Dolbeault Complex on $S^4\setminus \{\,\cdot\,\}$ and $S^6\setminus\{\,\cdot\,\}$ through Supersymmetric Glasses

$S^4$ is not a complex manifold, but it is sufficient to remove one point to make it complex. Using supersymmetry methods, we show that the Dolbeault complex (involving the holomorphic exterior derivative $\partial$ and its Hermitian conjugate) can be perfectly well defined in this case. We calculate the spectrum of the Dolbeault Laplacian. It involves $3$ bosonic zero modes such that the Dolbeault index on $S^4\setminus\{\,\cdot\,\}$ is equal to $3$.