Projective fibration over a projective manifold Let $X$ be a complex manifold, $B$ be a complex projective manifold, consider a smooth fibration $\pi:X\rightarrow B$ such that all the fibers of $\pi$ are projective manifolds, then is $X$ a projective manifold?
As we know, if $X,Y$ are projective manifolds, then the product $X\times Y$ is also a projective manifold, see for example Griffiths&Harris 《principles of algebraic geometry》 p192. And this example can be treated as a special case of the question stated above, I doubt that the projective fibration over a projective manifold is also a projective manifold, is that right? May someone prove it or construct a counter-example? Any comment is welcome!
Added: in《principles of algebraic geometry》p191-192, the corollary after the Kodaira embedding theorem, Griffiths gave a proof that if $X,Y$ is projective, then $X \times Y$ is also projective. His method can be stated as follows: choose closed, integral, positive (1,1)-form $\omega,\omega^\prime$ of $X,Y$, and $\pi:X\times Y\rightarrow X$, $\pi\prime:X\times Y\rightarrow Y$ are the projection maps, then $\pi^*\omega+\pi^{\prime*}\omega^\prime$ is again closed, integral, and positive of type (1,1), which proved that $X\times Y$ is also projective by Kodaira embedding theorem. I think the same method may work also for the projective fibration case? isn't it?
 A: As alluded to in the comments, the original Hopf surface $H$ provides a counterexample. Recall, $H$ is defined to be the quotient $(\mathbb{C}^2\setminus\{(0,0)\})/\mathbb{Z}$ where the $\mathbb{Z}$ action is generated by the map $(z_1, z_2) \mapsto (2z_1, 2z_2)$. The map $\pi : H \to \mathbb{CP}^1$ given by $[(z_1, z_2)] \mapsto [z_1, z_2]$ is a holomorphic submersion with fibers one-dimensional complex tori, as is explained in my answer to this similar question.
Note that one-dimensional complex tori and $\mathbb{CP}^1$ are algebraic, but $H$ is not even Kahler as it is diffeomorphic to $S^1\times S^3$. Topologically, we have the fiber bundle $S^1\times S^1 \to S^1\times S^3 \to S^2$ obtained by crossing the Hopf fibration with $S^1 \to S^1 \to *$.
One can obtain similar examples in higher dimensions, namely Calabi-Eckmann manifolds which are complex manifolds $X$ diffeomorphic to $S^{2n+1}\times S^{2m+1}$ and admit a holomorphic submersion $X \to \mathbb{CP}^n\times\mathbb{CP}^m$ with fibers one-dimensional complex tori. Topologically, we have $S^1\times S^1 \to S^{2n+1}\times S^{2m+1} \to \mathbb{CP}^n\times\mathbb{CP}^m$, the product of the standard fibre bundles $S^1 \to S^{2n+1} \to \mathbb{CP}^n$ and $S^1 \to S^{2m+1} \to \mathbb{CP}^m$.
