Is the tensor product of a complex line bundle with itself trivial?

Question

Let $\xi$ be a complex line bundle over a manifold $M$. Then $\xi\otimes \xi$ is a trivial complex line bundle.

Is my statement right?

Answer 1

No, this is not true in general. In fact, as $c_1(\xi\otimes\xi) = c_1(\xi) + c_1(\xi)$ and a complex line bundle is trivial if and only if its first Chern class vanishes, $\xi\otimes\xi$ is trivial if and only if $c_1(\xi) \in H^2(M, \mathbb{Z})$ is a two-torsion element.

Maybe you're thinking of the following result: the tensor product of a (real or complex) line bundle $\xi$ and its dual $\xi^*$ is trivial. To see this, note that $\xi^*\otimes\xi = \operatorname{Hom}(\xi, \xi)$ which is a line bundle that admits a nowhere zero section, namely $\operatorname{id}_{\xi}$, so it is trivial. For real line bundles, we have $\xi \cong \xi^*$ so $\xi\otimes\xi$ is trivial, but for complex line bundles $\xi^* \cong \overline{\xi}$ so $\xi\otimes\overline{\xi}$ is trivial.