Geometric description of a certain sphere bundle

It appears to be a standard fact in topology that $mathbbCmathbbP^2#-mathbbCmathbbP^2$ has a structure of a $mathbbS^2$ bundle over $mathbbS^2$. 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.

2 Answers
2

Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ 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 $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):

Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ 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 $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ 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 $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, 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 $S^2n-2$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

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