As we know, Kodaira's embedding theorem can be put as:
A compact Kähler manifold $X$ is projective if and only if $\mathcal K_X\cap H^2(X,\mathbb Z)\neq\emptyset$.
Where $\mathcal K_X$ denotes the Kähler cone of $X$.
So if we have $H^{1,1}(X)\cap H^2(X,\mathbb Z)=0$, we can make the conclusion that $X$ is not projective. And my question is:
Is there indeed exist any compact Kähler manifolds which satisfies $H^{1,1}(X)\cap H^2(X,\mathbb Z)=0$? To seek for such examples we should first limit ourselves to non-projective Kähler manifolds, for example, in dimension 2, we have K3 surfaces and complex tori, we know they have $h^{1,1}>0$, and I don't know whether part of them satisfy $H^{1,1}(X)\cap H^2(X,\mathbb Z)=0$. So can anybody provide some examples? Any dimension is ok. Any comments are welcome, thanks!
For some constructions of non-projective Kähler manifolds, check this MO question: https://mathoverflow.net/questions/257147/are-most-k%C3%A4hler-manifolds-non-projective
and the following article: https://encyclopediaofmath.org/wiki/K%C3%A4hler_manifold
I quote the second article: For example, this is the case for the torus $\mathbb{C}^2/\Gamma$, where $\Gamma$ is the lattice spanned by the vectors $(1,0), (0,1), (−\sqrt 2,−\sqrt 3), (−\sqrt 5,−\sqrt 7)$.