Let M be a compact complex manifold. Then is it true that if the first Chern class of M is even, then M admits a spin structure?
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Yes. An oriented real vector bundle is spin if and only if its second StiefelWhitney class vanishes. If $E_\mathbb{C}$ is a complex vector bundle and $E_\mathbb{R}$ is the underlying real bundle then the second StiefelWhitney class is given by $w_2(E_\mathbb{R}) = c_1(E_\mathbb{C})$ mod 2. The details appear somewhere in chapter 2 of Spin Geometry by Lawson and Michelsohn. 

