap \overline{U'_ l}$, and $|v_{j,m}-v_{l,m}|\leq C_ B/m$ there outside these common zeros.
Proof.
Fix $j,l$ with $\overline{U'_ j}\cap \overline{U'_ l}\neq \varnothing $, and work in the coordinates of the ball $\mathcal B_{jl}$. By lemma 6.14.2 choose a smooth function $g$ on $\mathcal B_{jl}$ with $\mathrm{dd}^{\mathrm{c}}g=\theta $. The functions $h_ j-g$ and $h_ l-g$ are pluriharmonic on the coordinate balls $B_ j^+$ and $B_ l^+$, so by the same lemma $h_ j-g=\operatorname{Re}b_ j$ and $h_ l-g=\operatorname{Re}b_ l$ with $b_ j,b_ l$ holomorphic on these balls (the constants are absorbed into $b_ j,b_ l$). Put $u=\varphi +g$, a plurisubharmonic function on $\mathcal B_{jl}$. Since $|\mathrm e^{-mb_ j/2}|^2=\mathrm e^{-m\operatorname{Re}b_ j}$, multiplication by $\mathrm e^{-mb_ j/2}$ is an isometry from the weighted space defining $K_{j,m}$ onto the space of holomorphic functions on $B_ j$ with weight $mu$ and volume form $\omega ^ n/n!$. If $K^{mu}_{B_ j}$ denotes the Bergman kernel of the latter space, then $K^{mu}_{B_ j}=\mathrm e^{-m\operatorname{Re}b_ j}K_{j,m}$, and therefore
\[ a_{j,m}=\mathrm e^{-mg}K^{mu}_{B_ j},\qquad a_{l,m}=\mathrm e^{-mg}K^{mu}_{B_ l}. \]
Hence it suffices to prove $K^{mu}_{B_ j}\leq C_0^2K^{mu}_{B_ l}$ on $\overline{U'_ j}\cap \overline{U'_ l}$ with $C_0\geq 1$ depending only on the fixed geometry.
Choose a smooth function $\rho \colon X\to [0,1]$ with compact support in $B_ j\cap B_ l$, equal to one on a neighborhood of the compact set $\overline{U'_ j}\cap \overline{U'_ l}$ (proposition 5.2.6); this choice belongs to the fixed geometry. Let $r>0$ be the distance, in the coordinates of $\mathcal B_{jl}$, from $\overline{U'_ j}\cap \overline{U'_ l}$ to the support of $\bar\partial \rho $, and let $c>0$ and $C''$ be lower and upper bounds for the density of $\omega ^ n/n!$ with respect to Lebesgue measure on $B_ j\cup B_ l$ in these coordinates. The ball $B_ l$ is pseudoconvex also in the coordinates of $\mathcal B_{jl}$, since a biholomorphism transports a continuous plurisubharmonic exhaustion function (proposition 5.9.3).
To compare the kernels at a point $x\in \overline{U'_ j}\cap \overline{U'_ l}$, we correct $\rho f$, for $f$ holomorphic on $B_ j$, to a holomorphic function on $B_ l$ with the same value at $x$. Let $f$ be holomorphic on $B_ j$ with norm at most one in the weighted space, and extend $\rho f$ by zero to a smooth function on $B_ l$. We apply theorem 6.4.9 on $B_ l$ to the smooth $\bar\partial $-closed form $f\bar\partial \rho $, with the plurisubharmonic function $p=mu+n\log |z-x|^2$ and the smooth function $q=|z|^2$, whose complex Hessian matrix is the identity. On the support of $\bar\partial \rho $ we have $|z-x|^{-2n}\leq r^{-2n}$, so
\[ \int _{B_ l}|f\bar\partial \rho |^2\mathrm e^{-p-q}\, \mathrm dV\leq r^{-2n}\sup |\bar\partial \rho |^2\int _{B_ j}|f|^2\mathrm e^{-mu}\, \mathrm dV\leq \frac{\sup |\bar\partial \rho |^2}{cr^{2n}}. \]
Hence there is a solution of $\bar\partial w=f\bar\partial \rho $ on $B_ l$ with
\[ \int _{B_ l}|w|^2|z-x|^{-2n}\mathrm e^{-mu-|z|^2}\, \mathrm dV\leq C=\frac{\sup |\bar\partial \rho |^2}{cr^{2n}}, \]
and $C$ depends only on the fixed geometry. The function $|z-x|^{2n}\mathrm e^{|z|^2}$ is bounded on the bounded set $B_ l$ by a constant of the same kind, and the density of $\omega ^ n/n!$ is at most $C''$. Hence the squared weighted norm of $w$, for the weight $mu$ and the volume form $\omega ^ n/n!$, is at most a constant $C'$ depending only on the fixed geometry.
Near $x$ we have $\bar\partial w=f\bar\partial \rho =0$, because $\rho =1$ there; hence $\Delta w=0$ near $x$, and by theorem 3.1.11 $w$ is smooth, hence holomorphic, near $x$. Suppose that $w(x)\neq 0$. Then $|w|^2\geq |w(x)|^2/4$ on a ball $B(x,\delta )$ on which $u\leq M$ for some constant $M$, since $u$ is upper semicontinuous, and the last integral is at least
\[ \frac{|w(x)|^2}4\, \mathrm e^{-mM-\sup _{B_ l}|z|^2}\int _{B(x,\delta )}|z-x|^{-2n}\, \mathrm dV=\infty , \]
because $\int _{B(x,\delta )}|z-x|^{-2n}\, \mathrm dV=s_{2n-1}\int _0^\delta t^{-1}\, \mathrm dt$ in polar coordinates (3.1.2). Hence $w(x)=0$. The function $F=\rho f-w$ satisfies $\bar\partial F=0$ on $B_ l$, so it is holomorphic by theorem 3.1.11, and $F(x)=f(x)$. Since $0\leq \rho \leq 1$ and $\rho $ vanishes outside $B_ j$, the weighted norm of $F$ on $B_ l$ is at most $1+C'^{1/2}=C_0$. By the definition of the kernel, $|f(x)|^2=|F(x)|^2\leq C_0^2K^{mu}_{B_ l}(x)$, and taking the supremum over $f$ gives $K^{mu}_{B_ j}(x)\leq C_0^2K^{mu}_{B_ l}(x)$.
No bound on $u$, $\theta $ or $m$ enters these constants. Taking $C_ B$ to be the maximum of $2\log C_0$ over the finitely many pairs $(j,l)$, with the roles of $j$ and $l$ also exchanged, proves the lemma.
$\square$
Theorem 6.16.3.E9EBCA5A681D There is a constant $c>0$, depending only on $(X,\omega )$ through the fixed geometry, with the following property. Let $\theta $ be a smooth closed real $(1,1)$-form on $X$, let $\varphi \in \mathrm{PSH}(X,\theta )$, and let $m\geq 1$ be an integer. Then there is a quasi-plurisubharmonic function $\varphi _ m$ on $X$ such that
\[ \theta +\mathrm{dd}^{\mathrm{c}}\varphi _ m\geq -\frac cm\omega , \qquad \varphi _ m\geq \frac1m\log \sum _ i|g_ i|^2+O(1)\quad \text{locally}, \]
where the $g_ i$ are local generators of $\mathcal I(m\varphi )$. The bounded term may depend on $m$, $\theta $, $\varphi $ and the point.
Proof.
By lemma 6.16.2, the functions $v_{j,m}\in \mathrm{PSH}(B_ j,\theta )$ satisfy $v_{j,m}\leq v_{l,m}+C_ B/m$ on $\overline{U'_ j}\cap \overline{U'_ l}$, also at their common zeros. Hence lemma 6.16.1 with $a=C_ B/m$ gives a quasi-plurisubharmonic function $\varphi _ m$ with $\theta +\mathrm{dd}^{\mathrm{c}}\varphi _ m\geq -(C_ BC_\chi /m)\omega $, so the first conclusion holds with $c=C_ BC_\chi $, increased if necessary to make it positive. On $U_ j$ the function $\varphi _ m$ dominates $v_{j,m}=m^{-1}\log K_{j,m}-h_ j$. Since $h_ j$ is smooth, $\mathcal I(m(\varphi +h_ j))=\mathcal I(m\varphi )$, and the comparison of lemma 6.15.3 gives $v_{j,m}\geq m^{-1}\log \sum _ i|g_ i|^2+O(1)$ near each point of $U_ j$. This proves the second conclusion.
$\square$
Pome
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