## ECH capacities revisited

Continuing the previous post, here is the promised new definition of ECH capacities. These capacities are denoted by $\overline{c}_k$ where $k$ is a nonnegative integer.

We first define a filtered version of completed ECH: if $L$ is a real number, define $\overline{ECC}^L(Y,\lambda,J)$ to be the subcomplex (over ${\mathbb Z}$) consisting of formal sums in which every exponent of $t$ is at least $-L$. Define $\overline{ECH}^L(Y,\lambda)$ to be its homology. This is related to the usual filtered ECH as follows: there is a chain map (over ${\mathbb Z}$) $ECC^L\to\overline{ECC}^L$ sending $\alpha\mapsto t^{-\mathcal{A}(\alpha)}\alpha$, and this induces a map $ECH^L\to\overline{ECH}^L$.

If $(X,\omega)$ is a strong symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$, then completing the cobordism maps as usual defines maps

$\overline{\Phi}^L(X,\omega):\overline{ECH}^L(Y_+,\lambda_+)\to\overline{ECH}^L(Y_-,\lambda_-).$

Also let $\Lambda^-$ be the subset of the Novikov ring $\Lambda$ in which every exponent of $t$ is less than or equal to zero.

Now suppose $(X,\omega)$ is a strong symplectic filling, by which we mean a strong symplectic cobordism from $(Y,\lambda)$ to the empty set, in which every component of $X$ has nonempty boundary. We then define $\overline{c}_k(X,\omega)$ to be the infimum over $L\in{\mathbb R}$ such that there exists $\eta\in\overline{ECH}^L(Y,\lambda)$ such that

$\overline{\Phi}^L(X,\omega)\overline{U}^k\eta\in\Lambda^-\setminus\{0\},$

where $\overline{U}^k$ ranges over all $k$-fold compositions of completed $U$ maps associated to the components of $Y$. It follows directly from the definition that $\overline{c}_k\le\overline{c}_{k+1}$.

By comparison, the definition of the old ECH capacities $c_k(X,\omega)$, for a strong symplectic filling with $\omega$ exact, is the infimum over $L$ such that there exists $\eta\in ECH^L(Y,\lambda,0)$ with $U^k\eta=[\emptyset]$. 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 $\overline{\Phi}^L$ that the new capacities are monotone with respect to symplectic embeddings of strong symplectic fillings.

Now if $(X,\omega)$ is an arbitrary symplectic four-manifold, define $\overline{c}_k(X,\omega)$ to be the supremum of $\overline{c}_k(X',\omega')$ where $(X',\omega')$ is a symplectic filling which can be symplectically embedded into $(X,\omega)$. 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 ${\mathbb C}P^2$ or $S^2\times S^2$ and so get obstructions to symplectic embeddings into these manifolds.

The key lemma is the following: let $(X,\omega)$ be a closed symplectic four-manifold. Suppose $Gr_k(X,\omega)\neq 0$. Let $g_k(X,\omega)$ denote the largest exponent of $t$ that appears in $Gr_k(X,\omega)$ (or infinity if there is no such largest exponent). Then

$\overline{c}_k(X,\omega) \le g_k(X,\omega).$

Proof: Let $L=g_k(X,\omega)$ and suppose this is finite. Suppose $(X_-,\omega_-)$ is a strong symplectic filling with boundary $(Y,\lambda)$ that symplectically embeds into $(X,\omega)$. Let $(X_+,\omega_+)$ 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 $t^{-L}$. We get

$t^{-L}Gr_k(X,\omega) = \overline{\Phi}^L(X_-,\omega_-)\circ\overline{U}^k\circ\overline{\Phi}^L(X_+,\omega_+)t^{-L}.$

(We can put the $\overline{\Phi}^L$ on the right hand side instead of $\overline{\Phi}$ because the maps $\overline{\Phi}^L$ are suitably compatible with the maps $\overline{\Phi}$.) Now by definition, the left hand side is in $\Lambda^-\setminus\{0\}$. If we let

$\eta=\overline{\Phi}^L(X_+,\omega_+)t^{-L},$

then this is in $\overline{ECH}^L(Y,\lambda)$ and satisfies the required property to show that $\overline{c}_k(X_-,\omega_-)\le L$. Since $(X_-,\omega_-)$ was arbitrary, it follows that $\overline{c}_k(X,\omega)\le L$. QED

Now let’s consider the example of ${\mathbb C}P^2$ with the standard symplectic form $\omega$ with $\langle\omega,A\rangle=1$, where $A$ is the homology class of a line. It follows from wall crossing formulas that $Gr(X,\omega,dA)$ is an odd integer whenever $d\ge 0$. Furthermore $(dA)\cdot (dA) + c_1(dA) = d^2+3d$. It follows that if $k=(d^2+3d)/2$ then

$Gr_k({\mathbb C}P^2,\omega) = n_dt^d$

where $n_d$ is some odd integer. So by the above lemma, we have

$\overline{c}_{(d^2+3d)/2}({\mathbb C}P^2,\omega) \le d.$

On the other hand, the reverse inequality holds by monotonicity, because if $a<1$, then the ball $B(a)$ symplectically embeds into ${\mathbb C}P^2$, and $\overline{c}_k(B(a))=da$. So this shows that $\overline{c}_k({\mathbb C}P^2,\omega)=\overline{c}_k(B(1))$ when $k=(d^2+3d)/2$, and it follows easily using monotonicity that the same is true for all $k$.

A similar calculation shows that $S^2\times S^2$ 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 ${\mathbb C}P^2$ or $S^2\times S^2$). 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.