A finite divisorial sum is pulled back through the local equations of its divisors. For an infinite sum we need pullback to be continuous on closed positive currents whose Bott–Chern classes vary. Solving a scalar equation for the Gauduchon trace selects a smooth representative in each class; the complementary normalized potentials then converge in $L^1$, and lemma 5.43.19 shows that their pullbacks converge as well.
In this section, $Y$ denotes an arbitrary connected compact complex manifold of dimension $n\geq 1$, and $\gamma $ is a Gauduchon metric on $Y$; in dimension one, any Hermitian metric will do. Fix smooth norms on tensor fields induced by $\gamma $. Put $V_\gamma =\int _ Y\gamma ^ n$ and, for a real $(1,1)$-form or current $B$, define its scalar trace $\sigma _\gamma (B)$ by
\[ \sigma _\gamma (B)\, \gamma ^ n=B\wedge \gamma ^{n-1}. \]
For a current this is an identity of distributions. As in the proof of theorem 11.4.2, we use the classical elliptic Fredholm theorem, elliptic regularity, and the continuity on distributions of the normalized elliptic inverse.
Lemma 20.4.20.721E66DF8C88 Every closed real $(1,1)$-current $T$ has a unique decomposition
\[ T=B_\gamma (T)+\mathrm{dd}^{\mathrm{c}}v_\gamma (T),\qquad \int _ Yv_\gamma (T)\, \gamma ^ n=0, \]
where $v_\gamma (T)$ is a real distribution and $B_\gamma (T)$ is a smooth closed real form with constant scalar trace. Both assignments are linear and continuous from weak convergence of currents to convergence of distributions. The forms $B_\gamma (T)$ lie in a finite-dimensional space, and their convergence takes place in $C^\infty $.
This space maps isomorphically onto $H^{1,1}_{\mathrm{BC}}(Y,\mathbb R)$. If $T\geq 0$, then $v_\gamma (T)$ is quasi-plurisubharmonic. On the cone of positive closed currents, $v_\gamma (T)$ depends continuously on $T$ in $L^1$, and for every nonnegative integer $l$,
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20.4.21
\begin{equation} \| B_\gamma (T)\| _{C^ l}+\| v_\gamma (T)\| _{L^1(\gamma ^ n)} \leq C_ l\int _ YT\wedge \gamma ^{n-1}, \end{equation}
where the constants $C_ l$ depend only on the fixed metric and norms.
Proof.
Set $L_\gamma f=\sigma _\gamma (\mathrm{dd}^{\mathrm{c}}f)$. We use the inverse $G_\gamma $ of (11.31.3), normalized with the volume form $\gamma ^ n/V_\gamma $. It acts on top-degree distributions of total mass zero and is continuous by the elliptic estimates for the operator and its transpose.
Step 1: construction. Define
\[ c_ T=\frac{\int _ YT\wedge \gamma ^{n-1}}{V_\gamma },\qquad v_\gamma (T)=G_\gamma \bigl ((\sigma _\gamma (T)-c_ T)\gamma ^ n\bigr ),\qquad B_\gamma (T)=T-\mathrm{dd}^{\mathrm{c}}v_\gamma (T). \]
The current $B_\gamma (T)$ is closed and has scalar trace $c_ T$. We now prove its smoothness and the finite-dimensionality assertion without using any Kähler identity.
Step 2: an elliptic system. Let $J$ be the complex structure acting on real covectors. On real $(1,1)$-forms consider the first-order operator
\[ \mathscr P B=(\mathrm{d}B,\mathrm{d}\sigma _\gamma (B)). \]
Its principal symbol at a nonzero real covector $\xi $ is $B\mapsto (\xi \wedge B,\xi \, \sigma _\gamma (B))$, and we claim that it is injective. If $\xi \wedge B=0$, then the $(1,1)$ condition forces $B$ to be a real multiple of $\xi \wedge J\xi $. To check this, choose unitary real coordinates with $\xi =\mathrm{d}x_1$: every term of $B$ contains $\mathrm{d}x_1$, and invariance under $J$ forces every term to contain $\mathrm{d}y_1$ as well. If moreover $\xi \, \sigma _\gamma (B)=0$, then $\sigma _\gamma (B)=0$; since the scalar trace of $\xi \wedge J\xi $ is nonzero, the multiple is zero.
Let $\mathscr P^*$ be the formal adjoint of $\mathscr P$ for the fixed metric and volume. Since the symbol of $\mathscr P$ is injective, $\mathscr P^*\mathscr P$ is elliptic. By elliptic regularity and compactness, its distributional kernel is a finite-dimensional space of smooth forms. Since $\mathscr P B_\gamma (T)=0$, the form $B_\gamma (T)$ belongs to this space. Conversely, the kernel consists exactly of the closed forms with constant scalar trace: integrating $\langle \mathscr P^*\mathscr P B,B\rangle $ gives $\mathscr P B=0$, and connectedness makes the trace constant. Denote this kernel by $\mathscr H_\gamma $.
Step 3: Bott–Chern representatives. The construction is linear and continuous on distributions. Convergence of the components $B_\gamma (T)$ therefore takes place in the finite-dimensional space $\mathscr H_\gamma $, hence in every smooth norm. If $T=\mathrm{dd}^{\mathrm{c}}f$ for a distribution $f$, then $c_ T=0$ and the inverse gives $v_\gamma (T)=f-V_\gamma ^{-1}\int _ Yf\gamma ^ n$, so $B_\gamma (T)=0$. Applied to smooth closed forms, this shows that $\mathscr H_\gamma $ contains exactly one representative of each Bott–Chern class. Uniqueness of the decomposition can also be seen directly: if two decompositions are given, $L_\gamma $ of the difference of the two potentials is a constant, integration forces this constant to be zero, and the normalized kernel of $L_\gamma $ is zero.
Step 4: positive currents. Suppose now that $T\geq 0$. The inequality $\mathrm{dd}^{\mathrm{c}}v_\gamma (T)\geq -B_\gamma (T)$ gives a quasi-plurisubharmonic representative of the distribution $v_\gamma (T)$. Let positive closed currents $T_ j$ converge weakly to $T$. Their components $B_\gamma (T_ j)$ converge smoothly, so the potentials $v_\gamma (T_ j)$ have a common smooth lower curvature bound. By theorem 5.25.2, after subtracting their suprema they form an $L^1$-precompact family. If $a_ j=\sup _ Yv_\gamma (T_ j)$, the zero-mean normalization gives
\[ 0\leq a_ j V_\gamma =\int _ Y\bigl (a_ j-v_\gamma (T_ j)\bigr )\, \gamma ^ n, \]
and the right-hand side is uniformly bounded. Hence the potentials $v_\gamma (T_ j)$ themselves are $L^1$-precompact. Every cluster point equals their distributional limit $v_\gamma (T)$, so the whole sequence converges in $L^1$.
Finally, the positive closed currents with trace mass $\int _ YT\wedge \gamma ^{n-1}=1$ form a weakly compact set: positivity bounds their coefficient measures, weak limits remain positive and closed, and testing with $\gamma ^{n-1}$ preserves the trace mass. The smooth and $L^1$ continuity just proved bounds both norms on this set, and positive homogeneity gives (20.4.21). A positive current of trace mass zero is zero, so this case is included.
$\square$
The smooth representatives give a common curvature bound for converging normalized potentials. This is exactly the hypothesis needed to pull back their $L^1$ convergence.
Proposition 20.4.22.D7373D3639FA Let $g:Z\to Y$ be a smooth compact modification, and fix a Gauduchon metric $\gamma '$ on $Z$. Pullback of positive closed $(1,1)$-currents, defined through their local potentials, is weakly continuous. Moreover, there is a constant $C_ g$, depending only on the modification and the fixed metrics, such that
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20.4.23
\begin{equation} \int _ Zg^*T\wedge (\gamma ')^{n-1} \leq C_ g\int _ YT\wedge \gamma ^{n-1} \qquad (T\geq 0,\ \mathrm{d}T=0). \end{equation}
Proof.
Write $T=B_\gamma (T)+\mathrm{dd}^{\mathrm{c}}v_\gamma (T)$. The pullback is
\[ g^*T=g^*B_\gamma (T)+\mathrm{dd}^{\mathrm{c}}(v_\gamma (T)\circ g). \]
This current is positive and closed, since it is locally $\mathrm{dd}^{\mathrm{c}}$ of plurisubharmonic functions, and the composed potential is not identically $-\infty $ because $g$ is surjective. The formula agrees with the definition through local potentials, so it does not depend on the decomposition.
If $T_ j\to T$, the smooth parts converge in $C^\infty $, and the potentials converge in $L^1(Y)$ with a common lower curvature bound. By lemma 5.43.19, their compositions with $g$ converge in $L^1(Z)$, and taking $\mathrm{dd}^{\mathrm{c}}$ proves the weak convergence of the pullbacks.
Integration by parts on $Z$ gives
\[ \int _ Zg^*T\wedge (\gamma ')^{n-1} =\int _ Zg^*B_\gamma (T)\wedge (\gamma ')^{n-1}. \]
The absolute value of the right-hand side is at most a fixed constant times $\| B_\gamma (T)\| _{C^0}$. Since the left-hand side is nonnegative, (20.4.21) gives (20.4.23).
$\square$
Corollary 20.4.24.AC7947F570DB Let $g:Z\to Y$, $\gamma $ and $\gamma '$ be as in proposition 20.4.22. Let $D_ j$ be prime divisors on $Y$, with integration currents $[D_ j]$, and let $c_ j>0$. If $S=\sum _ j c_ j[D_ j]$ is a positive closed current of finite mass, then
\[ g^*S=\sum _ j c_ j[g^*D_ j] \]
as positive currents of finite mass. The trace masses on $Z$ of the positive tails of this series tend to zero. In particular, $g^*S$ is again purely divisorial.
Proof.
Apply the proposition to the increasing finite partial sums $S_ N$ and to their positive closed tails $S-S_ N$. Pullback is additive on positive closed currents, by addition of local potentials. Thus $g^*S-g^*S_ N=g^*(S-S_ N)\geq 0$, and its trace mass is at most $C_ g$ times the trace mass of $S-S_ N$, which tends to zero. The pullback of a finite divisor is the usual pullback of its local equations, which identifies $g^*S_ N$ with the corresponding partial sum of the series, and hence $g^*S$ with its limit. Its terms are integration currents of prime divisors with nonnegative coefficients, and regrouping this countable positive sum of finite mass does not change it.
$\square$
Pome
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