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Let $X$ be a Kähler manifold. Associated to any hermitian metric $h$ on $X$ is a smooth real $(1,1)$-form $\omega = -\text{Im } h$, called the Kähler form of $h$. One of several equivalent conditions …

You need some more hypotheses for the existence of the $(n,0)$-form, in general it will exist only up to finite torsion. For example, an Enriques surface does not admit a nowhere zero holomorphic $(2, …

Has anyone translated Erich Kähler's "Über eine bemerkenswerte Hermitesche Metrik" into English or French? (Preferably, but I'll take anything.)

The short answer: No. The longer answer: The $(2,2)$-form $\operatorname{Tr}(F_h^2)$ represents the second Chern class of $E$, or $c_2(E)$. In general, the integral of that class over the manifold doe …

Let $X$ be an $n$-dimensional complex manifold, let $\omega$ be a Kahler metric on $X$ and let $R$ be the $(4,0)$ curvature tensor of $\omega$. We can simplify the tensor $R$ in different ways, two of …

I'm looking for an introduction to holomorphic foliations and foliations of complex manifolds. Any little helps, but I'm particularily interested in problems of the type where we have a hermitian man …

This is not true, as we can check by calculating the intersection number $\int_X c_4(\operatorname{End}(T_X)) \cup \omega^{n-4}$ for some easy examples of Kahler-Einstein spaces: This intersection nu …

Let $X$ be a compact complex Kahler manifold with first real Chern class $c_1 = 0$. Consider a family $\pi : \mathcal X \to \Delta$ over the unit disc in $\mathbb C$, where the fibers $X_s$ are compac …

I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau …

(1) Yes. If $M$ is $n$-dimensional and $K$ is the Kahler form of a Kahler metric $g$ (or just the Kahler form of a hermitian metric) we have $$ *_g \frac{K^p}{p!} = \frac{K^{n-p}}{(n-p)!} $$ for any $ …

Let $X$ be a compact Kahler manifold and let $\mathcal M$ denote the space of hermitian metrics on $X$. We'll identify a hermitian metric with a smooth, real and positive $(1,1)$-form $\omega$. Let $\ …

It's possible to approach the question in a slightly different way if $X$ is compact. Donu and Spiro are of course right in that the condition for a smooth closed $(1,1)$-form $\omega$ to be a Kahler …

Let $X$ be a non-compact complex manifold of dimension at least 2 equipped with a Kahler metric $\omega$. Take a smooth positive function $f : X \to \mathbb R$, and define a new hermitian metric on $X …

I thought this had already been answered here on MO but my searches didn't turn anything up. It might be good to have a general purpose answer here somewhere. I'll restrict myself to compact manifold …

Unfortunately this does not work, because the Hodge $*$ operator commutes with the Levi-Civita connection. Indeed, we have $$ \nabla (\langle u,v \rangle dV) = \langle \nabla u, v \rangle dV + (-1) …