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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?

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    $\begingroup$ 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). $\endgroup$
    – Robert Bryant
    Commented Nov 23, 2022 at 15:55
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    $\begingroup$ Robert's example is often called a Hopf manifold. $\endgroup$
    – Ben McKay
    Commented Nov 23, 2022 at 21:12

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There are many such examples, for instance, all complex nilmanifolds (and most complex solvmanifolds) have tangent bundle which is topologically trivial. The Hopf manifolds also have topologically trivial tangent bundle, as Robert Bryant noticed. Compact Lie groups with invariant complex structures also have trivial tangent bundle. My favourite example, outside of holomorphically symplectic, is this: https://arxiv.org/abs/0710.4492, manifolds with holomorphic Riemannian form. For such manifolds, obviously, the tangent bundle is holomorphically isomorphic to the cotangent.

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