## Obstructing symplectic embeddings into CP^2 and S^2xS^2

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, $c_k$ denotes the $k^{th}$ ECH capacity.

Theorem. Let $(X,\omega)$ be a disjoint union of finitely many star-shaped domains in ${\mathbb R}^4$. If $(X,\omega)$ can be symplectically embedded into ${\mathbb C}P^2(1)$, then $c_k(X,\omega)\le c_k(B(1))$ for all $k$. Likewise, if $(X,\omega)$ can be symplectically embedded into $S^2(a)\times S^2(b)$, then $c_k(X,\omega)\le c_k(P(a,b))$ for all $k$.

That is, as far as ECH capacities are concerned, embedding into ${\mathbb CP}^2$ is no easier than embedding into a ball; and embedding into $S^2\times S^2$ is no easier than embedding into a polydisk. (One can prove things for more general Liouville domains $(X,\omega)$, for example using the completed ECH capacities introduced here last August, but assuming that $X$ is a union of star-shaped domains makes life simpler.)

First, recall that if $(X,\omega)$ is a strong symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$, then for each real number $L$ and each class $A\in H_2(X,\partial X)$ there is an induced map

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

Here $\rho:H_2(X,\partial X)\to{\mathbb R}$ is a homomorphism which measures the failure of $\omega$ and the contact forms to obey Stokes’s theorem. (I explained this here last May.) This cobordism map commutes with the $U$ maps on the top and bottom and preserves the contact invariant. If $X$ is closed and $c_1(A)+A\cdot A=0$, then $\Phi^L(X,\omega,A)(\emptyset)\in {\mathbb Z}/2$ is the Gromov/Seiberg-Witten invariant $Gr(X,\omega,A)$ mod $2$ for any $L>0$. One can recover Gromov invariants $Gr(X,\omega,A)$ with $c_1(A)+A\cdot A=2k>0$ as follows: suppose $Y$ is a contact type hypersurface splitting $X$ into two unions of components $X_+$ and $X_-$ so that there is a contact form $\lambda$ on $Y$ making $X_+$ into a cobordism from the empty set to $(Y,\lambda)$ and $X_-$ into a cobordism from $(Y,\lambda)$ to the empty set. Suppose for simplicity that $H_2(Y)$ maps to zero in $H_2(X)$. Let $A_\pm$ denote the restriction of $A$ to $H_2(X_\pm,Y)$. Let $U^k$ denote any composition of $k$ of the $U$ maps (possibly repeated) associated to the components of $Y$. Then

$\Phi^{L+\rho(A_+)}(X_-,\omega,A_-)\circ U^k \circ \Phi^L(X_+,\omega,A_+)(\emptyset) = Gr(X,\omega,A) \mod 2$

for any $L>0$. The idea here is that $Gr(X,\omega,A)$ counts holomorphic curves in $X$ in the class $A$ with $k$ point constraints, and to prove the formula we can stretch the neck along $Y$ 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 $(X_-,\omega_-)$ be a disjoint union of star-shaped domains and suppose that $(X_-,\omega_-)$ symplectically embeds into ${\mathbb C}P^2(1)$. Recall that the ECH capacities of the ball $B(1)$ are given by $c_k(B(1))=d$ whenever

$\frac{d^2+d}{2} \le k \le \frac{d^2+3d}{2}.$

So it is enough to show that if $2k=d^2+3d$ then $c_k(X_-,\omega_-)\le d$. For this purpose let $X_+$ denote the complement of the image of $X_-$ in ${\mathbb C}P^2$, and let $A=dH\in H_2({\mathbb C}P^2)$ where $H$ denotes the homology class of a line. Then $c_1(A)+A\cdot A=d^2+3d$ and $Gr({\mathbb C}P^2,A)=1$ mod 2 (by the wall crossing formula for Seiberg-Witten invariants). Also, $\rho(A_+)=d$. So for any $L>0$, the above composition property implies that $\eta = \Phi^L(X_+,A_+)(\emptyset)$ is a class in $ECH^{L+d}(Y,\lambda)$ such that $\Phi^{L+d}(X_-,\omega_-)U^k\eta=1$. Note that $\Phi^{L+d}(X_-,\omega_-)$ sends the contact invariant to $1$ 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 $c_k(X_-,\omega_-)\le L+d$. Since $L>0$ was arbitrary, we are done.

The argument for embeddings into $S^2\times S^2$ is similar. Recall that

$c_k(P(a,b))=\min\{am+bn\mid (m+1)(n+1)\ge k+1\}$

where $m$ and $n$ are nonnegative integers. Now suppose that $(X_-,\omega_-)$ is a disjoint union of star-shaped domains that embeds into $S^2(a)\times S^2(b)$. It is enough to show that if $k,m,n$ are nonnegative integers with $(m+1)(n+1)= k+1$ then $c_k(X_-,\omega_-)\le am+bn$. For this purpose let $A=(m,n)\in H_2(S^2\times S^2)$. Then $c_1(A)+A\cdot A = 2(mn+m+n)=2k$, and $Gr(S^2(a)\times S^2(b),A)=1$ 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.