I’ve been working through Hutchings’ lovely survey paper on symplectic embedding problems and, as the next episode of the symplectic embeddings series, I enclose my notes. To be honest, Hutchings’ exposition is pretty accessible even for someone who doesn’t know a huge amount of symplectic geometry; I mostly just expand on some of the main arguments and computations. That said, I did intend for my notes to be compatible with some of my other posts on symplectic [toric] geometry, as well as carrying especial emphasis on the points that might be particularly enjoyable for an algebraic geometer or algebraic combinatorist. In particular, there are some interesting methods at play in some of the proofs alluding to the sort of ‘functoriality’ of the moment polytope of a toric symplectic manifold that I have been pondering of late.
The general approach taken is that, given this sequence of ECH capacities or necessary conditions for symplectic embeddings to exist, one wants to show that these conditions are also sufficient to guarantee that an embedding exists. The methodology passes through symplectic blowups to convert the problem into finding certain classes in (co)homology, which turns out to have the same numerics as the ECH capacities. Before that one also translates the problem into one of ‘ball packing’: embedding a disjoint union of balls into a single ball that, like many tales in symplectic geometry, has classical origins.