I am currently reading Dusa McDuff's paper "Blow ups and symplectic embedding in dimension 4" and had a few questions regarding the paper.
In the paper McDuff uses the following notation. $X = \mathbb{C}P^2 \# ~\overline{\mathbb{C}P^2}$,we fix an embedded sphere $S_L$ of $X$ in class $L$ (where $L$ is the homology class of the $\mathbb{C}P^1 \subset \mathbb{C}P^2$) and an embedded sphere $S_E$ in the class $E$ of $X$ (where $E$ is the class of the exceptional divisor in $X$).
In the paper (Theorem 1.6) she wants to prove that given a form $\omega$ that restricts to an area form on $S_L$(with area $\pi$) and has area $\pi \lambda^2$ on the class $E$, there exists a form $\alpha$ on $X$ with the property that $\alpha$ restricts to an area form of area $\pi$ on $S_L$ and an area form on area $\pi \lambda^2$ on $S_E$ and a diffeomorphism $\phi : X \to X$ such that $\phi^*(\alpha) = \omega$.
The sketch of how she does this is as follows. Choose a $\omega$-compatible almost complex structure such that $S_L$ is $J$-holomorphic. Through standard arguments we show that, given any compactible $J$ there is a $J$-curve in the class $L-E$ through each point in $X$ and there exists a curve $\tilde{S_E}$ in class $E$. Pick any $\alpha$ with the property as above, and get $J^\prime$ which is $\alpha$-compatible and such that $S_L$ and $S_E$ are $J^\prime$ holomorphic. Now she asks us to choose the diffeomorphism $\phi$ of $X$ to be the one such that:
- $\phi(S_L) = S_L$
- $\phi(\tilde{S_E}) = {S_E}$
- and $\phi$ takes the $L-E$ foliation due to $J$ to the $L-E$ foliation due to $J^\prime$.
1) So why does such a diffeomorphism exist? Also why does this diffeomorphism satisfy the required condition that $\phi^*(\alpha) = \omega$?
2) In the next part of the theorem she wants to prove that given two symplectic forms $\alpha$ and $\beta$ such that they both restrict to an area form of area $\pi$ on $S_L$ and an area form on area $\pi \lambda^2$ on $S_E$, there exists a diffeomorphism $\Psi$ such that:
- $\Psi(S_L) = S_L$
- $\Psi({S_E}) = {S_E}$
- $\Psi^*(\beta) = \alpha$
Why wouldn't the same proof as above work word for word to prove the second part as well? (In the paper she uses a completely different idea to prove the second part).
Any help would be appreciated. Thanks!
