# Complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles

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?

• Well, it doesn't have to be Kähler. For example, if you consider the compact manifold that you get by removing the origin from $\mathbb{C}^n$ ($n>1$) and dividing by the $\mathbb{Z}$-action of $n\cdot z = 2^n\,z$, that manifold, diffeomorphic to $S^1\times S^{2n-1}$, has $TX\simeq T^*X$ as complex bundles, but it is not holomorphic symplectic (or even Kähler). Nov 23, 2022 at 15:55
• Robert's example is often called a Hopf manifold. Nov 23, 2022 at 21:12