Let $X$ is a complex analytic space of (complex) dimension $m$.
Question 1) Is there a complex analogue of a covering dimension, which is equal to the complex dimension $m$ of $X$?
Question 2) Is there a vanishing theorem which says that, under some conditions, $H^i(X,A) = 0$ for all $i > m$, where $A$ is an analytic sheaf on $X$, possibly satisfying some conditions too?
I know these are 2 different questions, but I believe they may be related, namely that one could use a complex analogue of a covering dimension, if it exists, in order to prove a vanishing theorem such as the one outlined in Question 2). I could be wrong though. Any helpful comments (or better, full answers) will be appreciated.