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Let $(X,\omega)$ be a complex hermitian manifold, and call $\Theta$ its Chern curvature tensor. Out of this we can consider different notions of curvature, namely the holomorphic bisectional curvature …

As Dima said, it is much more: in fact it is projective. But let me give you some more insights on this kind of questions. I shall give you the definition of four different classes of compact comple …

Since Artie Prendergast-Smith is not expanding his comment in an answer, let me do it. As I said in the comments, his comment is essentially THE answer to the OP question. But let me give some more de …

Let me recall you briefly how to obtain such a Kähler metric on the total space of the projectivized bundle. Start with any given hermitian metric $h$ on $E$ and consider on the projectivized bundle …

Could you give me an example of a compact Kähler manifold which analytically deforms to a non Kähler one? For example, there is no hope to find a complex structure on a Hopf manifold in order to make …

It is well known, as well as absolutely intuitive, that the Riemannian holonomy of a generic Riemannian manifold is $O(n)$, the Riemannian holonomy of a generic orientable Riemannian manifold is $SO(n …

On a $n$-dimensional Kähler manifold $(X,\omega)$, the Ricci form is (minus) the curvature of the canonical bundle $K_X$ endowed with the induced metric. Thus, if $X$ has zero Ricci curvature then its …

I don't know if this answers to your question, and very likely you already know it, but one easy fact is the following. First of all an obvious necessary condition is that at the level of cohomology …

Examples of compact Kähler manifolds with non positive holomorphic bisectional curvature are given by: Closed submanifolds of complex tori. Smooth compact quotients of bounded symmetric domains. M …

Maybe you already were aware of that, or maybe it really doesn't answer to your question, but I'll try anyhow... Take a look at this, Subsection 7.5 on page 39. The construction you talk about in you …

Just for fun, here is an answer in the purely algebraic setting. So, suppose that $X$ is irreducible projective algebraic of dimension $n$, $\alpha=c_1(\mathcal O_X(D))$ is the class of a nef diviso …