Let $(X,g)$ be a $m$-dimensional complex, hermitian, spin manifold and let us denote by $S_{\mathbb{C}}$ its complex spinor bundle. Then:

$S_{\mathbb{C}}\simeq \Lambda_{\mathbb{C}}(X)$

Let $\nabla$ be any metric connection on $T_{\mathbb{C}}X$ that lifts to a connection $\nabla^{S_{\mathbb{C}}}$ on $S_{\mathbb{C}}$. Suppose that there is a complex form:

$\Omega\in \Lambda_{\mathbb{C}}(X)$

such that $\nabla\Omega = 0$. Under the isomorphism of the first equation, $\Omega$ corresponds to a spinor $\eta\in \Gamma(S_{\mathbb{C}})$. My question is, does it holds then that $\nabla^{S_{\mathbb{C}}}\eta = 0$? I guess this should be in fact and "if and only if".

Thanks.