Lemma 10.9.404C8B0EA1481

Proof. After computing $\mathrm{dd}^{\mathrm{c}}T$ (Step 1), we prove a version of (10.9.5) in which $\mathbf1_ E$ is smoothed by a convex cutoff, integrating by parts against regularizations of $u$ and $v$ (Step 2). The products $T$, $T_{\hat\imath }$ and $T_{\hat\imath \hat\imath '}$ are positive currents: for products with at least one factor this is lemma 10.9.3(2), and the empty product is the constant $1$. (If $n=1$, then $T=1$, $\mathrm{dd}^{\mathrm{c}}T=0$, and both sums in (10.9.6) are empty; the sum over pairs is empty also if $n=2$.)

Step 1. The measure $\mathrm{dd}^{\mathrm{c}}T$. Work on an open subset of a chart with $q$ and $\sigma $ as in (10.9.2). Each factor of $T$ has the form $\Theta _ i+\sigma $, where $\Theta _ i$ is $\mathrm{dd}^{\mathrm{c}}q$, $\mathrm{dd}^{\mathrm{c}}(u+q)$ or $\mathrm{dd}^{\mathrm{c}}(v+q)$; thus $T=\sum _ J\Theta _ J\wedge \sigma ^{n-1-|J|}$, where $\Theta _ J$ is the Bedford–Taylor product of the $\Theta _ i$ with $i\in J$. The currents $\Theta _ J$ are closed of pure bidegree, so $\partial \Theta _ J=\bar\partial \Theta _ J=0$, and hence $\mathrm d\Theta _ J=\mathrm d^{\mathrm c}\Theta _ J=0$, where $\mathrm d^{\mathrm c}$ is the operator (5.2.10). By the Leibniz rule for currents (section 5.3), $\mathrm{dd}^{\mathrm{c}}(\Theta _ J\wedge \sigma ^ m)=\Theta _ J\wedge \mathrm{dd}^{\mathrm{c}}(\sigma ^ m)$. Since the closed $(1,1)$-form $\mathrm{dd}^{\mathrm{c}}q$ satisfies $\partial \mathrm{dd}^{\mathrm{c}}q=\bar\partial \mathrm{dd}^{\mathrm{c}}q=0$, we have $\mathrm d\sigma =\mathrm d\omega $, $\mathrm d^{\mathrm c}\sigma =\mathrm d^{\mathrm c}\omega $ and $\mathrm{dd}^{\mathrm{c}}\sigma =\mathrm{dd}^{\mathrm{c}}\omega $, and the Leibniz rule gives

Expand $T_{\hat\imath }$ and $T_{\hat\imath \hat\imath '}$ in the same way. For a fixed $J$, with $m=n-1-|J|$, there are $m$ indices $i\notin J$ and $m(m-1)/2$ pairs $i<i'$ outside $J$; comparing coefficients, we obtain

In particular $\mathrm dT=\sum _ J\Theta _ J\wedge \mathrm d(\sigma ^{n-1-|J|})$, $\mathrm d^{\mathrm c}T$ and $\mathrm{dd}^{\mathrm{c}}T$ are currents of order zero, being sums of Bedford–Taylor products multiplied by smooth forms. The form $\mathrm{dd}^{\mathrm{c}}\omega $ is real of bidegree $(2,2)$. Since $\partial \omega $ and $\bar\partial \omega $ have odd degree, $\partial \omega \wedge \partial \omega =\bar\partial \omega \wedge \bar\partial \omega =0$ and $\bar\partial \omega \wedge \partial \omega =-\partial \omega \wedge \bar\partial \omega $, so $\mathrm d\omega \wedge \mathrm d^{\mathrm c}\omega =\frac{\mathrm i}{2\pi }\partial \omega \wedge \bar\partial \omega $ is real of bidegree $(3,3)$; these forms vanish if $n<2$ and $n<3$, respectively.

We claim that there is $C$ such that $C\omega ^2\pm \mathrm{dd}^{\mathrm{c}}\omega $ and $C\omega ^3\pm \mathrm d\omega \wedge \mathrm d^{\mathrm c}\omega $ are strongly positive at every point of $X$. Consider $\omega ^2$ and $\mathrm{dd}^{\mathrm{c}}\omega $, with $n\geq 2$; the other case, with $n\geq 3$, is the same. Let $D$ be a coordinate ball whose closure lies in a chart, and let $\mathcal C$ be the cone of strongly positive $(2,2)$-forms on $\mathbb C^ n$. For $x\in \overline D$, a complex linear change of coordinates takes $\omega (x)$ to $\beta $; since such changes preserve strong positivity in both directions, they map the interior of $\mathcal C$ onto itself, and $\beta ^2$ is an interior point of $\mathcal C$ (proposition 5.3.2). Hence $\omega (x)^2$ lies in the interior of $\mathcal C$. As $x\mapsto \omega (x)^2$ is continuous, these forms make up a compact subset of the interior of $\mathcal C$, which has a positive distance $d_0$ from the closed complement of the interior, for a fixed norm on the real $(2,2)$-forms. The coefficients of $\mathrm{dd}^{\mathrm{c}}\omega $ are bounded on $\overline D$, so for $C$ large the forms $\omega (x)^2\pm C^{-1}\mathrm{dd}^{\mathrm{c}}\omega (x)$ lie at distance less than $d_0$ from $\omega (x)^2$, hence in $\mathcal C$, for all $x\in \overline D$. Covering the compact manifold $X$ by finitely many such balls proves the claim.

The currents $T_{\hat\imath }$ and $T_{\hat\imath \hat\imath '}$ are positive, so $-CT_{\hat\imath }\wedge \omega ^2\leq \mathrm{dd}^{\mathrm{c}}\omega \wedge T_{\hat\imath }\leq CT_{\hat\imath }\wedge \omega ^2$, and similarly for the terms with $\mathrm d\omega \wedge \mathrm d^{\mathrm c}\omega $. A real measure $\nu $ with $-\mu \leq \nu \leq \mu $ for a positive measure $\mu $ satisfies $|\nu |\leq \mu $, so (10.9.6), with the constant $2C$, follows from (10.9.7).

Step 2. A convex cutoff. Put $w=v-u$, a bounded Borel function on $X$. Fix a smooth increasing function $\lambda \colon \mathbb R\to [0,1]$ with $\lambda =0$ on $(-\infty ,0]$ and $\lambda =1$ on $[1,\infty )$, and for $\delta >0$ put $F_\delta (t)=\int _0^ t\lambda (s/\delta )\, \mathrm ds$. Then $F_\delta $ is smooth and convex, $F_\delta =0$ on $(-\infty ,0]$, $0\leq F_\delta '\leq 1$, and as $\delta \downarrow 0$ we have $F_\delta (t)\to t^+$ and $F_\delta '(t)\to \mathbf1_{(0,\infty )}(t)$ for every real $t$; indeed $|F_\delta (t)-t|\leq \delta $ for $t>0$. We shall prove that, for every $\delta >0$,

Granting this, let $\delta \downarrow 0$. The integrands converge pointwise and boundedly to $\mathbf1_ E$ and to $(v-u)\mathbf1_ E$; note that $F_\delta '(0)=0$, so the first limit is the indicator of $\{ w>0\} =E$. Both measures have finite total variation, so dominated convergence turns (10.9.8) into (10.9.5).

To prove (10.9.8), choose finitely many coordinate balls $B_ k$ covering $X$, smooth functions $q_ k$ with $\mathrm{dd}^{\mathrm{c}}q_ k\geq \omega $ on a neighborhood of $\overline{B_ k}$, and a smooth partition of unity $(\chi _ k)$ on $X$ with $\operatorname{Supp}\chi _ k\subset B_ k$ (proposition 5.2.6). Fix $k$, write $\sigma _ k=\omega -\mathrm{dd}^{\mathrm{c}}q_ k$, and choose an open set $\Omega _ k$ with $\operatorname{Supp}\chi _ k\subset \Omega _ k\Subset B_ k$. Convolution of $u+q_ k$ and $v+q_ k$ with radial mollifiers whose radii decrease to zero and are smaller than the distance from $\Omega _ k$ to $\partial B_ k$ gives smooth plurisubharmonic functions on $\Omega _ k$, which we write as $u_ l+q_ k$ and $v_ l+q_ k$. They decrease to $u+q_ k$ and $v+q_ k$ as $l\to \infty $, by proposition 5.9.2(2), and they take values in a fixed compact interval $[-M',M']$, because $u+q_ k\leq u_ l+q_ k\leq u_1+q_ k$ and similarly for $v$. Then $w_ l=v_ l-u_ l\to w$ pointwise on $\Omega _ k$, with $|w_ l|\leq 2M'$.

The function $f=F_\delta (w_ l)$ is smooth on $\Omega _ k$, which contains the support of $\chi _ kT$, so the definition of derivatives of currents and the Leibniz rule give

where $\Lambda _ k=\mathrm{dd}^{\mathrm{c}}\chi _ k\wedge T+\mathrm d\chi _ k\wedge \mathrm d^{\mathrm c}T-\mathrm d^{\mathrm c}\chi _ k\wedge \mathrm dT$, and the chain rule gives

The last term is a smooth strongly positive $(1,1)$-form, since $F_\delta ''\geq 0$, and $T$ is positive, so the product of this term with $\chi _ kT$ is a positive measure. Consequently

Now let $l\to \infty $. By Step 1, $\mathrm{dd}^{\mathrm{c}}T$ and $\Lambda _ k$ are measures of finite total variation, so the right side tends to $\int F_\delta (w)(\chi _ k\mathrm{dd}^{\mathrm{c}}T+\Lambda _ k)$ by dominated convergence.

On the left, write $T=\sum _ J\Theta _ J\wedge \sigma _ k^{m_ J}$ as in Step 1, with $q=q_ k$ and $m_ J=n-1-|J|$. The exterior product of the smooth form $\mathrm{dd}^{\mathrm{c}}(v_ l+q_ k)$ with $\Theta _ J$ is the Bedford–Taylor product $\mathrm{dd}^{\mathrm{c}}(v_ l+q_ k)\wedge \Theta _ J$, by (7.2.3), so the left side of (10.9.9) is the sum over $J$ of

Fix $J$ and consider the first integral; the second is treated in the same way. Let $\vartheta >0$. By the Weierstrass approximation theorem there is a real polynomial $P$ of one variable with $|F_\delta '-P|\leq \vartheta $ on $[-2M',2M']$. The weights $P(w_ l)=P\bigl ((v_ l+q_ k)-(u_ l+q_ k)\bigr )$ are polynomials in two decreasing sequences of bounded plurisubharmonic functions on $\Omega _ k$, so corollary 7.3.8, applied on $\Omega _ k$ with the closed positive base current $\Theta _ J$ and the decreasing factors $v_ l+q_ k$, gives the weak convergence of $P(w_ l)\, \mathrm{dd}^{\mathrm{c}}(v_ l+q_ k)\wedge \Theta _ J$ to $P(w)\, \mathrm{dd}^{\mathrm{c}}(v+q_ k)\wedge \Theta _ J$, which we test with the smooth form $\chi _ k\sigma _ k^{m_ J}$ with compact support in $\Omega _ k$. On the other hand, by theorem 7.2.5 and theorem 5.3.7, the total variations on $\operatorname{Supp}\chi _ k$ of the coefficient measures of the positive currents $\mathrm{dd}^{\mathrm{c}}(v_ l+q_ k)\wedge \Theta _ J$ and $\mathrm{dd}^{\mathrm{c}}(v+q_ k)\wedge \Theta _ J$ are bounded uniformly in $l$, so replacing $F_\delta '(w_ l)$ by $P(w_ l)$, or $F_\delta '(w)$ by $P(w)$, changes the integrals by at most a constant times $\vartheta $, with a constant independent of $l$. As $\vartheta $ was arbitrary,

By (10.9.2), $\omega _ v\wedge T=\sum _ J\mathrm{dd}^{\mathrm{c}}(v+q_ k)\wedge \Theta _ J\wedge \sigma _ k^{m_ J}+\sigma _ k\wedge T$ on $B_ k$, and similarly for $u$. Summing over $J$, we conclude that the left side of (10.9.9) tends to $\int \chi _ kF_\delta '(w)\, (\omega _ v-\omega _ u)\wedge T$. Finally we sum the limiting inequalities over $k$. Since $\sum _ k\chi _ k=1$, we have $\sum _ k\Lambda _ k=\mathrm{dd}^{\mathrm{c}}T-\sum _ k\chi _ k\mathrm{dd}^{\mathrm{c}}T=0$, and (10.9.8) follows. $\square$