PomeLemma 13.2.1.6632853A40C9 Let $R$ be a positive current of bidegree $(p,p)$ on an open set $W\subset \mathbb C^ n$, and let $E\subset W$ be a Borel set. Then the current $\mathbf1_ ER$, obtained by multiplying the coefficient measures of $R$ by $\mathbf1_ E$, is positive. If $R$ is strongly positive, that is, $R(\varphi )\geq 0$ for every test form $\varphi $ that is weakly positive at every point, then so is $\mathbf1_ ER$.
13.2 Truncation and plurifine localityA56454EDE5B3
A56454EDE5B3
Let $X$ be a complex manifold of dimension $n\geq 1$, let $U\subset X$ be a coordinate domain, and fix an integer $1\leq p\leq n$. Let $u_1,\ldots ,u_ p$ be plurisubharmonic functions on $U$, and recall that $\{ u_ j=-\infty \} $ is the pole set of $u_ j$. For each integer $k\geq 1$ put
\[ \begin{gathered} u_{j,k}=\max \{ u_ j,-k\} ,\qquad O_ k=\bigcap _{j=1}^ p\{ u_ j>-k\} ,\\ S_ k=\mathbf1_{O_ k}\mathrm{dd}^{\mathrm{c}}u_{1,k}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{p,k}. \end{gathered} \]
The functions $u_{j,k}$ are locally bounded and plurisubharmonic, so the product in $S_ k$ is the Bedford–Taylor product of definition 7.2.1, and multiplication by $\mathbf1_{O_ k}$ restricts its coefficient measures to $O_ k$, where every $u_{j,k}$ coincides with $u_ j$. By proposition 5.47.23, the trace measures of the currents $S_ k$ increase with $k$ to a Borel measure $\mu $ for which every pluripolar subset of $U$ has outer measure zero, but $\mu $ may be infinite on compact sets (example 13.2.9 below).
The set $O_ k$ is Borel, because plurisubharmonic functions are upper semicontinuous, and it is plurifine open, because they are plurifine continuous; it need not be open in the ordinary topology. For instance, the countable set $A$ of points with rational coordinates in the unit disc $\Delta \subset \mathbb C$ is polar, each point $a$ being the pole of $\log |z-a|$, so by proposition 4.1.11 there is a finite positive measure $\nu $ with compact support such that the fundamental potential $E_2*\nu $ equals $-\infty $ on $A$. The function $u_1(z)=(E_2*\nu )(z_1)$ is plurisubharmonic on $\mathbb C^ n$ by proposition 3.4.11, proposition 5.9.3, and its pole set contains the dense subset $A\times \mathbb C^{n-1}$ of $\Delta \times \mathbb C^{n-1}$. For $p=1$ and large $k$, the set $O_ k\cap (\Delta \times \mathbb C^{n-1})$ is nonempty, because $E_2*\nu $ is locally integrable, but it has empty interior, so $O_ k$ is not open. We shall often restrict positive currents to Borel sets such as $O_ k$, and we first note that this preserves positivity.
Proof. By theorem 5.3.7, $R$ has measure coefficients. Let $\varphi $ be a test form of bidegree $(n-p,n-p)$ that is strongly positive, respectively weakly positive, at every point. The complex measure $R\wedge \varphi $, whose integral against a test function $\chi $ is $R(\chi \varphi )$, defines a positive distribution, because $\chi \varphi $ has the same positivity as $\varphi $ whenever $\chi \geq 0$; by the uniqueness in theorem 2.5.6, it is a positive measure. Hence $(\mathbf1_ ER)(\varphi )=(R\wedge \varphi )(E)\geq 0$. $\square$
The next lemma shows that local finiteness of $\mu $ is exactly what is needed for the currents $S_ k$ to converge.
Lemma 13.2.2.D59675D9F6D8 For $l\geq k$ the current $S_ l-S_ k$ is positive, and $\mathbf1_{O_ k}S_ l=S_ k$. The currents $S_ k$ converge weakly to a positive current $S$ if and only if
E88E33ACAD5B
13.2.3
\begin{equation} \sup _ k\int _ K S_ k\wedge \beta ^{n-p}<\infty \quad \text{for every compact set }K\subset U, \end{equation}
where $\beta $ is the Euclidean Kähler form (5.2.8) of the coordinates. In that case $\mathbf1_{O_ k}S=S_ k$ for every $k$, the coefficient measures of $S_ k$ converge to those of $S$ in total variation on compact subsets of $U$, and the trace measure of $S$ is the measure $\mu $ above. Every pluripolar subset of $U$ has outer measure zero for $\mu $ and for the total variations of the coefficient measures of $S$; consequently every plurifine open subset of $U$ is measurable for the completions of these measures. Moreover, if $E_ k\subset O_ k$ are Borel sets such that $\mathbf1_{E_ k}\to 1$ almost everywhere for the trace measure of $S$, then the coefficient measures of $\mathbf1_{E_ k}S_ k$ converge to those of $S$ when integrated against any bounded Borel function with compact support in $U$. This is the case, in particular, if the sets $E_ k$ increase and $U\setminus \bigcup _ kE_ k$ is pluripolar.
Proof. Let $l\geq k$. By proposition 5.47.23, the level-$l$ and level-$k$ Bedford–Taylor products have the same restriction to $O_ k$; since $O_ k\subset O_ l$, this is the identity $\mathbf1_{O_ k}S_ l=S_ k$. Consequently $S_ l-S_ k=\mathbf1_{O_ l\setminus O_ k}\mathrm{dd}^{\mathrm{c}}u_{1,l}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{p,l}$, and this current is positive by lemma 13.2.1.
Suppose that (13.2.3) holds. The trace measures $S_ k\wedge \beta ^{n-p}$ increase with $k$ and, by (13.2.3), converge to a locally finite measure. Hence the trace mass of the positive current $S_ l-S_ k$ on a compact set $K$ tends to zero as $k,l\to \infty $. By theorem 5.3.7, the total variation on $K$ of each coefficient measure of $S_ l-S_ k$ is at most a fixed multiple of this trace mass. Therefore each coefficient measure of $S_ k$ is a Cauchy sequence in total variation on compact sets, and the limits are the coefficients of a current $S$, which is positive as a weak limit of positive currents. Conversely, suppose that $S_ k\to S$ weakly for a positive current $S$. For every smooth compactly supported $\chi \geq 0$, the numbers $\int \chi S_ k\wedge \beta ^{n-p}$ increase with $k$ and converge to $\int \chi S\wedge \beta ^{n-p}$, which therefore bounds them; choosing $\chi =1$ on $K$ gives (13.2.3).
Assume now that (13.2.3) holds. Letting $l\to \infty $ in $\mathbf1_{O_ k}S_ l=S_ k$ gives $\mathbf1_{O_ k}S=S_ k$. By the convergence in total variation, the trace measure of $S$ is the increasing limit of the trace measures of the $S_ k$, which is $\mu $. By proposition 5.47.23, every pluripolar subset of $U$ has outer $\mu $-measure zero. The total variation of each coefficient measure of $S$ is at most $C_{n,p}\mu $ (theorem 5.3.7), so pluripolar sets have outer measure zero for these total variations as well, and theorem 5.47.20(2), applied to $\mu $ and to these total variations, shows that plurifine open subsets of $U$ are measurable for their completions. In particular $S$ vanishes on $U\setminus \bigcup _ kO_ k$, the union of the pole sets of the $u_ j$, which is a pluripolar Borel set.
Finally, let $E_ k$ be as in the statement. Since $E_ k\subset O_ k$, we have $\mathbf1_{E_ k}S_ k=\mathbf1_{E_ k}S$. By theorem 5.3.7, the total variation of each coefficient measure of $S$ is locally finite and absolutely continuous with respect to the trace measure. Hence $\mathbf1_{E_ k}\to 1$ almost everywhere for each of these variations, and dominated convergence proves the assertion. If the $E_ k$ increase and $U\setminus \bigcup _ kE_ k$ is pluripolar, then $\mathbf1_{E_ k}\to 1$ outside this pluripolar Borel set, which $S$ does not charge. $\square$
Definition 13.2.4.D79952D7E272 If (13.2.3) holds, we say that the non-pluripolar product of $u_1,\ldots ,u_ p$ is locally finite, and we call the limit $S$ of lemma 13.2.2 the non-pluripolar product
\[ \langle \mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle =S. \]
The angle brackets distinguish this construction from a product that retains mass on pluripolar sets. The empty product ($p=0$) is $1$.
Proposition 13.2.5.1FAE60CC76F9 Every locally finite non-pluripolar product $S$ is strongly positive: $S(\varphi )\geq 0$ for every compactly supported smooth weakly positive form $\varphi $ of bidegree $(n-p,n-p)$.
Proof. First let all potentials be locally bounded. Near each point they are limits of decreasing sequences of smooth plurisubharmonic functions, for instance of their mollifications (proposition 5.9.2(2)). The complex Hessians of these functions are semipositive $(1,1)$-forms (proposition 5.9.2(1)), hence strongly positive, since the two notions of positivity coincide in bidegree $(1,1)$; by proposition 5.3.2, their exterior products are strongly positive, and the product of such a product with a weakly positive form of complementary bidegree is a nonnegative multiple of the volume form. Thus the smooth products pair nonnegatively with every smooth compactly supported weakly positive form. By theorem 7.3.7, the Bedford–Taylor product is their weak limit, and therefore it pairs nonnegatively with these forms as well.
By lemma 13.2.1, applied to the bounded products in the definition of $S_ k$, every $S_ k$ is strongly positive. Since $S_ k\to S$ weakly by lemma 13.2.2, so is $S$. $\square$
Proposition 13.2.6.7F7123F92478 Let $u_1,\ldots ,u_ p$ and $v_1,\ldots ,v_ p$ be plurisubharmonic on $U$. If $\mathrm{dd}^{\mathrm{c}}u_ j=\mathrm{dd}^{\mathrm{c}}v_ j$ for every $j$, then the non-pluripolar product of the $u_ j$ is locally finite if and only if that of the $v_ j$ is, and then the two products are equal; thus local finiteness and the product depend only on the currents $\mathrm{dd}^{\mathrm{c}}u_ j$. The product is symmetric in its factors and local in the plurifine topology: if $u_ j=v_ j$ on a plurifine open set $O$ for every $j$ and both products are locally finite, then they have the same restriction to $O$. Finally, the product is multilinear: if $u_1'$ is plurisubharmonic on $U$, then the product of $u_1+u_1',u_2,\ldots ,u_ p$ is locally finite if and only if the products of $u_1,u_2,\ldots ,u_ p$ and of $u_1',u_2,\ldots ,u_ p$ are, and in that case
\[ \begin{aligned} \langle \mathrm{dd}^{\mathrm{c}}(u_1+u_1')\wedge \mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle & =\langle \mathrm{dd}^{\mathrm{c}}u_1\wedge \mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle \\ & \quad +\langle \mathrm{dd}^{\mathrm{c}}u_1'\wedge \mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle . \end{aligned} \]
Similarly, for real $c>0$ the product of $cu_1,u_2,\ldots ,u_ p$ is locally finite if and only if that of $u_1,u_2,\ldots ,u_ p$ is, and then it is $c$ times the latter.
Proof. In this proof we write $P^{(m)}(w_1,\ldots ,w_ p)=\mathrm{dd}^{\mathrm{c}}\max \{ w_1,-m\} \wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\max \{ w_ p,-m\} $ for plurisubharmonic $w_1,\ldots ,w_ p$ and an integer $m\geq 1$.
Symmetry and locality. Each truncated product $S_ k$ is symmetric in its factors, hence so is the limit. Suppose that $u_ j=v_ j$ on the plurifine open set $O$ for every $j$, and that both products are locally finite. By lemma 13.2.2, plurifine open subsets of $U$ are measurable for the completions of the total variations of the coefficient measures of both products, so the restrictions of the products to $O$ and to the plurifine open sets $W_ k=O\cap \bigcap _ j\{ u_ j>-k\} =O\cap \bigcap _ j\{ v_ j>-k\} $ are well defined; the sets $W_ k$ increase with $k$. On $W_ k$ the truncations $\max \{ u_ j,-k\} $ and $\max \{ v_ j,-k\} $ coincide, so the two level-$k$ products agree on $W_ k$ by theorem 5.47.22. Since $W_ k$ is contained in the level-$k$ sets of both tuples, lemma 13.2.2 shows that the two non-pluripolar products also agree on $W_ k$. The union of the $W_ k$ is $O$ minus the union of the pole sets of the $u_ j$, a pluripolar set, which has outer measure zero for these total variations by the same lemma. Letting $k\to \infty $, we conclude that the two products have the same restriction to $O$.
Dependence on the currents $\mathrm{dd}^{\mathrm{c}}u_ j$. Suppose that $\mathrm{dd}^{\mathrm{c}}u_ j=\mathrm{dd}^{\mathrm{c}}v_ j$ for every $j$. The distribution $v_ j-u_ j$ then has zero Laplacian, so by Weyl's lemma (theorem 3.1.11) it is a smooth function $w_ j$, and its complex Hessian vanishes: $w_ j$ is pluriharmonic. The plurisubharmonic functions $v_ j$ and $u_ j+w_ j$ agree almost everywhere, hence everywhere, since both are subharmonic (theorem 5.9.1, theorem 3.4.12). Fix a relatively compact open set $V\subset U$ and an integer $C$ such that $|w_ j|\leq C$ on $V$ for all $j$. Write $O_ k(u),S_ k(u)$ and $O_ k(v),S_ k(v)$ for the level sets and truncated products of the two tuples. On $V\cap O_ k(v)$ we have $u_ j=v_ j-w_ j>-k-C$, hence
\[ \max \{ v_ j,-k\} =v_ j=\max \{ u_ j,-k-C\} +w_ j . \]
The functions $\max \{ v_ j,-k\} $ and $\max \{ u_ j,-k-C\} +w_ j$ are bounded and plurisubharmonic on $V$ and coincide on $V\cap O_ k(v)$. Moreover, adding the pluriharmonic function $w_ j$ to a factor does not change Bedford–Taylor products. Indeed, by additivity (proposition 7.2.2) and symmetry (theorem 7.3.7), the difference is a sum of products having $\mathrm{dd}^{\mathrm{c}}w_ j$ as first factor; by proposition 7.2.2(1), such a product is the exterior product of the remaining closed positive current with the smooth form $\mathrm{dd}^{\mathrm{c}}w_ j=0$. As $V\cap O_ k(v)\subset O_{k+C}(u)$, plurifine locality gives on $V$
\[ S_ k(v)=\mathbf1_{O_ k(v)}P^{(k+C)}(u_1,\ldots ,u_ p)=\mathbf1_{O_ k(v)}S_{k+C}(u)\leq S_{k+C}(u). \]
Thus (13.2.3) for the $u_ j$ implies (13.2.3) for the $v_ j$ on compact subsets of $V$, and exchanging the two tuples (that is, replacing $w_ j$ by $-w_ j$) gives the converse. If both products are locally finite, the display and lemma 13.2.2 give $\mathbf1_{O_ k(v)}\langle \mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ p\rangle =\mathbf1_{O_ k(v)}\langle \mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle $ on $V$ for every $k$. The sets $O_ k(v)$ exhaust the complement of the union of the pole sets, which are the same for the two tuples and carry no mass for either product. Hence the two products coincide.
Additivity. Let $u_1'$ be plurisubharmonic and put
\[ E_ k=\{ u_1>-k\} \cap \{ u_1'>-k\} \cap \bigcap _{j=2}^ p\{ u_ j>-k\} . \]
These plurifine open sets increase with $k$ and exhaust the complement of the union of the pole sets of $u_1,u_1',u_2,\ldots ,u_ p$. On $E_ k$ we have $\max \{ u_1+u_1',-2k\} =u_1+u_1'=\max \{ u_1,-k\} +\max \{ u_1',-k\} $ and $\max \{ u_ j,-2k\} =\max \{ u_ j,-k\} $ for $j\geq 2$. Plurifine locality and the additivity of bounded products (proposition 7.2.2) therefore give
491D0E94E07A
13.2.7
\begin{equation} \begin{aligned} \mathbf1_{E_ k}P^{(2k)}(u_1+u_1',u_2,\ldots ,u_ p) & =\mathbf1_{E_ k}P^{(k)}(u_1,u_2,\ldots ,u_ p)\\ & \quad +\mathbf1_{E_ k}P^{(k)}(u_1',u_2,\ldots ,u_ p). \end{aligned} \end{equation}
Before local finiteness is known we argue with trace measures. By proposition 5.47.23, for each of the three tuples the trace measures of the truncated products increase to a Borel measure, possibly infinite on compact sets, for which every pluripolar set has outer measure zero; for a locally finite product it is the trace measure of the product. The set $E_ k$ is contained in the level-$2k$ set of the first tuple and in the level-$k$ sets of the other two, so by the identity $\mathbf1_{O_ k}S_ l=S_ k$ of lemma 13.2.2, the restriction of each limit measure to $E_ k$ is the restriction of the corresponding trace measure in (13.2.7). Hence, on $E_ k$, the limit measure of the first tuple is the sum of those of the other two. Letting $k\to \infty $, monotone convergence gives this equality on every Borel subset of $\bigcup _ kE_ k$, and hence everywhere, because the complement of $\bigcup _ kE_ k$ is pluripolar. Since all three measures are positive, the first is locally finite if and only if the other two are; this is the asserted equivalence of local finiteness.
When the products are locally finite, (13.2.7) and lemma 13.2.2 give the additivity identity on each $E_ k$, and letting $k\to \infty $ proves it. For $c>0$ the same argument applies to the sets $\{ u_1>-k\} \cap \{ cu_1>-k\} \cap \bigcap _{j\geq 2}\{ u_ j>-k\} $, on which $\max \{ cu_1,-k\} =c\max \{ u_1,-k\} $, and proves the homogeneity assertion. $\square$
Let now $T_1,\ldots ,T_ p$ be closed positive $(1,1)$-currents on the complex manifold $X$. By theorem 5.4.3, every point of $X$ has a coordinate neighborhood on which $T_ j=\mathrm{dd}^{\mathrm{c}}u_ j$ with $u_ j$ plurisubharmonic. By proposition 13.2.6, local finiteness and the local non-pluripolar product do not depend on the choice of these potentials, and the construction is compatible with restriction to smaller open sets.
They do not depend on the coordinates either. Indeed, let $F$ be a biholomorphism between coordinate domains. Then $\max \{ u_ j,-k\} \circ F=\max \{ u_ j\circ F,-k\} $ and $O_ k(u\circ F)=F^{-1}(O_ k(u))$. Since $F$ is a submersion, the pullback $F^*$ of proposition 5.6.2 satisfies $F^*(\mathbf1_ ES)=\mathbf1_{F^{-1}(E)}F^*S$ for every current $S$ of order zero and every Borel set $E$ (corollary 5.6.4(2)), so proposition 7.2.2(5) gives $F^*S_ k(u)=S_ k(u\circ F)$ for the truncated products. To compare trace measures, note that if $\alpha $ and $\alpha '$ are smooth semipositive $(1,1)$-forms on an open set with $\alpha '\leq c\alpha $, then
\[ (c\alpha )^{n-p}-\alpha '^{n-p}=(c\alpha -\alpha ')\wedge \sum _{i=0}^{n-p-1}(c\alpha )^ i\wedge \alpha '^{n-p-1-i} \]
is a sum of products of semipositive $(1,1)$-forms, hence strongly positive (proposition 5.3.2). Since the product of a positive current with a smooth strongly positive form of complementary bidegree is a positive measure, it follows that
591B39D2BEA0
13.2.8
\begin{equation} R\wedge \alpha '^{n-p}\leq c^{n-p}R\wedge \alpha ^{n-p} \end{equation}
on that open set, for every positive current $R$ of bidegree $(p,p)$. On a neighborhood of a compact set, the Euclidean forms $\beta $ and $\beta '$ of the two coordinate systems satisfy $\beta '\leq c\beta $ for some $c$ (lemma 6.14.1), so by (13.2.8) the trace measures of a positive current in the two coordinate systems are comparable on compact sets, and local finiteness does not depend on the coordinates. Since $F^*$ is weakly continuous, $F^*\langle \mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle =\langle \mathrm{dd}^{\mathrm{c}}(u_1\circ F)\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}(u_ p\circ F)\rangle $.
We say that the non-pluripolar product of $T_1,\ldots ,T_ p$ is locally finite if this holds near every point; the local products then glue to a positive current $\langle T_1\wedge \cdots \wedge T_ p\rangle $ on $X$, which is closed by theorem 13.3.1 below. When $X$ is compact we also say simply that the product is finite. Repeated factors are written as powers, as in $\langle T^ j\wedge S^{r-j}\rangle $, and for $T_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}\varphi _ j$ we also write $\langle (\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1)\wedge \cdots \wedge (\theta _ p+\mathrm{dd}^{\mathrm{c}}\varphi _ p)\rangle $. The assertions of proposition 13.2.6 carry over to these products, with multilinearity in the factors $T_ j$.
Example 13.2.9.6F4C920FEDCA Let $U=\{ (x,y)\in \mathbb C^2:|x|^2<1/2,\ |y|<\mathrm e^{-1}\} $, put $L=-\log |y|$, and define
\[ u(x,y)=(|x|^2-1)\sqrt L\quad (y\ne 0),\qquad u(x,0)=-\infty . \]
On $U\setminus \{ y=0\} $ we have $L>1$, the function $u$ is smooth, and its complex second derivatives are
\[ u_{x\bar x}=\sqrt L,\qquad u_{y\bar y}=\frac{1-|x|^2}{16|y|^2L^{3/2}},\qquad u_{x\bar y}=-\frac{\bar x}{4\bar y\sqrt L},\qquad \det (u_{j\bar k})=\frac{1-2|x|^2}{16|y|^2L}. \]
Since $u_{x\bar x}>0$ and $\det (u_{j\bar k})>0$, the complex Hessian is positive definite, so $u$ is plurisubharmonic off the axis $\{ y=0\} $ (proposition 5.9.2(1)). Moreover $u<-\sqrt L/2$ is bounded above and tends to $-\infty $ at the axis. By theorem 5.11.1, the plurisubharmonic extension across the axis takes the value $\varlimsup u=-\infty $ there, which is the displayed value; hence $u$ is plurisubharmonic on $U$.
Off the axis, $(\mathrm{dd}^{\mathrm{c}}u)^2$ is the smooth measure with density a fixed positive multiple of $\det (u_{j\bar k})$, by (7.2.4). The sets $O_ k=\{ u>-k\} $ are open, since $u$ is continuous as an extended-real function, and they increase to $U\setminus \{ y=0\} $; on $O_ k$ the truncation $\max \{ u,-k\} $ equals $u$, so $S_ k=\mathbf1_{O_ k}(\mathrm{dd}^{\mathrm{c}}u)^2$. Take $K=\{ |x|^2\leq 1/4,\ |y|\leq r\} $ with $0<r<\mathrm e^{-1}$. On $K$ we have $1-2|x|^2\geq 1/2$, and polar coordinates in $y$ give
\[ \int _{|y|\leq r}\frac{\mathrm dA(y)}{|y|^2L}=2\pi \int _0^ r\frac{\mathrm dt}{t(-\log t)}=\infty . \]
Hence $\sup _ k\int _ KS_ k=\int _{K\setminus \{ y=0\} }(\mathrm{dd}^{\mathrm{c}}u)^2=\infty $, and (13.2.3) fails. Thus the non-pluripolar product of an arbitrary plurisubharmonic function need not be locally finite.
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