Is there an example of a smooth complex projective variety and an $S^{2}$-bundle over it which is not diffeomorphic to a complex projective variety?



$S^{2}$-bundles over complex projective varieties
Yes. Let $E\to T^2$ be the $S^2$-bundle over the torus $T^2$ obtained by gluing the trivial bundle over $[0,1]\times[0,1]$ by the identity map in the first coordinate and a reflection in the second coordinate. The torus admits the structure of a smooth complex projective variety, but $E$ isn't orientable, so cannot be diffeomorphic to a smooth complex variety.


