TITLE: first chern class and spin structures
QUESTION [5 upvotes]: 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?

REPLY [12 votes]: 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.