As is well-known (see here 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.