Lecture notes on ECH 9: Why is the differential well defined?

In defining the ECH chain complex, I skipped over the proofs that the differential $\partial$ is well defined and that $\partial^2=0$ . Let me now give you some idea of what is involved in the former. Below, fix a closed oriented three-manifold $Y$ , a nondegenerate contact form $\lambda$ on $Y$ , and a generic symplectization-admissible almost complex structure $J$ on ${\mathbb R}\times Y$ .

Symplectic action

The first question is: given an ECH generator $\alpha$ , why do only finitely many generators $\beta$ appear in $\partial\alpha$ ? To answer this question, define the symplectic action of a Reeb orbit $\gamma$ by

${\mathcal A}(\gamma) = \int_\gamma\lambda.$

Said differently, if we choose a metric on $Y$ such that the Reeb vector field has length $1$ , then ${\mathcal A}(\gamma)$ is the length of $\gamma$ . Our asumption the the contact form is nondegenerate implies that for any real number $L$ , there are only finitely many Reeb orbits with symplectic action less than $L$ . (Exercise.)

If $\alpha=\{(\alpha_i,m_i)\}$ is an orbit set, define its symplectic action by

${\mathcal A}(\alpha) = \sum_im_i{\mathcal A}(\alpha_i).$

Lemma. Let $\alpha$ and $\beta$ be orbit sets. If there exists $C\in{\mathcal M}(\alpha,\beta)$ , then ${\mathcal A}(\alpha)\ge {\mathcal A}(\beta)$ , with equality if and only if $C$ is a union of covers of trivial cylinders.

Proof. The conditions on $J$ imply that $d\lambda|_C$ is pointwise nonnegative, and equal to zero only where $C$ is singular or tangent to the span of the $\partial_s$ and Reeb directions. Consequently, if $s_-<s_+$ are real numbers such that $C$ is transverse to $\{s_\pm\}\times Y$ , then by Stokes’s theorem,

$\int_{C\cap(\{s_+\}\times Y)}\lambda \ge \int_{C\cap(\{s_-\}\times Y)}\lambda,$

with equality if and only if $C\cap([s_-,s_+])$ is translation-invariant as a current. Taking $s_\pm\to\pm\infty$ compltes the proof.

It follows immediately from this lemma that if $\beta$ appears in $\partial\alpha$ then ${\mathcal A}(\alpha) > {\mathcal A}(\beta)$ , so for a given $\alpha$ there are only finitely many possible $\beta$ .

Gromov compactness with currents

Next, we need to show that if $J$ is generic and if $\alpha,\beta$ are ECH generators, then the set of $J$ -holomorphic curves $C\in {\mathcal M}(\alpha,\beta)$ (where two such curves are considered equivalent if they are the same as currents, possibly after translation in the ${\mathbb R}$ direction) is finite.

To prove this, one might first try to apply Gromov compactness as in the paper by Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder [BEHWZ] on compactness in symplectic field theory. However their result, as well as most versions of Gromov compactness, requires an a priori genus bound on the holomorphic curves. And one of the strange features of ECH is that the definition of the differential does not explicitly specify the genus of the curves to be counted; it just says that they must have ECH index 1. (It turns out that the genus is more or less indirectly determined by the condition $I=1$ and the relative adjunction formula; I will discuss this more later.)

Fortunately, in dimension four, Taubes proved a special version of Gromov compactness using currents which does not assume a genus bound:

Theorem (Taubes) Let $(X,\omega)$ be a compact symplectic four-manifold, possibly with boundary, and let $J$ be an $\omega$ -compatible almost complex structure. Let $\{C_n\}_{n\ge 1}$ be a sequence of $J$ -holomorphic curves (possibly with boundary in $\partial X$ ) such that $\int_{C_n}\omega$ has an $n$ -independent upper bound. Then there is a subsequence which converges as a current and as a point set to a $J$ -holomorphic curve $C\subset X$ (possibly with boundary in $\partial X$ ).

For the full statement and proof, see Proposition 3.3 in Taubes’s paper “The structure of pseudoholomorphic subvarieties for a degenerate almost complex structure and symplectic form on $S^1\times B^3$ “. Here “converges as a current” means that if $\sigma$ is any 2-form then $\lim_{n\to\infty}\int_{C_n}\sigma=\int_C\sigma$ . And again, if a curve is multiply covered, then when we are regarding it as a current, all that we see is the underlying somewhere injective curve and the covering multiplicity. Finally, “convergence as a point set” means convergence with respect to the metric on compact sets defined by

$d(K_1,K_2) = \sup_{x_1\in K_1}\inf_{x_2\in K_2}d(x_1,x_2) + \sup_{x_2\in K_2}\inf_{x_1\in K_1}d(x_2,x_1).$

Compactness for ECH

Returning to our symplectization setup, let $\alpha$ and $\beta$ be orbit sets. Define a broken $J$ -holomorphic current from $\alpha$ to $\beta$ to be a $k$ -tuple of holomorphic curves $(\hat{C}_1,\ldots,\hat{C}_k)$ for some positive integer $k$ such that there exist orbit sets $\alpha=\gamma_0,\gamma_1,\ldots,\gamma_k=\beta_k$ for which $\hat{C}_i\in{\mathcal M}(\gamma_{i-1},\gamma_i)$ and $\hat{C}_i$ is not a union of covers of trivial cylinders. The curves $\hat{C}_i$ are called “levels”. The above is similar to the notion of “holomorphic building” from [BEHWZ], except that the latter allows levels consisting entirely of covers of trivial cylinders as long as they are not ${\mathbb R}$ -invariant (i.e. as long as there exists at least one branch point), while we are imposing an equivalence relation which makes them ${\mathbb R}$ -invariant, and so we ignore/disallow them.

Given $s_0\in{\mathbb R}$ , let $T_{s_0}:{\mathbb R}\times Y\to {\mathbb R}\times Y$ denote the translation $(s,y) \mapsto (s+s_0,y)$ . Let us say that a sequence $\{C_n\}_{n\ge 1}$ of curves in ${\mathcal M}(\alpha,\beta)$ converges to the broken curve $(\hat{C}_1,\ldots,\hat{C}_k)$ if there exist real numbers $\{s_{n,i}\}_{n\ge 1\le i \le k}$ with $s_{n,1}>s_{n,2}>\cdots>s_{n,k}$ for each $n$ such that for each $i\in\{1,\ldots,k\}$ , the sequence $T_{s_{n,i}}(C_n)$ converges as a current, and as a point set on compact sets, to $\hat{C}_i$ .

The version of Gromov compactness that is relevant to ECH is now the following:

Proposition. Fix orbit sets $\alpha$ and $\beta$ and let $\{C_n\}_{n\ge 1}$ be a sequence of $J$ -holomorphic curves in ${\mathcal M}(\alpha,\beta)$ . Then one can pass to a subsequence so that $\{C_n\}$ converges to a broken $J$ -holomorphic curve from $\alpha$ to $\beta$ .

Here is the idea of the proof. First, by applying translations we may assume that for each $n$ , the intersection $C_n\cap(\{0\}\times Y)$ contains a point which is not on any Reeb orbit of action $\le {\mathcal A}(\alpha)$ . Now recall that the symplectic form on ${\mathbb R}\times Y$ is $\omega = d(e^s\lambda)$ . Given real numbers $s_-<s_+$ such that $C_n$ is transverse to $\{s_\pm\}\times Y$ , by using Stokes’s theorem as in the previous lemma, we have an $n$ -independent upper bound

$\int_{C_n\cap ([s_-,s_+]\times Y)}\omega = e^{s_+}\int_{C_n\cap(\{s_+\}\times Y)}\lambda - e^{s_-}\int_{C_n\cap (\{s_-\}\times Y)}\lambda \le e^{s_+}{\mathcal A}(\alpha) - e^{s_-}{\mathcal A}(\beta).$

By exhausting ${\mathbb R}\times Y$ by compact sets of the form $[s_-,s_+]\times{\mathbb R}$ and applying Taubes’s theorem quoted above, we can pass to a subsequence such that $\{C_n\}$ converges as a current and as a point set on compact sets to some holomorphic curve $\hat{C}$ . By applying the same argument to translations of $\hat{C}$ , one can show that $\hat{C}\in{\mathcal M}(\gamma_+,\gamma_-)$ for some orbit sets $\gamma_+,\gamma_-$ with ${\mathcal A}(\alpha) \ge {\mathcal A}(\gamma_+)$ and ${\mathcal A}(\gamma_-)\ge {\mathcal A}(\beta)$ . The assumption that each $C_n\cap(\{0\}\times Y)$ contains a point not on any Reeb orbit of action less than or equal to ${\mathcal A}(\alpha)$ insures that $C_n$ is not a union of covers of trivial cylinders. The curve $\hat{C}$ is one level of the limiting broken holomorphic curve we are seeking.

If $\gamma_+=\alpha$ and $\gamma_-=\beta$ we are done; otherwise we need to repeat the above argument for different translations of $C_n$ to find the rest of the broken curve. A more detailed version of this argument (in the slightly different mapping torus case, and using outdated terminology) is in Lemma 9.8 of my paper “An index inequality for pseudoholomorphic curves in symplectizations” (and versions of this argument for more different contexts can be found in many places, since convergence to broken objects is a standard phenomenon).

Proof that the differential is defined

To prove that the ECH differential is well-defined, we need to show the following:

Lemma. Given ECH generators $\alpha,\beta$ , the set of curves $C\in{\mathcal M}(\alpha,\beta)$ with $I(C)=1$ , modulo translation and regarded as currents, is finite.

To prove this, suppose there is an infinite sequence $\{C_n\}$ of pairwise inequivalent curves in ${\mathcal M}(\alpha,\beta)$ . By the above compactness result, we can pass to a subsequence such that they converge to a broken curve $(\hat{C}_1,\ldots,\hat{C}_k)$ . By the key proposition from installment 7, using genericity of $J$ , we have $I(\hat{C}_i)\ge 0$ for each $i$ . Since the ECH index is additive under gluing (this was an exercise in installment 7), we have $\sum_{i=1}^k I(\hat{C}_i) = 1$ . Hence one of the levels $\hat{C}_i$ has $I=1$ , and all other levels have $I=0$ . But the proposition from installment 7 implies that any $I=0$ curve is a union of covers of trivial cylinders. Hence there is in fact only one level, and we know that $\{C_n\}$ converges, as a current and as a point set on compact sets, to a curve $\hat{C}\in{\mathcal M}(\alpha,\beta)$ .

Now there is a technical point, which is that in the sense of [BEHWZ], the sequence $\{C_n\}$ might still converge to a holomorphic building with some $I=0$ levels, i.e. the Euler characteristic of $C_n$ might be less than that of $\hat{C}$ . This can be ruled out, but we have not yet introduced the necessary machinery. I will do that later. (I am getting ever deeper into explanatory debt…)

So if we accept that the above technical point can be somehow resolved, then $\{C_n\}$ is converges in the usual sense to $C_n$ . (I never told what you what this “usual sense” of convergence is, e.g. when I was talking about moduli spaces of holomorphic curves, but it basically means that the domains converge in the moduli space of punctured Riemann surfaces, and the maps converge with appropriate exponential decay conditions on the ends.) However since $\hat{C}$ is somewhere injective and has Fredholm index 1 (by the proposition from installment 7), it follows that $\hat{C}$ is isolated in the moduli space of $J$ -holomorphic curves modulo translation. This is a contradiction.

Conclusion

That wasn’t so bad. But the proof that $\partial^2=0$ is much more subtle, for reasons that I will explain in the next installment.

Lecture notes on ECH 9: Why is the differential well defined?

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