I’d say that, just as for the plain Dirac operator, the obstruction is the existence of the topological structure. Given that, a connection may be chosen.

We don’t have an entry on geometric/differential Spin^c-structures, I believe.

]]>Don’t we need a *geometric* Spin^c structure to define the Spin^c Dirac operator?
Otherwise, how is the connection on the complex spinor bundle defined?

Does the nLab have an article about geometric Spin^c structures?

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