7.2 Bounded potentials and recursive products4272F622A446

Proof. We first record two consequences of the Leibniz rule of section 5.3. Let $R$ be a current of even degree on an open subset of $\mathbb C^ n$ such that $\partial R=\bar\partial R=0$; this holds if $R$ is closed and of pure bidegree. If $f$ is a smooth function, then $\mathrm d(fR)=\mathrm df\wedge R$ and $\mathrm d^{\mathrm c}(fR)=\mathrm d^{\mathrm c}f\wedge R$, and applying $\mathrm d$ to the latter gives

If $\gamma $ is a smooth closed form of pure bidegree, then $\partial \gamma $ and $\bar\partial \gamma $ are the components of $\mathrm d\gamma =0$ of two different bidegrees, so they vanish, and the Leibniz rule gives $\mathrm{dd}^{\mathrm{c}}(R'\wedge \gamma )=\mathrm{dd}^{\mathrm{c}}R'\wedge \gamma $ for every current $R'$.

We now prove positivity and closedness, first for a single factor $u$. By proposition 5.9.2, the regularizations $u_\varepsilon =u*\rho _\varepsilon $ by the radial mollifiers of section 2.5 are smooth plurisubharmonic functions on $\Omega _\varepsilon $ with positive semidefinite complex Hessians, and they decrease to $u$ as $\varepsilon \downarrow 0$. On each compact set the $u_\varepsilon $ are bounded uniformly in $\varepsilon $, because $u\le u_\varepsilon \le u_{\varepsilon _0}$ for $\varepsilon \le \varepsilon _0$ and $u$ is locally bounded. Dominated convergence with respect to the total variations of the coefficient measures of $T$ therefore gives $u_\varepsilon T\to uT$ weakly. By (7.2.3), $\mathrm{dd}^{\mathrm{c}}(u_\varepsilon T)=\mathrm{dd}^{\mathrm{c}}u_\varepsilon \wedge T$. Since weak and strong positivity agree in bidegree $(1,1)$ (section 5.3), the positive smooth form $\mathrm{dd}^{\mathrm{c}}u_\varepsilon $ is strongly positive, so its product with the positive current $T$ is positive; it is also closed. As $\mathrm{dd}^{\mathrm{c}}$ is continuous for weak convergence, $\mathrm{dd}^{\mathrm{c}}u_\varepsilon \wedge T\to \mathrm{dd}^{\mathrm{c}}(uT)$ weakly, and positivity and closedness pass to the limit. Thus $\mathrm{dd}^{\mathrm{c}}u\wedge T$ is positive and closed. Applying this with $T$ replaced by the closed positive current $\mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge T$, we obtain the first assertion by induction on $q$.

(1) If the $u_ j$ are smooth, (7.2.3) shows at each step of the recursion that the product is the exterior product of smooth forms with $T$.

(2), (3) The definition uses only multiplication of coefficient measures by functions and the operator $\mathrm{dd}^{\mathrm{c}}$; both operations are local and linear, and no product of two measures occurs. This gives locality, and additivity and positive homogeneity in $u_1$ and in $T$; for the other factors they follow by induction on $q$, since multiplication by $u_1$ and $\mathrm{dd}^{\mathrm{c}}$ are linear. If $c$ is a constant and $R=\mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge T$, then $\mathrm{dd}^{\mathrm{c}}((u_1+c)R)=\mathrm{dd}^{\mathrm{c}}(u_1R)+c\, \mathrm{dd}^{\mathrm{c}}R=\mathrm{dd}^{\mathrm{c}}(u_1R)$, since $R$ is closed; for the other factors the invariance under constants follows again by induction.

(4) The product $T\wedge \gamma $ is closed, and multiplication of coefficient measures by $u_1$ commutes with the exterior product by $\gamma $. By the identity $\mathrm{dd}^{\mathrm{c}}(R'\wedge \gamma )=\mathrm{dd}^{\mathrm{c}}R'\wedge \gamma $ and induction on $q$, every step of the recursion commutes with $\wedge \gamma $.

(5) The pullback $F^*T$ is closed and positive and commutes with $\partial $ and $\bar\partial $, hence with $\mathrm{dd}^{\mathrm{c}}$ (proposition 5.6.2), and the functions $u_ j\circ F$ are locally bounded and plurisubharmonic (proposition 5.9.3). We claim that $F^*(uT)=(u\circ F)F^*T$ for a locally bounded plurisubharmonic $u$. For a smooth function $f$ we have $F^*(fT)=(f\circ F)F^*T$: both sides are weakly continuous in $T$, they agree when $T$ is smooth, and on every relatively compact open subset of $\Omega $ the current $T$ is the weak limit of its smooth regularizations (discussion following theorem 5.3.7). Applying this to the regularizations $u_\varepsilon $ on relatively compact open subsets, and letting $\varepsilon \downarrow 0$, we obtain the claim: $u_\varepsilon T\to uT$ as shown above, so $F^*(u_\varepsilon T)\to F^*(uT)$, while $(u_\varepsilon \circ F)F^*T\to (u\circ F)F^*T$ by dominated convergence with respect to the coefficient measures of the positive current $F^*T$. Consequently every step of the recursion commutes with $F^*$, which proves (5). In particular, by (2) and (5), products of locally bounded plurisubharmonic functions with closed positive currents on a complex manifold can be computed in holomorphic charts: the products defined in two charts agree on their overlap. $\square$

Proof. For $q=0$ the estimate holds with constant $1$, since $K\subset L$. Let $q=1$ and write $u=u_1$. By lemma 2.5.1 there is a smooth function $0\le \chi \le 1$ with compact support in $\operatorname{Int}L$ which equals $1$ on a neighborhood of $K$; it depends only on $K$ and $L$. Since $\mathrm{dd}^{\mathrm{c}}u\wedge T\wedge \beta ^{r-1}$ is a positive measure, $\beta $ is closed, and $(\mathrm{dd}^{\mathrm{c}}S)(\chi )=S(\mathrm{dd}^{\mathrm{c}}\chi )$ for currents $S$ of bidimension $(1,1)$, we have

The form $\mathrm{dd}^{\mathrm{c}}\chi \wedge \beta ^{r-1}$ is smooth with support in $L$, though in general not positive; its coefficients are bounded by a constant depending only on $\chi $ and $n$. The last inequality therefore holds because the total variations of the coefficient measures of $T$ on $L$ are bounded by a constant multiple of $\| T\| _ L$, the multiple depending only on $n$ and $r$ (theorem 5.3.7). Dividing by $(r-1)!$ and taking the largest of the constants obtained for $r=1,\ldots ,n$, we obtain the case $q=1$ with a constant depending only on $K$, $L$ and $n$.

For $q>1$, let $\delta >0$ be the distance from $K$ to $\mathbb C^ n\setminus \operatorname{Int}L$ (if $K=\varnothing $ there is nothing to prove), and put $L_ j=\{ z:\operatorname {dist}(z,K)\le j\delta /(q+1)\} $ for $0\le j<q$ and $L_ q=L$. These are compact sets, depending only on $K$, $L$ and $q$, with $K=L_0$ and $L_{j-1}\subset \operatorname{Int}L_ j$ for $1\le j\le q$. By proposition 7.2.2, the currents $S_ j=\mathrm{dd}^{\mathrm{c}}u_{j+1}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge T$, $0\le j\le q$, are positive and closed, and $S_{j-1}=\mathrm{dd}^{\mathrm{c}}u_ j\wedge S_ j$. The case $q=1$, applied to $S_ j$ and the pair $L_{j-1}\subset \operatorname{Int}L_ j$, gives $\| S_{j-1}\| _{L_{j-1}}\le C_ j\| u_ j\| _{L^\infty (L)}\| S_ j\| _{L_ j}$ for $j=1,\ldots ,q$, since $\| u_ j\| _{L^\infty (L_ j)}\le \| u_ j\| _{L^\infty (L)}$. Combining these $q$ inequalities, we obtain the estimate with $C_{K,L,q}=C_1\cdots C_ q$. $\square$

Proof. We first show that $u_ j$ is plurisubharmonic for $1\le j\le q$. Since plurisubharmonicity is a local property, it suffices to prove this on every ball $W\Subset \Omega $. Choose $k_1<k_2<\cdots $ such that $|u_{j,k_ m}-u_ j|\le 2^{-m}$ on $W$, and put $w_ m=u_{j,k_ m}+3\cdot 2^{-m}$. These functions are plurisubharmonic on $W$, and they decrease, because

Their limit $u_ j$ is finite, so it is plurisubharmonic on $W$ by theorem 5.10.1.

Since the $u_{j,k}$ converge locally uniformly to continuous functions, they are bounded on each compact set uniformly in $k$. Since $T_ k\to T$ weakly, the masses $\| T_ k\| _ L$ are bounded for each compact set $L$: if $\chi \in C^\infty _ c(\Omega )$ satisfies $0\le \chi \le 1$ and $\chi =1$ on $L$, then $\| T_ k\| _ L\le T_ k(\chi \beta ^ r)/r!$, and the right-hand side converges. By theorem 7.2.5, the masses of all the products considered below are therefore bounded on compact sets uniformly in $k$, and so are the total variations of their coefficient measures (theorem 5.3.7).

We claim: if $S_ k\to S$ weakly, where $S_ k$ and $S$ are closed positive currents whose masses are bounded on compact sets uniformly in $k$, and if locally bounded Borel functions $v_ k$ converge locally uniformly to a continuous function $v$, then $v_ kS_ k\to vS$ weakly. Applying this to $S_ k=\mathrm{dd}^{\mathrm{c}}u_{2,k}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{q,k}\wedge T_ k$ and $v_ k=u_{1,k}$, and then $\mathrm{dd}^{\mathrm{c}}$, which is weakly continuous, proves the first assertion by induction on $q$; the case $q=0$ is the hypothesis. The second assertion is the claim with $v_ k=u_{0,k}$.

To prove the claim, fix a test form $\varphi $ with compact support $K$ and $\varepsilon >0$. By lemma 2.5.1(1), applied to $v$ on a relatively compact open neighborhood of $K$, there is a smooth function $v_\varepsilon $ on a neighborhood of $K$ with $|v-v_\varepsilon |\le \varepsilon $ on $K$. Write

Evaluated on $\varphi $, a term $fS_ k$ with a bounded Borel function $f$ is at most $C_\varphi \sup _ K|f|$ times the total variation on $K$ of the coefficient measures of $S_ k$, where $C_\varphi $ bounds the coefficients of $\varphi $; the same holds for $S$. Hence the first term tends to zero, since $v_ k\to v$ uniformly on $K$; the second and fourth terms are bounded by a constant times $\varepsilon $, uniformly in $k$; and the third tends to zero as $k\to \infty $, since $v_\varepsilon \varphi $ is a test form. Letting first $k\to \infty $ and then $\varepsilon \downarrow 0$ proves the claim. $\square$