MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## first chern class and spin structures

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?

-

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.

-
More generally, the total Stiefel-Whitney class of $E_{\mathbb R}$ is the reduction mod $2$ of the total Chern class of $E_{\mathbb C}$. This is Problem 14-B on page 171 of Characteristic Classes, by John Milnor and James Stasheff – Alex Suciu Jan 21 2012 at 18:44
thank you very much for the references! –  math3.14159 Jan 21 2012 at 20:50

Yes, this is true:

See for example Spin geometry -- H. Blaine Lawson, Marie-Louise Michelsohn Remark 2.2/ page 87.

-
 It's what you'd expect if the squaring map on Pic(M) has the image you'd first guess. The connected component being divisible, it's about the discrete part ... – Charles Matthews Jan 21 2012 at 14:28 Oops, you beat me to it! I guess that clarifies "somewhere in chapter 2" in my answer. – Paul Siegel Jan 21 2012 at 14:30