# 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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### 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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### de Rham closed harmonic form on a Kähler manifold

For a compact Kähler manifold, we say that a form is primitive if it is contaned in the kernel of the dual Lefschetz operator, or the co-Lefschetz operator. For all examples I know, a primitive form $\...

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### K-homology classes of Dirac operators on Hermitian manifolds

Given a compact Hermitian manifold $M$, we have three canonical pseudo-differential operators on the sections of complexified de Rham complex, namely
1) (d + d$^*,\Omega^{*})$
2) ($\partial$ + $\...

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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?
...

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### Geometrical meaning of admissible hermitian metric on a line bundle

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 $X$ ...

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### Hodge dual on hermitian manifold

Recently I have been reading a number of mathematical physics articles in which different definitions are given for the Hodge dual on a hermitian manifold. For the Hodge operator that transforms a (p,...