Let $P^2 \tilde \times \mathbb R^2$ be the $\mathbb Z_2$-quotient of $S^2 \times \mathbb R^2$, where the $\mathbb Z_2$ action on $S^2 \times \mathbb R^2$ is antipodal on $S^2$ and a reflection on $\mathbb R^2$. Similarly, $K^2 \tilde \times \mathbb R^2$ is the $\mathbb Z_2$-quotient of $T^2 \times \mathbb R^2$.

Can we find a Kahler surface $M$ so that $M$ is homeomorphic to $P^2 \tilde \times \mathbb R^2$ or $K^2 \tilde \times \mathbb R^2$?

Let $P^2 \mathrel{\tilde \times} \mathbb{R}^2$ be the $\mathbb{Z}_2$-quotient of $S^2 \times \mathbb{R}^2$, where the $\mathbb{Z}_2$ action on $S^2 \times \mathbb{R}^2$ is antipodal on $S^2$ and a reflection on $\mathbb{R}^2$. Similarly, $K^2 \mathrel{\tilde \times} \mathbb{R}^2$ is the $\mathbb{Z}_2$-quotient of $T^2 \times \mathbb{R}^2$.

Can we find a Kahler surface $M$ homeomorphic to $P^2 \mathrel{\tilde \times} \mathbb{R}^2$ or $K^2 \mathrel{\tilde \times} \mathbb{R}^2$?

Let $P^2 \tilde \times \mathbb R^2$ and $K^2 \tilde \times \mathbb R^2$ be the twisted plane bundle over the real projective plane and the Klein bottle so that their double covers are $S^2 \times \mathbb R^2$ and $T^2 \times \mathbb R^2$, respectively.

Can we find a Kahler surface $M$ so that $M$ is homeomorphic to $P^2 \tilde \times \mathbb R^2$ or $K^2 \tilde \times \mathbb R^2$?

Let $P^2 \tilde \times \mathbb R^2$ be the $\mathbb Z_2$-quotient of $S^2 \times \mathbb R^2$, where the $\mathbb Z_2$ action on $S^2 \times \mathbb R^2$ is antipodal on $S^2$ and a reflection on $\mathbb R^2$. Similarly, $K^2 \tilde \times \mathbb R^2$ is the $\mathbb Z_2$-quotient of $T^2 \times \mathbb R^2$.

Can we find a Kahler surface $M$ so that $M$ is homeomorphic to $P^2 \tilde \times \mathbb R^2$ or $K^2 \tilde \times \mathbb R^2$?