Proposition 7.2.2D9911E035563

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. $\square$