What is the relationship between a complex manifold being Kähler, projective, nonprojective, and nonKähler?

Question

I was wondering if this implication is true. I read a few places that $$\text{nonprojective} \Longrightarrow \text{nonKähler}$$ but I think I maybe have misunderstood. Equivalently, this is of course asking if $$\text{Kähler} \Longrightarrow \text{projective ?}$$

Answer 1

Both of these implications are incorrect.

As $\mathbb{CP}^n$ is Kähler and complex submanifolds of Kähler manifolds are Kähler, all projective manifolds are Kähler. The converse however is not true. That is, there exist Kähler manifolds which are not projective. For example, all two (complex) dimensional tori $\mathbb{C}^2/\Lambda$ are Kähler, but many of them are not projective. In order to be projective, the lattice must satisfy the so-called Riemann conditions (see Griffiths & Harris, Chapter 2, Section 6).

In summary, projective implies Kähler. Taking the contrapositive, we see that non-Kähler implies non-projective.