Continuing the previous post, here is the promised new definition of ECH capacities. These capacities are denoted by where is a nonnegative integer.
We first define a filtered version of completed ECH: if is a real number, define to be the subcomplex (over ) consisting of formal sums in which every exponent of is at least . Define to be its homology. This is related to the usual filtered ECH as follows: there is a chain map (over ) sending , and this induces a map .
If is a strong symplectic cobordism from to , then completing the cobordism maps as usual defines maps
Also let be the subset of the Novikov ring in which every exponent of is less than or equal to zero.
Now suppose is a strong symplectic filling, by which we mean a strong symplectic cobordism from to the empty set, in which every component of has nonempty boundary. We then define to be the infimum over such that there exists such that
where ranges over all -fold compositions of completed maps associated to the components of . It follows directly from the definition that .
By comparison, the definition of the old ECH capacities , for a strong symplectic filling with exact, is the infimum over such that there exists with . It is not hard to check that the old and new capacities agree for example for an ellipsoid. (I have no reason to believe that they always agree…)
It follows from the composition property for the maps that the new capacities are monotone with respect to symplectic embeddings of strong symplectic fillings.
Now if is an arbitrary symplectic four-manifold, define to be the supremum of where is a symplectic filling which can be symplectically embedded into . This is tautologically monotone with respect to symplectic embeddings. This may seem like a cheap trick for extending the definition of ECH capacities to arbitrary symplectic manifolds. I used this same trick for extending the definition of the old ECH capacities, but had no way of getting an upper bound on it for closed manifolds. But the cool thing is that for the new ECH capacities, we can use the functoriality of completed ECH to compute them for or and so get obstructions to symplectic embeddings into these manifolds.
The key lemma is the following: let be a closed symplectic four-manifold. Suppose . Let denote the largest exponent of that appears in (or infinity if there is no such largest exponent). Then
Proof: Let and suppose this is finite. Suppose is a strong symplectic filling with boundary that symplectically embeds into . Let denote the closure of the complenent of the image of the embedding. Now take the last formula from my previous post and apply it to . We get
(We can put the on the right hand side instead of because the maps are suitably compatible with the maps .) Now by definition, the left hand side is in . If we let
then this is in and satisfies the required property to show that . Since was arbitrary, it follows that . QED
Now let’s consider the example of with the standard symplectic form with , where is the homology class of a line. It follows from wall crossing formulas that is an odd integer whenever . Furthermore . It follows that if then
where is some odd integer. So by the above lemma, we have
On the other hand, the reverse inequality holds by monotonicity, because if , then the ball symplectically embeds into , and . So this shows that when , and it follows easily using monotonicity that the same is true for all .
A similar calculation shows that has the same new ECH capacities as a polydisk. It’s kind of neat that the lower and upper bounds agree, because they are computed by quite different methods (ECH capacities of the ball or a polydisk versus Gromov invariants of or ). On the other hand, at the end of the day, the ECH capacities and the Gromov invariants are counting more or less the same holomorphic curves…
Anyway, sorry if the above is a bit rushed; it is explained in a more slow and boring manner in the paper I am working on. But I wanted to quickly show the basic definition. Thanks to Olguta Buse, Chris Wendl, T-J. Li, and others for their questions and comments which inspired me to write this post and the previous one.