## Lecture notes on ECH 5: holomorphic curves in symplectizations

Sorry, I lied at the end of the last installment of lecture notes when I said that I would give the definition of ECH in the present installment. But I’m going to at least explain some of the ingredients in the definition of ECH. Better not to rush through everything. (By the way, my minicourses are probably not going to cover as much material as announced, for the same reason.)

Contact 3-manifolds and Reeb orbits

Let us begin by reviewing some basic definitions. Let $Y$ be a closed oriented three-manifold. Recall that a contact form on $Y$ is a 1-form $\lambda$ on $Y$ such that $\lambda\wedge d\lambda>0$ everywhere. Let $\xi = Ker(\lambda)$; this is a 2-plane field on $Y$, which is oriented by $d\lambda$. An oriented 2-plane field which can be obtained this way is called a contact structure. Two contact forms determine the same contact structure if and only if one is obtained from the other by multiplication by a positive function on $Y$. The 1-form $\lambda$ also determines the Reeb vector field $R$ characterized by $d\lambda(R,\cdot)=0$ and $\lambda(R)=1$. The Reeb flow preserves the contact form, i.e. ${\mathcal L}_R\lambda = 0$ (one-line exercise).

Reeb orbit is a closed orbit of $R$, i.e. a map $\gamma:{\mathbb R}/T{\mathbb Z}\to Y$ for some $T>0$ such that $\gamma'(t)=R(\gamma(t))$. We consider two Reeb orbits to be equivalent if they differ by translating the parameter. Every Reeb orbit is either embedded, or an $m$-fold cover of an embedded Reeb orbit for some integer $m > 1$. Given $\gamma$ as above, the time $T$ flow of $R$ determines a diffeomorphism $\psi_T:Y\to Y$, and if $y\in Im(\gamma)$, then $d\psi_T:T_yY\to T_yY$ restricts to a symplectic linear map $\xi_y\to\xi_y$. This is called the “linearized return map” of the Reeb flow along $\gamma$, and we denote it by $P_\gamma$. The eigenvalues of $P_\gamma$ do not depend on the choice of $y$. The Reeb orbit latex $\gamma$ is called nondegenerate if $P_\gamma$ does not have 1 as an eigenvalue. One can think of this as meaning that $\gamma$ is cut out transversely in an appropriate sense. If $\gamma$ is nondegenerate, then it is called elliptic, positive hyperbolic, or negative hyperbolic, according as the eigenvalues of $P_\gamma$ are on the unit circle, positive, or negative respectively.  The contact form $\lambda$ is called nondegenerate if all Reeb orbits are nondegenerate. It is a standard fact that generic contact forms are nondegenerate (although the only reference I have heard of for a proof is a paper by Cieliebak and Frauenfelder, I need to check this out). Assume below that the contact form is nondegenerate.

This geometric setup is very similar to the mapping torus setup we considered previously. We have just replaced the mapping torus flow $\partial_t$ with the Reeb vector field $R$, and the vertical tangent bundle of the mapping torus by the contact structure.

For reasons I explained previously, we are led to try to define a chain complex whose generators are unions of Reeb orbits, of which the elliptic ones but not the hyperbolic ones may be multiply covered. The differential should then count certain holomorphic curves in ${\mathbb R}\times Y$. (Embedded contact homology will be the homology of this chain complex.)  Let us now start to discuss the latter.

Holomorphic curves in symplectizations

The symplectization of the contact 3-manifold $(Y,\lambda)$ is the symplectic 4-manifold $({\mathbb R}\times Y,\omega=d(e^s\lambda))$. Here and below, $s$ denotes the ${\mathbb R}$ coordinate on ${\mathbb R}\times Y$.

We call an almost complex structure $J$ on ${\mathbb R}\times Y$ “symplectization admissible” (sorry for the clunky terminology, but we need some term) if it satisfies the following three conditions: First, $J(\partial_s)=R$. Second, $J(\xi)=\xi$, and $J$ rotates $\xi$ positively with respect to the orientation given by $d\lambda$. Third, $J$ is ${\mathbb R}$-invariant. (Symplectization-admissible almost complex structures are quite analogous to the almost complex structures we considered on $S^1$ cross a mapping torus.) Note that a symplectization-admissible $J$ is determined by its restriction to $\xi$, and the space of such $J$ (with e.g. the $C^\infty$ topology) is contractible. Exercise: every symplectization-admissible $J$ is $\omega$-compatible. Fix such a $J$.

Observe that if $\gamma$ is an embedded Reeb orbit, then ${\mathbb R}\times\gamma$ is an embedded $J$-holomorphic cylinder in ${\mathbb R}\times Y$. More generally, we consider $J$-holomorphic curves of the form $u:(\Sigma,j)\to({\mathbb R}\times Y,J)$ where the domain $(\Sigma,j)$ is a punctured compact Riemann surface (i.e. a Riemann surface with finitely points called “punctures” removed). If $\gamma$ is a (possibly multiply covered) Reeb orbit, a positive end of $u$ is a puncture near which $u$ is asymptotic to ${\mathbb R}\times\gamma$ as $s\to\infty$. What we mean by this is that a neighborhood of the puncture can be given coordinates $(\sigma,\tau)\in ({\mathbb R}/T{\mathbb Z})\times[0,\infty)$ with $j(\partial_\sigma)=\partial_\tau$ such that $\lim_{\sigma\to\infty}\pi_{\mathbb R}(u(\sigma,\tau))=\infty$ and $\lim_{\sigma\to\infty}\pi_Y(u(s,\cdot))=\gamma$. A negative end is defined analogously with $\sigma\in(-\infty,0]$ and $s\to -\infty$. We assume that all punctures are positive ends or negative ends as above. We mod out by the usual equivalence relation on holomorphic curves (namely composition with biholomorphic maps between domains).

The dimension formula

Recall that in a closed symplectic four-manifold $(X,\omega)$, if $J$ is $\omega$-compatible and generic, and if $C$ is a somewhere injective $J$-holomorphic curve, then the moduli space of $J$-holomorphic curves is a manifold near $C$ of dimension

$ind(C) = -\chi(C) + 2\langle c_1(TX),[C]\rangle.$

I didn’t explain this before, but when $C$ is immersed this is basically just the Riemann-Roch formula for the normal bundle, and the general case can be deduced from this (or a different argument using the tangent bundle instead of the normal bundle).

We will need an analogue of the above formula for holomorphic curves in symplectizations.  While there is a general version of this for symplectizations of any dimension, we will stick with the four-dimensional formula that we need. Here it is: If $J$ is generic, and if $C$ is a somewhere-injective $J$-holomorphic curve in ${\mathbb R}\times Y$ with positive ends at Reeb orbits $\gamma_1,\ldots,\gamma_m$ and negative ends at Reeb orbits $\delta_1,\ldots,\delta_n$, then the moduli space of $J$-holomorphic curves near $C$ is a manifold of dimension

$ind(C) = -\chi(C) + 2c_1(\xi|_C,\tau) + \sum_i CZ_\tau(\gamma_i)-\sum_j CZ_\tau(\delta_j).$

I need to explain the terms on the right hand side. First, $\tau$ is a trivialization of $\xi$ over the Reeb orbits $\gamma_i$ and $\delta_j$. Second, $c_1(\xi|_C,\tau)$ is the “relative first Chern class” of the complex line bundle $\xi$ over $C$ with respect to the trivialization $\tau$. To define this, one chooses a generic section $\psi$ of $\xi$ over $C$ which on each end is nonvanishing and constant with respect to the trivialization $\tau$. One then defines $c_1(\xi|_C,\tau)$ to be the algebraic count of zeroes of $\psi$. Note that this definition only uses the fact that $C$ is an oriented surface and not that it is $J$-holomorphic.

Exercise: $c_1(\xi|_C,\tau)$ depends only on the Reeb orbits $\gamma_i$ and $\delta_j$, the homotopy class of the trivialization $\tau$, and the relative homology class of $C$ (i.e. it is unchanged if you replace $C$ by $C'$ such that $C-C'$ is nullhomologous in $H_2({\mathbb R}\times Y)$). More generally, if $C'$ is any other oriented surface mapped into ${\mathbb R}\times Y$ with the same boundary conditions, so that $[C-C']\in H_2({\mathbb R}\times Y)$ is defined, then

$c_1(\xi|_C,\tau) - c_1(\xi|_{C'},\tau) = \langle c_1(\xi),[C-C']\rangle$

where on the right hand side, $c_1(\xi)$ denotes the usual first Chern class of the complex line bundle $\xi\to{\mathbb R}\times Y$ in $H^2({\mathbb R}\times Y;{\mathbb Z})$.

Continuing with the explanation of the index formula, if $\gamma$ is a Reeb orbit, and if $\tau$ is a trivialization of $\xi|_\gamma$ (which we can take to be symplectic with respect to the symplectic form $d\lambda$ on $\xi$), then $CZ_\tau(\gamma)\in{\mathbb Z}$ is the Conley-Zehnder index of $\gamma$ with respect to the trivialization $\tau$. To define this, pick a parametrization $\gamma:{\mathbb R}/T{\mathbb Z}\to Y$. Let $\{\psi_t\}_{t\in{\mathbb R}}$ denote the one-parameter group of diffeomorphisms given by the flow of $R$. Then $d\psi_t:T_{\gamma(0)}Y\to T_{\gamma(t)}Y$ induces a symplectic linear map $\phi_t:\xi_{\gamma(0)}\to\xi_{\gamma(t)}$, which using our trivialization $\tau$ we can regard as a $2\times 2$ symplectic matrix. Then $\phi_0=1$, and $\phi_T$ is the linearized return map (in our trivialization), which does not have 1 as an eigenvalue. Then $CZ_\tau(\gamma)\in{\mathbb Z}$ is the Conley-Zehnder index of the family of symplectic matrices $\{\phi_t\}_{t\in[0,T]}$, which roughly speaking is twice the rotation number of this family of matrices. While there is a general definition of the Conley-Zehnder index for families of symplectic matrices in any dimension (starting the identity and ending at a matrix which does not have $1$ as an eigenvalue), we will just give an explicit formula for the case of two-dimensional symmetric matrices which we need. If $\gamma$ is hyperbolic, let $v\in{\mathbb R}^2$ be an eigenvector of $\phi_T$; then the family of vectors $\{\phi_t(v)\}_{t\in[0,T]}$ rotates by angle $\pi k$ for some integer $k$, and this integer $k$ is the Conley-Zehnder index (which is even in the positive hyperbolic case and odd in the negative hyperbolic case). If $\gamma$ is elliptic, then we can change the trivialization so that each $\phi_t$ is rotation by angle $2\pi \theta_t\in{\mathbb R}$ where $\theta_t$ is a continuous function of $t$ and $\theta_0=0$. The number $\theta=\theta_T\in{\mathbb R}\setminus{\mathbb Z}$ is called the “rotation angle” of $\gamma$ with respect to $\tau$, and the Conley-Zehnder index is $2\lfloor\theta\rfloor+1$.

Exercise: the right hand side of the dimension formula does not depend on $\tau$ (even though the individual terms in it do).

The detailed proof of the dimension formula is beyond the scope of this course. There are two things to prove: first that for generic $J$ the moduli space of somewhere-injective $J$-holomorphic curves is a manifold, and second the dimension formula for the moduli space. The former is a standard transversality argument (not to say that this is trivial; it is worked out in a paper by Dragnev). To prove the second part, one considers general operators over punctured Riemann surface of the form that arise as deformations of holomorphic curves and shows that these are Fredholm and proves a general index formula for them. To prove the general index formula, the key is to show that the index is “additive under gluing”. This allows one to reduce to the case of closed Riemann surfaces, for which the general index formula reduces to the Riemann-Roch formula.

As usual, the somewhere injective assumption is necessary; there is no $J$ for which transversality holds for all multiply covered curves, and this causes a lot of headaches.

What’s next

The differential on the ECH chain complex will count certain $J$-holomorphic curves in moduli spaces of dimension $1$. Since $J$ is ${\mathbb R}$-invariant, there is an ${\mathbb R}$-action on the moduli spaces, which we can mod out by to obtain a $0$-dimensional space to count.

But we are not going to count all index 1 curves, but rather certain special ones, which are singled out analogously to the way one singles out certain index 0 curves to count in the definition of Taubes’s Gromov invariant. To explain which curves to single out, the next ingredient we need is a relative version of the adjunction formula.