Corollary 11.28.8`7AE0B102B55A`

Corollary 11.28.8.7AE0B102B55A Let $Y$ be a connected compact complex manifold admitting a finite unramified holomorphic covering $\pi :Z\to Y$ by a complex torus. Every smooth Hermitian tangent metric on $Y$ with nonpositive Chern holomorphic sectional curvature is Kähler and Chern flat.

Proof. The covering is locally biholomorphic, so the pullback metric has the same sectional-curvature sign. By proposition 11.28.7, its associated form is closed and its Chern curvature is zero. Both conclusions descend through the covering. $\square$

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@misc{pome-tag-7AE0B102B55A,
  author = {{Pome contributors}},
  title = {Corollary 11.28.8},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/7AE0B102B55A/}
}

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Corollary 11.28.87AE0B102B55A

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Corollary 11.28.8`7AE0B102B55A`