As the famous Kodaira's embedding theorem states:
A compact Kähler manifold $X$ is a projective manifold if and only if there a positive line bundle $L$ of $X$.
So, my question is if $X$ admits a negative line bundle, is $X$ still projective?
Particularly, if the first Chern class $c_1(X)$ is negative, is it still projective? I doubt the answer may be yes because by the proof in Morrow and Kodaira's book, the authors choose a Kähler form $\omega$ equals to a positive element of $c_1(X)\in H^{1,1}(X,\mathbb Z)$, so if $c_1(X)$ is negative, we may choose $\omega=-c_1(X)$ thus get a Hodge metric, so we proved that $X$ with a negative first Chern class is also projective, is that right? Any comments are welcome!