Bounded cohomology for coherent analytic sheaves over complex spaces

In this paper a certain continuous family $V_t=\{V_{ti}\}$, $0\leqslant t\leqslant1$, of finite covers by holomorphically complete domains is constructed for a compact complex space $X$ such that if $t_1<t_2$ then $V_{t_1i}\Subset V_{t_2i}$ and $\overline V_{ti}=\bigcap_{t'>t}V_{t'i}V_{ti}=\bigcup_{t'<t}V_{t'i}$ for all $i$ and $t$. It is proved that for each coherent sheaf $F$ over $X$ there exist positive constants $K$ and $\alpha$ such that for any $t_1,t_2$ with $t_1<t_2$, if $c\in C^p(V_{t_2},F)$ is a coboundary then one can find a cochain $c'\in C^{p-1}(V_{t_2},F)$ such that $\delta c'=c$ and
$$ \|c'\|_{t_1}<K\frac1{(t_2-t_1)^\alpha}\|c\|_{t_2}. $$

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