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A complex manifold which admits a positive line bundle is automatically Kähler. Furthermore, if the manifold is compact, then it is projective by the Kodaira Embedding Theorem. In particular, not ...

Are there examples of compact complex manifolds $X$ with $K_X$ nef, but $X$ is not Kähler? Perhaps even non-Moishezon examples? Here, nef can be defined as follows: For any $\varepsilon>0$ there is ...

In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...

I'm stuck on the proof that for a (compact) Kähler manifold $X$ (of complex dimension $n$), a spin structure on the tangent bundle $TX$ is equivalent to a line bundle $L$ together with an isomorphism $...

Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...

I try to understand the following version of the Kodaira embedding theorem: Let $X$ be a compact Kähler manifold. A line bundle $L$ is positiv if and only if it is ample. I have a problem with the '...