Let $X$ be a Kähler manifold, $B$ a projective manifold, there is a smooth fibration $\pi:X\rightarrow B$, such that all the fibers $F$ of $\pi$ satisfy $H^i(F,\mathcal O)=0,i>0$, then is $X$ a projective manifold?
I know there are some special cases: for example, if all the fibers are projective spaces, Kodaira had proved that the total space must be a projective manifold, but if we relax the condition to $H^i(F,\mathcal O)=0,i>0$, is it still true? This question has already been asked in the comment of a question in MO and Jason Starr commented it is true, but he did not give a proof, and I can't work the proof out by myself, so I ask again here, I need someone to provide more details, thanks!