Lemma 5.3.8376F5C171BED

Proof. If two currents on $X$ agree with $T_ a$ on $W_ a$ for every $a$, their difference vanishes on every $W_ a$, hence on $X$, since a current that vanishes on each member of a family of open sets vanishes on their union, as shown at the beginning of this section. For existence, choose a partition of unity $(\chi _ c)_{c\in C}$ with compact supports subordinate to the cover (proposition 5.2.6), and for each $c$ an index $a(c)$ with $\operatorname{Supp}\chi _ c\subset W_{a(c)}$. For a test form $\varphi $ of degree $2n-k$ on $X$, only finitely many $\chi _ c$ are not identically zero on the compact set $\operatorname{Supp}\varphi $, and $\chi _ c\varphi $ is a test form supported in $W_{a(c)}$; we put

This is a linear functional, and it is a current. Indeed, let $K$ be a compact subset of a chart domain $U$. Only finitely many indices $c_1,\ldots ,c_ N$ have $\chi _{c_ i}\not\equiv 0$ on $K$, and for test forms $\varphi $ supported in $K$ the form $\chi _{c_ i}\varphi $ is supported in the compact subset $K\cap \operatorname{Supp}\chi _{c_ i}$ of the chart domain $U\cap W_{a(c_ i)}$ of $W_{a(c_ i)}$. By the Leibniz rule, its seminorms $p_{K,m}$ are bounded by a constant times those of $\varphi $, so the estimates for the currents $T_{a(c_ i)}$ give the estimate in the definition of a current for $T$. If $\varphi $ is supported in some $W_ b$, then each $\chi _ c\varphi $ is supported in $W_{a(c)}\cap W_ b$, where $T_{a(c)}$ and $T_ b$ agree, so that $T(\varphi )=\sum _ cT_ b(\chi _ c\varphi )=T_ b(\varphi )$. Thus $T$ agrees with $T_ b$ on $W_ b$.

Multiplication by $\chi _ c$ preserves bidegrees, and $\chi _ c\varphi $ is strongly positive at every point if $\varphi $ is, because $\chi _ c\geq 0$; so the assertions about bidegrees and positivity follow from the formula for $T(\varphi )$. Finally, if all $T_ a$ are closed and $\psi $ is a test form supported in $W_ a$, then so is $\mathrm d\psi $, and $(\mathrm dT)(\psi )=(-1)^{k+1}T(\mathrm d\psi )=(-1)^{k+1}T_ a(\mathrm d\psi )=(\mathrm dT_ a)(\psi )=0$; hence $\mathrm dT$ vanishes on every $W_ a$, and therefore $\mathrm dT=0$. $\square$