**Question.** *What are examples of compact complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles?*

A family of examples are, of course, holomorphically symplectic manifolds (a.k.a. hyper-Kähler varieties), i.e. when we have a non-degenerate holomorphic closed $2$-form. But note that we do not require $TX \cong T^*X$ to be an isomorphism of holomorphic vector bundles (just *complex* vector bundles) so this is stronger than necessary.

Note that the canonical bundle would be isomorphic to its dual, so a necessary condition is that the first Chern class $c_1(X)$ is trivial, i.e. $X$ is Calabi-Yau.

Are there nice examples which are not holomorphically symplectic?