5.6 Maps of currents and local mass9F1C9C4A649E

Proof. Let $\varphi $ be a test form on $Y$. By properness, the set $\operatorname{Supp}T\cap F^{-1}(\operatorname{Supp}\varphi )$ is compact. Choose a smooth function $\chi \geq 0$ with compact support on $X$ that equals one on a neighborhood of this set (proposition 5.2.6), and define

The value does not depend on $\chi $: if $\chi '$ is another choice, then $(\chi -\chi ')F^*\varphi $ is a test form whose support is disjoint from $\operatorname{Supp}T$, since $\chi =\chi '$ near $\operatorname{Supp}T\cap F^{-1}(\operatorname{Supp}\varphi )$ and $F^*\varphi =0$ outside $F^{-1}(\operatorname{Supp}\varphi )$; so $T$ vanishes on this form. For all test forms supported in a fixed compact set $L\subset Y$ a single cutoff serves, namely one equal to one near the compact set $\operatorname{Supp}T\cap F^{-1}(L)$; in particular, $F_*T$ is linear. Now let $L$ lie in a chart of $Y$, fix such a cutoff $\chi $, cover $\operatorname{Supp}\chi $ by finitely many charts of $X$ and choose a partition of unity subordinate to this cover near $\operatorname{Supp}\chi $. By the chain rule, the coefficients of the pieces of $\chi F^*\varphi $ in these charts, together with their derivatives of order at most $m'$, are bounded by a constant times $\sum _ Jp_{L,m'}(\varphi _ J)$, where the $\varphi _ J$ are the coefficients of $\varphi $ (at points mapped outside $\operatorname{Supp}\varphi $ they vanish). The estimates for $T$ in these charts therefore give the estimate in the definition of a current for $F_*T$.

Pullback preserves bidegrees, and $T$ acts on test forms of bidegree $(m-p,m-q)$, so $F_*T$ has bidegree $(n-(m-p),n-(m-q))$. For differentiation, note that $\mathrm d(\chi F^*\varphi )=\chi F^*\mathrm d\varphi +\mathrm d\chi \wedge F^*\varphi $, and $T$ vanishes on the last term because its support is disjoint from $\operatorname{Supp}T$. Since the degrees of $T$ and $F_*T$ differ by the even integer $2(m-n)$, the signs in the definitions of $\mathrm dT$ and $\mathrm dF_*T$ agree, and $\mathrm d(F_*T)=F_*(\mathrm dT)$; the same argument applies to $\partial $ and $\bar\partial $. If $\varphi $ is strongly positive, so is $\chi F^*\varphi $ by proposition 5.3.2, which proves the last assertion. $\square$

Proof. Local construction. Let $x\in X$, and choose holomorphic coordinates $z=(z_1,\ldots ,z_ n)$ near $F(x)$. Since $F$ is a submersion, the complex Jacobian matrix at $x$ of the $n$ components of $z\circ F$ has rank $n$ in every chart around $x$. By proposition 5.2.4(3), these components are the first $n$ coordinates of holomorphic coordinates $(z,w)\in \mathbb C^ n\times \mathbb C^{m-n}$ near $x$, in which $F$ is the projection $(z,w)\mapsto z$; shrinking, we may assume that the coordinate domain is a product $U\times W$. For a distribution $a$ on $U$, define the distribution $F^*a$ on $U\times W$ by

Integration in $w$ maps test functions on $U\times W$ continuously to test functions on $U$: the $z$-derivatives of $\int _ W\psi (\cdot ,w)\, \mathrm dV_{2m-2n}(w)$ are the integrals of the $z$-derivatives of $\psi $, by differentiation under the integral sign, and its support lies in the projection of $\operatorname{Supp}\psi $. Hence $F^*a$ is a distribution, and if $a$ is locally integrable, then $F^*a$ is the function $(z,w)\mapsto a(z)$, by Fubini's theorem. For $T=\sum T_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$ on $U$ put $F^*T=\sum F^*T_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$. This is a linear map from currents on $U$ to currents on $U\times W$ preserving bidegrees, continuous for weak convergence, and equal to the pullback of forms for smooth $T$. It commutes with $\partial $ and $\bar\partial $, since these act on the coefficients by differentiation, $\partial _{z_ l}F^*a=F^*\partial _{z_ l}a$, and $\partial _{w_ l}F^*a=0$ because $\int _ W\partial _{w_ l}\psi (\cdot ,w)\, \mathrm dV_{2m-2n}(w)=0$ by the fundamental theorem of calculus; similarly for the conjugate derivatives. Continuity for weak convergence holds because the coefficients of a current depend continuously on it, by their explicit description in section 5.3.

Independence and gluing. Two such local constructions near $x$ agree. Indeed, mollifying the coefficients of $T$ in a chart near $F(x)$, we obtain smooth forms $T_\varepsilon \to T$ weakly on a neighborhood of $F(x)$, as in the discussion following theorem 5.3.7. Both constructions are weakly continuous and agree with the pullback of forms on each $T_\varepsilon $, so they agree on $T$ near $x$. Consequently the local pullbacks agree on the overlaps of their domains, and by lemma 5.3.8 they glue to a unique current $F^*T$ on $X$ that agrees with each local pullback on its domain. Explicitly, if $(\chi _ c)$ is a partition of unity with compact supports subordinate to the cover of $X$ by the domains of the local pullbacks, and $(F^*T)_ c$ denotes a local pullback whose domain contains $\operatorname{Supp}\chi _ c$, then $(F^*T)(\varphi )=\sum _ c(F^*T)_ c(\chi _ c\varphi )$.

The local pullbacks are linear, preserve bidegrees and commute with $\mathrm d$, $\partial $ and $\bar\partial $; they give the pullback of forms if $T$ is smooth, and the function $a\circ F$ if $T$ is given by a locally integrable function $a$. By the uniqueness in lemma 5.3.8, $F^*T$ has the same properties. Continuity for weak convergence follows from the explicit formula, since each of the finitely many nonzero terms $(F^*T)_ c(\chi _ c\varphi )$ depends continuously on $T$.

Positivity. If $T$ is positive, the forms $T_\varepsilon $ are positive near $F(x)$, hence weakly positive at every point. By proposition 5.3.2 their pullbacks are weakly positive at every point, hence positive currents, and positivity passes to the weak limit, the local pullback of $T$ near $x$. Hence $F^*T$ is positive, by lemma 5.3.8. $\square$

Proof. (1) The local construction in the proof of proposition 5.6.2 can be carried out with the coordinates $z$ on $U$ and $(z,w)$ on $U\times W$, and it is then defined on all of $U\times W$. By the independence of the local constructions shown in that proof, $F^*T$ agrees with it near every point, hence on $U\times W$. Thus the coefficients of $F^*T$ at the basis forms containing some $\mathrm dw_ l$ or $\mathrm d\bar w_ l$ vanish, and the coefficient at $\mathrm dz_ I\wedge \mathrm d\bar z_ J$ is the distribution $\psi \mapsto \langle T_{IJ},\widehat\psi \rangle $, where $\widehat\psi (z)=\int _ W\psi (z,w)\, \mathrm dV_{2m-2n}(w)$ is a test function on $U$. Since $T_{IJ}$ is a complex Radon measure, (5.6.3) gives $\langle T_{IJ},\widehat\psi \rangle =\int _ U\widehat\psi \, \mathrm dT_{IJ}=\int \psi \, \mathrm d(T_{IJ}\otimes \lambda _ W)$ for every test function $\psi $ on $U\times W$. Hence this coefficient is the complex Radon measure $T_{IJ}\otimes \lambda _ W$.

(2) Let $x\in X$. As in the proof of proposition 5.6.2, there are holomorphic coordinates $z$ on a chart of $Y$ around $F(x)$ and holomorphic coordinates $(z,w)$ on a neighborhood $N$ of $x$, with coordinate domain a product $U\times W$, in which $F$ is the projection $(z,w)\mapsto z$. On $N$, the current $F^*T$ is the pullback of the restriction of $T$ to the open set $U$ by this projection. By (1), the coefficients of $F^*T$ in the chart $N$ are complex Radon measures. The charts $N$ obtained in this way form an atlas of $X$, so $F^*T$ has order zero, by the discussion following (5.3.6).

Now let $f$ be a locally bounded Borel function on $Y$; then $f\circ F$ is a locally bounded Borel function on $X$, since $F$ is continuous. In the coordinates above, the coefficient measures of $fT$ are the measures $fT_{IJ}$, where $f$ is written in the coordinates $z$, so by (1) the coefficient measures of $F^*(fT)$ on $N$ are the measures $(fT_{IJ})\otimes \lambda _ W$, while those of $(f\circ F)F^*T$ are the measures $(f\circ F)(T_{IJ}\otimes \lambda _ W)$; the coefficients at the basis forms containing some $\mathrm dw_ l$ or $\mathrm d\bar w_ l$ vanish for both currents. Let $g$ be a continuous function with compact support in $U\times W$. The function $g\cdot (f\circ F)$ is bounded, Borel and compactly supported, and $\widehat g(z)=\int _ Wg(z,w)\, \mathrm dV_{2m-2n}(w)$ is a bounded Borel function on $U$ with compact support. Applying (5.6.3) twice, we obtain

Hence the two currents have the same coefficient measures on $N$, and therefore agree on $N$. Since the sets $N$ cover $X$, the currents $F^*(fT)$ and $(f\circ F)F^*T$ coincide. The last assertion is the case $f=\mathbf1_ E$, since $\mathbf1_ E\circ F=\mathbf1_{F^{-1}(E)}$. $\square$

Proof. Canonical local potentials are defined on manifolds of positive dimension (theorem 5.4.3), so $Y$ has positive dimension. If $X$ has dimension zero, every $(1,1)$-current on $X$ vanishes, and $F^*T=0$. Assume that $X$ has positive dimension, let $W$ and $u$ be as in the hypothesis, and let $W_0$ be a connected component of $W$. Each point of $W_0$ has a connected neighborhood $W'\subset W_0$ in a chart of $X$ such that $F(W')$ lies in a chart of $Y$ contained in the domain of $u$. In that chart, $u$ is the canonical subharmonic representative of its class, by the remark following proposition 5.4.4, so by that proposition $u\circ F$ is either identically $-\infty $ on $W'$, or locally integrable on $W'$ with $\mathrm{dd}^{\mathrm{c}}(u\circ F)\geq 0$. No point has neighborhoods $W'$ of both kinds, because a locally integrable function is finite almost everywhere. Hence the points of $W_0$ having a neighborhood of the first kind and those having one of the second kind form two disjoint open sets covering $W_0$. Since $W_0$ is connected and $u\circ F$ is not identically $-\infty $ on $W_0$, the first set is empty. Thus $u\circ F$ is locally integrable on $W$, and $\mathrm{dd}^{\mathrm{c}}(u\circ F)\geq 0$ there.

Let $(\widetilde W,\widetilde u)$ be another pair as in the hypothesis. Near each point of the intersection of the domains of $u$ and $\widetilde u$ we have $\widetilde u=u+h$ with $h$ smooth and pluriharmonic, by theorem 5.4.3 and the remark following proposition 5.4.4. Locally $h=\operatorname{Re}g$ with $g$ holomorphic, so $h\circ F=\operatorname{Re}(g\circ F)$ is smooth and pluriharmonic, and $\widetilde u\circ F=u\circ F+h\circ F$ near each point of $W\cap \widetilde W$. Consequently $\mathrm{dd}^{\mathrm{c}}(\widetilde u\circ F)=\mathrm{dd}^{\mathrm{c}}(u\circ F)$ near each point of $W\cap \widetilde W$, hence on $W\cap \widetilde W$, since a current that vanishes near each point vanishes. By lemma 5.3.8, the local currents $\mathrm{dd}^{\mathrm{c}}(u\circ F)$, which are positive and closed because $\mathrm d\mathrm{dd}^{\mathrm{c}}=0$, glue to a unique closed positive current $F^*T$ on $X$. It is independent of the choices: the currents obtained from two systems of choices agree near each point, by what we have just shown, and hence coincide by the uniqueness in lemma 5.3.8.

If $T$ is smooth, lemma 5.4.2 gives smooth real local potentials $v$ with $T=\mathrm{dd}^{\mathrm{c}}v$; they are subharmonic, by the matrix criterion and proposition 3.1.5, hence their own canonical representatives, so we may use them as the potentials $u$. For a smooth potential $u$ the chain rule gives $\mathrm{dd}^{\mathrm{c}}(u\circ F)=F^*\mathrm{dd}^{\mathrm{c}}u=F^*T$. Now let $F$ be a submersion, and let $V$ be the domain of $u$. The restriction of $F$ to $F^{-1}(V)$ is a submersion to $V$, and its submersion pullback agrees on $F^{-1}(V)$ with that of $F$, since both are glued from the same local constructions. By proposition 5.6.2, the locally integrable function $u\circ F$ represents the submersion pullback $F^*u$, and since this pullback commutes with $\partial $ and $\bar\partial $, we obtain $\mathrm{dd}^{\mathrm{c}}(u\circ F)=\mathrm{dd}^{\mathrm{c}}F^*u=F^*\mathrm{dd}^{\mathrm{c}}u=F^*T$ on $F^{-1}(V)$, in particular on $W$. $\square$