13.5 Global finiteness and bounded cohomological mass48A11CF18369

Proof. Fix $j$, let $R$ be the closed positive current $\bigwedge _{i>j}(\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i)$, and let $\eta '$ be a smooth closed form of pure complementary bidegree. Locally $\theta _ j=\mathrm{dd}^{\mathrm{c}}h_ j$ with $h_ j$ smooth, and since $R$ is closed, the recursion gives

Here $\mathrm{dd}^{\mathrm{c}}(h_ jR)=\mathrm{dd}^{\mathrm{c}}h_ j\wedge R$ because $h_ j$ is smooth and $R$ is closed of pure bidegree, so that $\partial R=\bar\partial R=0$. Both sides are globally defined, because $\varphi _ j$ is a global bounded function. Moreover $\int _ X\mathrm{dd}^{\mathrm{c}}(\varphi _ jR)\wedge \eta '=\int _ X\varphi _ jR\wedge \mathrm{dd}^{\mathrm{c}}\eta '=0$, since $\eta '$ is closed of pure bidegree, so that $\partial \eta '=\bar\partial \eta '=0$ and $\mathrm{dd}^{\mathrm{c}}\eta '=0$. Hence $\int _ X(\theta _ j+\mathrm{dd}^{\mathrm{c}}\varphi _ j)\wedge R\wedge \eta '=\int _ X\theta _ j\wedge R\wedge \eta '$. Apply this successively for $j=1,2,\ldots ,p$, with $\eta '=\theta _1\wedge \cdots \wedge \theta _{j-1}\wedge \eta $ at the $j$th step, which is smooth and closed. After $p$ steps the integral becomes $\int _ X\theta _1\wedge \cdots \wedge \theta _ p\wedge \eta $. $\square$

Proof. The measures. Near each point of $X$, a bounded quasi-plurisubharmonic function $b$ is the difference $b_1-b_2$ of two bounded plurisubharmonic functions: if $b+g$ is plurisubharmonic on a coordinate ball with $g$ smooth, and $q=C|z|^2$ with $C$ so large that $g+q$ is plurisubharmonic on a smaller ball, take $b_1=b+g+q$ and $b_2=g+q$. Hence $\mathrm{dd}^{\mathrm{c}}(bR)=\mathrm{dd}^{\mathrm{c}}b_1\wedge R-\mathrm{dd}^{\mathrm{c}}b_2\wedge R$ is locally the difference of two Bedford–Taylor products, which are positive measures. Covering the compact manifold $X$ by finitely many such neighborhoods, we see that $\mathrm{dd}^{\mathrm{c}}(bR)$ is a finite signed measure; the same holds for $a$.

For bounded plurisubharmonic functions $A$ and $B$ on an open set, additivity of Bedford–Taylor products (proposition 7.2.2) and the formula (7.14.4) give, after expanding $(A+B)^2$,

so that $\frac12(\mathrm{dd}^{\mathrm{c}}(ABR)-A\, \mathrm{dd}^{\mathrm{c}}B\wedge R-B\, \mathrm{dd}^{\mathrm{c}}A\wedge R)=\frac12(E_ R(A+B)-E_ R(A)-E_ R(B))$ is a signed measure, by lemma 7.14.3. Since the expression defining the mixed gradient is bilinear in $(a,b)$, writing $a=a_1-a_2$ and $b=b_1-b_2$ locally shows that the mixed gradient is a signed measure near each point, hence a finite signed measure on $X$. It is symmetric by definition. For $a=b$ plurisubharmonic it equals $\frac12\mathrm{dd}^{\mathrm{c}}(a^2R)-a\, \mathrm{dd}^{\mathrm{c}}a\wedge R=E_ R(a)$. For smooth $a$ and $b$ we have $\mathrm{dd}^{\mathrm{c}}(ab)=a\, \mathrm{dd}^{\mathrm{c}}b+b\, \mathrm{dd}^{\mathrm{c}}a+\mathrm da\wedge \mathrm d^{\mathrm c}b+\mathrm db\wedge \mathrm d^{\mathrm c}a$, and the $(1,1)$-parts of $\mathrm da\wedge \mathrm d^{\mathrm c}b$ and $\mathrm db\wedge \mathrm d^{\mathrm c}a$ both equal $\frac{\mathrm i}{4\pi }(\partial a\wedge \bar\partial b+\partial b\wedge \bar\partial a)$; since only $(1,1)$-parts contribute in products with $R$ and $\mathrm{dd}^{\mathrm{c}}(abR)=\mathrm{dd}^{\mathrm{c}}(ab)\wedge R$, the mixed gradient is $\mathrm da\wedge \mathrm d^{\mathrm c}b\wedge R$.

A local identity. Let $\chi $ be a test function with support in a coordinate ball $W$, and let $A$ and $B$ be bounded plurisubharmonic functions on a neighborhood of $\overline W$. We claim that

where $\mathrm d(BR)$ is the current of order zero of lemma 7.14.3 and its value on the bounded Borel one-form $A\, \mathrm d^{\mathrm c}\chi $ with compact support is computed with its coefficient measures. Suppose first that $A$ is smooth. Then $\int \chi A\, \mathrm{dd}^{\mathrm{c}}(BR)=(BR)(\mathrm{dd}^{\mathrm{c}}(\chi A))$ and

The first term contributes $\int \chi B\, \mathrm{dd}^{\mathrm{c}}A\wedge R$. Since $BR$ has bidegree $(n-1,n-1)$, it sees only the $(1,1)$-parts of the last two terms, which agree as above; so these terms contribute $2(BR)(\mathrm dA\wedge \mathrm d^{\mathrm c}\chi )$. Finally $\mathrm dA\wedge \mathrm d^{\mathrm c}\chi =\mathrm d(A\, \mathrm d^{\mathrm c}\chi )-A\, \mathrm{dd}^{\mathrm{c}}\chi $, and $(BR)(\mathrm d\psi )=-\mathrm d(BR)(\psi )$ for one-forms $\psi $, because $BR$ has even degree. Collecting terms gives (13.5.3).

In general, let $A_ l$ be smooth plurisubharmonic functions decreasing to $A$ on a neighborhood of $\overline W$, for instance mollifications (proposition 5.9.2(2)). Then (13.5.3) holds for $A_ l$, and every term converges to the corresponding term for $A$. For the first, third and last terms this follows by dominated convergence against the fixed signed measures $\mathrm{dd}^{\mathrm{c}}(BR)$, $\mathrm{dd}^{\mathrm{c}}\chi \wedge R$ and the coefficient measures of $\mathrm d(BR)$, since the $A_ l$ are uniformly bounded on $\operatorname{Supp}\chi $ and converge pointwise to $A$. For the second term it follows from the weighted form of theorem 7.3.7, with the constant sequence of weights $B$. This proves the claim. Since both sides of (13.5.3) are bilinear in $(A,B)$, the identity also holds for differences of bounded plurisubharmonic functions, and in particular for $A=a$ and $B=b$ near $\overline W$.

Conclusion. Let $(\chi _ i)$ be a finite smooth partition of unity on $X$ subordinate to coordinate balls as above (proposition 5.2.6). Summing (13.5.3) for $\chi =\chi _ i$, $A=a$ and $B=b$ over $i$, and using that $abR$, $\mathrm d(bR)$ and $\mathrm{dd}^{\mathrm{c}}(aR)$, $\mathrm{dd}^{\mathrm{c}}(bR)$ are global currents, we obtain

because $\sum _ i\chi _ i=1$. This is the first equality. For the second, note that $\mathrm{dd}^{\mathrm{c}}(abR)$ is a finite signed measure, being a combination of the mixed gradient and the measures $a\, \mathrm{dd}^{\mathrm{c}}b\wedge R$ and $b\, \mathrm{dd}^{\mathrm{c}}a\wedge R$, and that its total integral is $\mathrm{dd}^{\mathrm{c}}(abR)(1)=(abR)(\mathrm{dd}^{\mathrm{c}}1)=0$, because the constant function $1$ is a test function on the compact manifold $X$. Integrating the definition of the mixed gradient over $X$ and using the first equality, we get $\int _ X\mathrm da\wedge \mathrm d^{\mathrm c}b\wedge R=-\int _ Xa\, \mathrm{dd}^{\mathrm{c}}b\wedge R$. $\square$

Proof. The decomposition $T_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}\varphi _ j$ exists by theorem 10.2.15. Since $\tau _ j\geq \theta _ j$ and $\tau _ j\geq 0$, $\varphi _ j$ is $\tau _ j$-plurisubharmonic, and so is every constant; hence $\max \{ \varphi _ j,-k\} $ is a bounded $\tau _ j$-plurisubharmonic function, and

is a closed positive current. By proposition 13.5.1, the total mass $\int _ XR_{1,k}\wedge \cdots \wedge R_{p,k}\wedge \omega ^{n-p}$ equals the right side of (13.5.5) for every $k$.

Local finiteness. Let $W$ be a relatively compact coordinate chart, and choose smooth functions $h_ j$ and $h$ near $\overline W$ with $\mathrm{dd}^{\mathrm{c}}h_ j=\theta _ j$ and $\mathrm{dd}^{\mathrm{c}}h=\omega $. Then $u_ j=\varphi _ j+h_ j$ is a plurisubharmonic potential of $T_ j$ on $W$. Let $B$ be an integer with $h_ j\leq B$ on $W$ for all $j$, fix an integer $l\geq 1$, and put $O_ l(u)=\bigcap _ j\{ u_ j>-l\} $. On $W\cap O_ l(u)$ we have $\varphi _ j>-l-B$, so for every integer $k\geq l+B$ the bounded plurisubharmonic functions $\max \{ u_ j,-l\} +Ch$ and $\max \{ \varphi _ j,-k\} +h_ j+Ch$ both equal $u_ j+Ch$ there; the second is a local potential of $R_{j,k}$. By plurifine locality,

Expanding the left side into positive terms gives

Let $K\subset W$ be compact. On a neighborhood of $K$ in $W$ we have $\beta \leq c\, \omega $ for some $c>0$ (lemma 6.14.1), so by the last display and (13.2.8), the trace masses on $K$ of the truncated local products are at most $c^{n-p}$ times the right side of (13.5.5), uniformly in $l$. Hence the non-pluripolar product is locally finite everywhere, and $\langle T_1\wedge \cdots \wedge T_ p\rangle $ is a closed positive current by theorem 13.3.1.

Mass bound. By the same argument, the currents $T_ j+C\omega =\tau _ j+\mathrm{dd}^{\mathrm{c}}\varphi _ j$, with local potentials $u_ j+Ch$, have a locally finite non-pluripolar product. The sets $G_ k=\bigcap _ j\{ \varphi _ j>-k\} $ increase and exhaust the complement of the union of the pole sets of the $\varphi _ j$. On $G_ k$ the global truncations $\max \{ \varphi _ j,-k\} $ coincide with $\varphi _ j$. If $B'$ is an integer with $h_ j+Ch\geq -B'$ on $W$ for all $j$, then on $G_ k\cap W$ all local potentials satisfy $u_ j+Ch>-k-B'$, so they coincide there with their truncations at level $k+B'$, and also with the local potentials $\max \{ \varphi _ j,-k\} +h_ j+Ch$ of $R_{j,k}$. By plurifine locality and lemma 13.2.2, $\mathbf1_{G_ k}\bigwedge _ jR_{j,k}=\mathbf1_{G_ k}\langle \bigwedge _ j(T_ j+C\omega )\rangle $, and these restrictions increase to $\langle \bigwedge _ j(T_ j+C\omega )\rangle $. Their total masses against $\omega ^{n-p}$ are at most the right side of (13.5.5), hence so is that of the limit. Finally, expanding $\langle \bigwedge _ j(T_ j+C\omega )\rangle $ by multilinearity (proposition 13.2.6) into a sum of positive currents, one of which is $\langle \bigwedge _ jT_ j\rangle $, we obtain $\langle \bigwedge _ j T_ j\rangle \leq \langle \bigwedge _ j(T_ j+C\omega )\rangle $, which proves (13.5.5). $\square$

Proof. For $n=1$ the conclusion is the hypothesis, so let $n\geq 2$. Work on a relatively compact coordinate domain on which $T_ i=\mathrm{dd}^{\mathrm{c}}a_ i$ with plurisubharmonic potentials $a_ i$ bounded above, and put

Since $G_ j$ is contained in the level-$j$ set $\{ a_ i>-j\} $ of the tuple $(a_ i,\ldots ,a_ i)$, lemma 13.2.2 shows that the bounded measure $(\mathrm{dd}^{\mathrm{c}}a_{i,j})^ n$ agrees with $\langle T_ i^ n\rangle $ on $G_ j$. By positivity, therefore, on the whole coordinate domain

Now theorem 7.12.1, applied to the densities $\mathbf1_{G_ j}f_ i$, gives $\mathrm{dd}^{\mathrm{c}}a_{1,j}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}a_{n,j}\geq \mathbf1_{G_ j}\prod _ if_ i^{1/n}\nu $. Since $G_ j$ is the level-$j$ set of the tuple $(a_1,\ldots ,a_ n)$, the left side agrees on $G_ j$ with the non-pluripolar product of the $T_ i$, by lemma 13.2.2. Restricting to $G_ j$ and using positivity (lemma 13.2.1), we obtain

The complement of $\bigcup _ jG_ j$ is the union $P$ of the pole sets of the $a_ i$. Every $\langle T_ i^ n\rangle $ gives zero mass to the whole pluripolar set $P$, not only to the pole set of $a_ i$, so the inequality $\langle T_ i^ n\rangle \geq f_ i\nu $ forces $f_ i=0$ $\nu $-almost everywhere on $P$, for every $i$. Hence $\mathbf1_{G_ j}\prod _ if_ i^{1/n}$ increases to $\prod _ if_ i^{1/n}$ $\nu $-almost everywhere as $j\to \infty $, and monotone convergence on every Borel subset of the coordinate domain removes the indicator. Since finitely many such coordinate domains cover $X$, and an inequality of measures that holds on each set of a finite open cover holds on $X$ (decompose a Borel set into Borel pieces, each contained in one set of the cover), the asserted inequality follows.

Finally, $\int _ Xf_ i\, \mathrm d\nu \leq \int _ X\langle T_ i^ n\rangle <\infty $. To integrate the product, we use the inequality between weighted geometric and arithmetic means: for $y_1,\ldots ,y_ m\in [0,\infty ]$ and weights $\lambda _ i>0$ with $\sum _ i\lambda _ i=1$,

If all $y_ i$ are positive and finite, this is lemma 2.2.4 for the exponential function on the probability space $\{ 1,\ldots ,m\} $ with masses $\lambda _ i$, on which the integrand takes the values $\log y_ i$; otherwise the left side vanishes or the right side is infinite. With $\lambda _ i=1/n$, it gives $\int _ X\prod _ if_ i^{1/n}\, \mathrm d\nu \leq \frac1n\sum _ i\int _ Xf_ i\, \mathrm d\nu <\infty $. $\square$

Proof. Apply proposition 13.5.8 with $k$ factors equal to $T$, with density $f$, and $n-k$ factors equal to $S$, with density $g$. $\square$

Proof. For $t=0$ and $t=1$ this is the hypothesis. Otherwise, multilinearity (proposition 13.2.6) expands the left side as $\sum _{k=0}^ n\binom nk(1-t)^ kt^{n-k}\langle T^ k\wedge S^{n-k}\rangle $. Bounding the terms with $1\leq k\leq n-1$ by corollary 13.5.10 and the terms with $k=0,n$ by the hypotheses, we obtain

by the binomial formula. The measure on the right is finite, because $f$ and $g$ are $\nu $-integrable by the hypotheses and each density $f^{k/n}g^{(n-k)/n}$ is at most $\frac knf+\frac{n-k}ng\leq f+g$: for $k=0$ and $k=n$ this is an equality, and for $1\leq k\leq n-1$ it is (13.5.9). $\square$