TITLE: Embeddings of $S^2$ in $\mathbb{CP}^2$
QUESTION [29 upvotes]: Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line?
Note: I suspect that either it is known that there is such a diffeomorphism, or the problem is open. This is because if there was an embedding for which no such diffemorphism existed, you could use it to produce an exotic 4-sphere. To see this, reverse the orientation on $\mathbb{CP}^2$ then blow down the sphere.
EDIT: for a counter-example, it is tempting to look for the connect-sum of a line and a knotted $S^2$. The problem is to prove that the result cannot be taken to a complex line. For example, the fundamental group of the complement $C$ is no help, since it must be simply connected. This is because the boundary of a small neighbourhood $N$ of the sphere is $S^3$ and so $\mathbb{CP}^2$ is the sum of $N$ and $C$ across $S^3$ and so in particuar $C$ must be simply-connected.

REPLY [4 votes]: Take a self-intersection one $S^2$ in $\mathbb CP^2$ -- it appears to me that several people have made the observation in this thread that the complement is simply-connected.  If you don't see this right away then a nice way to see this is that the unit normal bundle is $S^3$ and the Hopf fibration $S^3 \to S^2$ then gives a relator that kills the $S^2$-linking elements of $\pi_1$ but these generate $\pi_1 (\mathbb CP^2 \setminus S^2)$ (generalized Wirthinger presentation). So Poincare/Alexander duality tells you $\mathbb CP^2 \setminus S^2$ is contractible.  So "blowing-down" on this embedded $S^2$ basically amounts to a twisted sphere construction -- gluing two discs $D^4$ together along their common boundary.   There's the issue of whether or not one of the discs has an exotic smooth structure or not, but that's the only issue. 
So I guess there's a question lurking in this -- is there a "natural" way to unknot a self-intersection one embedded $S^2$ in $\mathbb CP^2$?   If there were, you couldn't produce any interesting $S^4$'s via this construction.
I've seen this kind of "unnatural unknotting" phenomena with other constructions -- deform-spinning / twist-spinning knots is another example.