# Questions tagged [kahler-manifolds]

A complex manifold with a Hermitian metric is called a Kähler manifold if the (1,1) form that gives its Hermitian metric is a closed differential form.

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### All Kahler structures on a compact Riemann surface

Let $X$ be a compact Riemann surface, is it possible to have a description for the space of all Khaler forms on $X$? More concretely, given two Kahler forms $\Omega$ and $\Omega'$ is it known what is ...
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### Question on specific properties of inner product on complex manifolds

I am having trouble understanding part of a proof within Principles of Algebraic Geometry; GRIFFITHS / HARRIS. The proof I am struggling with is located at page 112 under the subitem The Hodge ...
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### A small doubt in Kähler geometry

We often use this lemma in Kähler geometry. For a compact Kähler manifold $(X,\omega)$ and a function $f$ on $X$ we have \begin{align*} \langle\partial\bar\partial f,\omega\rangle=\frac{i}{2}\Delta(f) ...
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### Kähler metrics on holomorphic vector bundles

Let $(X,\omega)$ be a Kähler manifold not necessarily compact of complex dimension $n$. Let $\pi:E\to X$ be a holomorphic vector bundle of rank $r$, then $E$ can be seen as a complex manifold of ...
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### Non-Kähler $\partial\bar{\partial}$-manifold

As we know, a compact Kähler manifold always satisfies the $\partial\bar{\partial}$-lemma (see for example Huybrechts' book 《complex geometry》p128), thus we call it a $\partial\bar{\partial}$-manifold,...