For every subnormal
$m$ -variable weighted shift
$S$ (with bounded positive weights), there is a corresponding positive Reinhardt measure
$\mu $ supported on a compact Reinhardt subset of
${{\mathbb{C}}^{m}}$ . We show that, for
$m\,\ge \,2$ , the dimensions of the 1-st cohomology vector spaces associated with the Koszul complexes of
$S$ and its dual
$\widetilde{S}$ are different if a certain radial function happens to be integrable with respect to μ (which is indeed the case with many classical examples). In particular,
$S$ cannot in that case be similar to
$\widetilde{S}$ . We next prove that, for
$m\,\ge \,2$ , a Fredholm subnormal
$m$ -variable weighted shift
$S$ cannot be similar to its dual.