A compact Kähler manifold X with $H^{1,1}(X; \mathbb Z)=0$ cannot be embedded in a projective space?

Question

Why is it impossible to embed a compact Kähler manifold X with $H^{1,1}(X; \mathbb Z)=0$ in a projective space?

I suspect this can be shown using the Lefschetz theorem on (1,1)-classes, but I don't see how.

Answer 1

Supposing that $X$ is compact and that by $H^{1,1}(X,\mathbb Z)=0$ you mean $H^{1,1}(X)=0$: if $X\subset\mathbb CP^N$ and if $\omega\in\Omega^{1,1}(\mathbb CP^N)$ is the (Fubini-Study) symplectic form, then $\omega$ is a symplectic form also on $X$ (it is non-degenerate on $X$ as it is the imaginary of the Fubini-Study hermitian metric), and thus the cohomology class of $\omega$ gives you a non-zero element of $H^{1,1}(X)$.