I have a few questions regarding the existence of a spin structure on Kaehlerian and hyperKaehlerian manifolds. I cannot seem to provide a reference for proofs or counterexamples, so references are more than welcome.
Q1: Does every hyperKaehler manifold admit a spin structure?
Q2: Does every Kaehler manifold admit a spin structure?
Q3: Are there any dimension constraints on the existence of a spin structure on such manifolds?
Remark: According to page 85 of Jost's Riemannian Geometry and Geometric Analysis (6th edition), every orientable Riemannian manifold in dimension 4 carries a spin$^c$ structure. Since complex manifolds are orientable, this offers a partial to answer to the above questions. Jost, however, provides no proof or reference.