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.

up vote 14 down vote accepted

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.

I would like to add another geometric description of $\mathbb C\mathbb P^n \#\overline{\mathbb C\mathbb P^n}$ as sphere bundle (same works for $\mathbb C\mathbb P^n \# \mathbb C\mathbb P^n$):

Let $S^1 \to S^{2n+1} \to \mathbb C\mathbb P^n$ be the Hopf fibration. Then $S^1$ acts on $\mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $\mathbb C\mathbb P^{n-1}$ in $\mathbb C\mathbb P^n$. Thus the total space of $E$ is diffeomorphic to $\mathbb C\mathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $\overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $\mathbb C\mathbb P^n\# \overline{\mathbb C\mathbb P^n}$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^{2n-1}\times S^2)/S^1$, where $S^1$ acts on $S^{2n-1}$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $\mathbb C\mathbb P^{n-1}$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

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