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.


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One Response to Obstructing symplectic embeddings into CP^2 and S^2xS^2

  1. Ian Agol says:

    Hey, I’m at the University of Montreal too!

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