Lemma 13.5.205FB10FAF0D8

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 $\int _ X\mathrm da\wedge \mathrm d^{\mathrm c}b\wedge R=-\int _ Xa\, \mathrm{dd}^{\mathrm{c}}b\wedge R$. $\square$