It appears to be a standard fact in topology that $\mathbb{C}\mathbb{P}^2\#-\mathbb{C}\mathbb{P}^2$ has a structure of a $\mathbb{S}^2$ bundle over $\mathbb{S}^2$. Is there a nice geometric description of the projection to the sphere?


This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one. 



Geometric description of a certain sphere bundle
Yes. If $p\in\mathbb{CP}^2$ is a point, you can consider the blowup $X\_p$ of $\mathbb{CP}^2$ at $p$ as the space of pairs $(L,q)$ such that $L\subset\mathbb{CP}^2$ is a line passing through $p$ and $q\in L$ is any point. Now let $M\subset\mathbb{CP}^2$ be any line *not* passing through $p$. Then one can define a map $\pi:X\_p\to M$ by letting $\pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $\mathbb{CP}^2$), this gives a submersion of $X\_p$ onto $M\simeq S^2$ whose fibers are $2$-spheres. In fact, $\pi$ is a holomorphic submersion, as is easy to verify in local coordinates.


