# 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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### Reference on Complex Manifold with Smooth Boundary

On a smooth $2d$-dimensional real manifold $M$ with $\mathcal{C}^{\infty}$ boundary $\partial M$, the most common model for the boundary are via boundary charts $U$ which are homeomorphic to the upper ...
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### Algebraic (?) proof that Ricci form is closed

Let $(M,\omega, J, g)$ be a Kähler manifold. The Ricci form of $M$ is defined as $\rho(X,Y)={\rm Ric}(JX,Y)$. I wanted to give a possibly coordinate-free proof that ${\rm d}\rho=0$. From the condition ...
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### Definition of being nef and big

On a compact Kähler manifold $M$, given a real $(1,1)$ form $\alpha$ we say that it is nef if it is in the closure of the Kähler cone. Moreover, if $\int_{M} \alpha^n > 0$ we say that it is big. ...
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### Existence of a spin structure in Kaehlerian manifolds

I have a few questions regarding the existence of a spin structure on Kaehlerian and hyperKaehlerian manifolds. I cannot seem to provide a reference for proofs or counterexamples, so references are ...
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### Show Positive-Definiteness of Symplectic Form

Here's a baby example I'm working on: Consider the manifold $\mathbb{R}^2$ and the symplectic form $dp\wedge dx$ where clearly the space is parameterized by $(x,p)$. For fixed $\alpha > 0$ ...
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### component of the covariant derivative is a tensor

Let $M$ be a complex manifold with a Kahler metric $g$, define the covariant derivative of a smooth complex valued $T^{(1,0)}M$ vector field $X = X^i\partial_i$ to be such that its i^{th} component is ...
### Question regarding the Taub-NUT metric on $\mathbb{S}^3 \times \mathbb{R}^+$.
I have a question regarding Claude Lebrun's paper Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat. In the introduction of the paper, he writes that the Taub-NUT metric is given ...