I am reading Chapter 3 (Moduli Spaces and Transversality) of "J-holomorphic curves and symplectic topology" by McDuff & Salamon. Fixing $k\in\mathbb N,p>1$ such that $kp>2$, they consider $\mathcal B^{k,p} = W^{k,p}(\Sigma,M)$. I understand the Banach manifold structure on $\mathcal B^{k,p}$ (as explained in Palais's "Foundations of Global Non-linear Analysis").

McDuff & Salamon then define a vector bundle $\pi:\mathcal E^{k-1,p}\to\mathcal B^{k,p}$ by its fibres: for $u\in\mathcal B^{k,p}$, they define $\mathcal E_u^{k-1,p} = W^{k-1,p}(\Sigma,\Lambda^{0,1}T^*\Sigma\otimes_Ju^*TM)$. They also describe trivializations of $\mathcal E^{k-1,p}$, but it is not clear to me their transition functions are smooth. Where can I find a proof of why $\pi:\mathcal E^{k-1,p}\to\mathcal B^{k,p}$ is a smooth Banach vector bundle?

Next, they define the section $\bar\partial_J$ of $\mathcal E^{k-1,p}$ by $u\mapsto \bar\partial_J(u) := \frac12(du+J\circ du\circ j)$. They claim that this is a smooth section. Where can I find a proof of this?

I think these are essential to understand the application of the implicit function and Sard-Smale theorem to construct the moduli space of $J$-holomorphic maps $\Sigma\to M$. (I would of course be grateful if somebody could tell me of a way to avoid these considerations and still construct the moduli space.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.