Theorem 5.3.71C212E45A497

Proof. Fix a chart and put $q=n-p$. By proposition 5.3.2, $\beta ^ q$ is an interior point of the cone of strongly positive $(q,q)$-forms. Hence for every constant real $(q,q)$-form $\tau $ there is $C_\tau >0$ such that the forms $C_\tau \beta ^ q\pm \tau $ are strongly positive. For a nonnegative test function $\chi $, the forms $\chi (C_\tau \beta ^ q\pm \tau )$ are strongly positive test forms, so positivity of $T$ gives $T(\chi (C_\tau \beta ^ q\pm \tau ))\geq 0$. Therefore $T(\chi \tau )$ is real and

The distributions $C_\tau \mu _ T\pm T(\, \cdot \, \tau )$ are nonnegative on nonnegative test functions, hence positive Radon measures by the remark after definition 5.3.5 and theorem 2.5.6. Consequently $S_\tau =T(\, \cdot \, \tau )$ is a real signed Radon measure, the difference of the positive measures $(C_\tau \mu _ T\pm S_\tau )/2$, and $|S_\tau (A)|\leq C_\tau \mu _ T(A)$ for every relatively compact Borel set $A$, so that $|S_\tau |\leq C_\tau \mu _ T$.

Choose real constant $(q,q)$-forms $\tau _1,\ldots ,\tau _ N$ forming a basis of the real $(q,q)$-forms; they also form a complex basis of all constant $(q,q)$-forms, and they can be chosen depending only on $n$ and $p$. Up to a constant factor depending only on $n$, $I$ and $J$, the coefficient $T_{IJ}$ is the distribution $\chi \mapsto T(\chi \, \mathrm dz_{I^ c}\wedge \mathrm d\bar z_{J^ c})$, where $I^ c$ and $J^ c$ are the complementary multi-indices. Expanding $\mathrm dz_{I^ c}\wedge \mathrm d\bar z_{J^ c}=\sum _ rc_ r\tau _ r$, we see that $T_{IJ}$ is a fixed complex linear combination of the measures $S_{\tau _ r}$, and the coefficient bound follows from $|S_{\tau _ r}|\leq C_{\tau _ r}\mu _ T$. To prove that $T$ is real, let $\varphi $ be a test form supported in the chart; only its $(q,q)$-component contributes, and we write it as $\sum _ rf_ r\tau _ r$ with complex test functions $f_ r$. Since $T(f\tau _ r)=\int f\, \mathrm dS_{\tau _ r}$ is real for real $f$, we obtain $T(\bar\varphi )=\overline{T(\varphi )}$, and a partition of unity extends this identity to all test forms.

Now let $X$ have a countable basis, and let $(T_ j)$ be as in the statement. Then $X$ is covered by countably many charts $U_1,U_2,\ldots $: for each basic open set contained in some chart, choose one such chart. On $U_1$ the positive Radon measures $C_{\tau _ r}\mu _{T_ j}\pm S^ j_{\tau _ r}$, where $S^ j_\tau =T_ j(\, \cdot \, \tau )$, have mass at most $2C_{\tau _ r}\sup _ j\mu _{T_ j}(K)$ on every compact $K\subset U_1$. Applying theorem 2.6.5 successively to these finitely many sequences, we find a subsequence along which all of them converge vaguely on $U_1$; then each $S^ j_{\tau _ r}$ converges vaguely, and $T_ j(\varphi )$ converges for every test form $\varphi $ supported in $U_1$. Repeating the extraction for $U_2,U_3,\ldots $ and passing to the diagonal subsequence, we may assume that $T_ j(\varphi )$ converges for every test form $\varphi $ supported in some $U_ k$. A partition of unity subordinate to $(U_ k)$ writes every test form as a finite sum of such forms, so $T(\varphi )=\lim _ jT_ j(\varphi )$ exists for every test form $\varphi $.

The functional $T$ is linear, and it is a current. Indeed, if $\varphi =\sum _ rf_ r\tau _ r$ is supported in a compact set $K\subset U_ k$, then $|T_ j(\varphi )|\leq \sum _ rC_{\tau _ r}\| f_ r\| _\infty \sup _ i\mu _{T_ i}(K)$ for all $j$, and this bound passes to the limit. On compact subsets of the intersection of two charts, the coefficients of a form in the two charts are related by a matrix of continuous functions. Hence, covering a compact subset of an arbitrary chart by finitely many $U_ k$ and using a partition of unity, we obtain the estimate in the definition of a current, with $m=0$. Thus $T_ j\to T$ weakly. If $\varphi $ is a strongly positive test form, then $T(\varphi )=\lim _ jT_ j(\varphi )\geq 0$; and if all $T_ j$ are closed, then $T(\mathrm d\psi )=\lim _ jT_ j(\mathrm d\psi )=0$ for every test form $\psi $. $\square$