Hello from the CRM in Montreal, where I am at a very nice conference this week. In my talk I attempted (among other things) to present the following theorem. Unfortunately my explanation was not very organized and I said a couple of things slightly wrong. (Sorry audience!) So let me attempt to explain all this more clearly here. Below, denotes the ECH capacity.
Theorem. Let be a disjoint union of finitely many star-shaped domains in . If can be symplectically embedded into , then for all . Likewise, if can be symplectically embedded into , then for all .
That is, as far as ECH capacities are concerned, embedding into is no easier than embedding into a ball; and embedding into is no easier than embedding into a polydisk. (One can prove things for more general Liouville domains , for example using the completed ECH capacities introduced here last August, but assuming that is a union of star-shaped domains makes life simpler.)
First, recall that if is a strong symplectic cobordism from to , then for each real number and each class there is an induced map
Here is a homomorphism which measures the failure of and the contact forms to obey Stokes’s theorem. (I explained this here last May.) This cobordism map commutes with the maps on the top and bottom and preserves the contact invariant. If is closed and , then is the Gromov/Seiberg-Witten invariant mod for any . One can recover Gromov invariants with as follows: suppose is a contact type hypersurface splitting into two unions of components and so that there is a contact form on making into a cobordism from the empty set to and into a cobordism from to the empty set. Suppose for simplicity that maps to zero in . Let denote the restriction of to . Let denote any composition of of the maps (possibly repeated) associated to the components of . Then
for any . The idea here is that counts holomorphic curves in in the class with point constraints, and to prove the formula we can stretch the neck along and put the constraining points into the neck. (The real proof of course uses Seiberg-Witten.)
Now to prove the first part of the theorem, let be a disjoint union of star-shaped domains and suppose that symplectically embeds into . Recall that the ECH capacities of the ball are given by whenever
So it is enough to show that if then . For this purpose let denote the complement of the image of in , and let where denotes the homology class of a line. Then and mod 2 (by the wall crossing formula for Seiberg-Witten invariants). Also, . So for any , the above composition property implies that is a class in such that . Note that sends the contact invariant to and all classes of positive grading to zero. It follows from the previous two sentences, similarly to the proof of Proposition 4.5 in my paper “Quantitative ECH”, that . Since was arbitrary, we are done.
The argument for embeddings into is similar. Recall that
where and are nonnegative integers. Now suppose that is a disjoint union of star-shaped domains that embeds into . It is enough to show that if are nonnegative integers with then . For this purpose let . Then , and mod 2 by the wall crossing formula for Seiberg-Witten invariants. The rest of the argument is the same as before.
I’ll continue my series about rational SFT shortly; I have done some calculations and while there were some initial disappointments, things are now getting very interesting.