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?



first chern class and spin structures
Yes. An oriented real vector bundle is spin if and only if its second Stiefel-Whitney class vanishes. If $E\_\mathbb{C}$ is a complex vector bundle and $E\_\mathbb{R}$ is the underlying real bundle then the second Stiefel-Whitney 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.


