# dg.differential geometry – Pull back of Spin\$^{mathbb{C}}\$ bundle

Let $$M$$ be a closed $$4$$-d Riemannian manifold and $$Z$$ be its twistor space of $$M$$, i.e., the bundle of almost complex structures on $$M$$. Let $$V$$ be a Spin$$^{mathbb{C}}$$ bundle, $$V_+$$ denote the positive spin bundle. Now we know $$Z$$ admits more than one almost complex structure. So it can have canonical Spin$$^{mathbb{C}}$$ bundle once we fix an almost complex structure and we can have other Spin$$^{mathbb{C}}$$ bundles by twisting it with a complex line bundle. Now let $$pi:Zrightarrow M$$ be the projection. Can we realize $$pi^*(V_+)$$ as a sub-bundle of some Spin$$^{mathbb{C}}$$ bundle of positive spinors on $$Z$$ and if yes how?