# Questions tagged [hermitian-manifolds]

A Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each holomorphic tangent space. Hermitian manifolds were introduced by Cartan in 1922.

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### Warped product manifold with real and complex parts

Is possible to define a warped product manifold $M=(N,g_N) \times f(F, g_F)$ where $(N, g_N)$ is a Riemannian manifold with Riemannian metric (i.e., real manifold with real structure) and $(F, g_F)$ ...
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### Confusion about complex differential forms

I follow Kobayashi "Differential Geometry of Complex Vector Bundles", pages 11-12, prop. 4.9. Given a rank-$r$ Hermitian holomorphic vector bundle $(E,h)$ over a complex manifold $M$, there exists a ...
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### Lie super algebra presentation of the Kähler identities

For any Kähler manifold $(M,h)$, with Lefschetz operators $L$ and $\Lambda$, and counting operator $H$, we have the following the well-known Kähler-Hodge identities: \begin{align*} [\partial,L] = 0, ...
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### Where do the (Akizuki)-Nakano Identities First Appear

The answers to this M.O. question give a history of the Kaehler identities. The identities can be extended to the vector bundle-valued setting, and play a central role in the proof of the Kodaira ...
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### When is a compact Lie group endowed with a left-invariant complex structure a Kähler manifold of balanced manifold?

Let $G$ be a compact Lie group having a left-invariant complex structure $J$. Is there a hermitian metric $h$ in $G$, compatible with the complex structure $J$, such that $G$ is a Kähler manifold? ...
Let $(X,\Omega)$ be a complex compact Kahler manifold, where $\Omega$ is the fundamental $(1,1)$-form. Moreover let $L$ be a holomorphic line bundle on $X$. A (smooth) hermitian metric $h$ on $X$ ...