As is well-known (see [here](https://mathoverflow.net/questions/220502/spin-structures-for-quaternionic-kaehler-and-hyper-kaehler-manifolds) for a M.O. question) all Kahler manifolds are $spin^c$. I would like to ask which are in fact $spin$. 


Taking my motivation from the case of complex projective space, I make the following (naive) conjecture:



> 
> Conjecture: A compact $2n$-dimensional Kahler manifold $M$ is spin, if there exists a line bundle $L$ over $M$ such that 
>  $$
> L \otimes L \simeq \Omega^{(0,n)},
> $$
>  where $\Omega^{(a,b)}$ denotes the space of $(a,b)$-forms.
> 
> 
> 


Can someone tell me if this is true or nor, and if not, what is an easy counter-example. 



Which Kahler Manifolds Are Spin?
By a classical paper of Atiyah (<http://www.maths.ed.ac.uk/~aar/papers/atiyahspin.pdf>)
the spin structures on a **compact** complex manifold $(M^{2n},J)$ are in bijective correspondence with isomorphism classes of holomorphic line bundles $\cal{L}$ such that $\cal{L}\otimes\cal{L}=K$ where ${\cal{K}}=\Lambda^{n}(T^{\*}M^{1, 0})$ is the canonical line bundle. In particular, an almost complex manifold admits a spin structure if and only if $\cal{K}$ admits a square root, i.e. there exists a complex line bundle $L$ such that $L^{\otimes 2}=\cal{K}$.  

Of course not any Kahler manifold is spin, for example there are several flag manifolds which are not spin, since their first Chern class is not even. 


**added** Example: For the coset $M=G/K=SU(n)/S(U(p)\times U(n−p))$ one can show that admits a unique $SU(n)$-invariant spin structure, if and only if $n$ is even.


