## Lecture notes on ECH 2: Taubes’s SW=Gr theorem

In the previous installment of these notes, we briefly reviewed the definition of Seiberg-Witten invariants of four-manifolds. This definition is not very easy to understand. However I will now review Taubes’s SW=Gr theorem, which shows that for a symplectic four-manifold, the Seiberg-Witten invariant agrees with Taubes’s “Gromov invariant”, which is a certain count of holomorphic curves.

Holomorphic curves

Before stating the theorem, we now briefly review what we will need to know about holomorphic curves.

Let $(X^{2n},\omega)$ be a closed symplectic manifold.  Let $J$ be an $\omega$-compatible almost complex structure: this means that $J:TX\to TX$ is a bundle map such that $J^2=-1$ and $g(v,w) = \langle Jv,w\rangle$ defines a Riemannian metric on $X$. Given $\omega$, the space of compatible almost complex structures $J$ is contractible; see e.g. McDuff-Salamon, “Introduction to symplectic topology”. (Taubes’s theorem would presumably work with tame almost compatible structures instead of compatible ones, but I don’t know if anyone has checked this.)

A $J$-holomorphic curve in $(X,\omega)$ is a holomorphic map $u:(\Sigma,j)\to(X,J)$ where $(\Sigma,j)$ is a compact Riemann surface (i.e. $\Sigma$ is a compact surface and $j$ is an almost complex structure on $\Sigma$), $u:\Sigma\to X$ is a smooth map, and $J\circ du=du\circ j$.  The curve $u$ is considered equivalent to $u':(\Sigma',j')\to (X,J)$ if there exists a holomorphic bijection $\phi:(\Sigma,j)\to(\Sigma',j')$  such that $u'\circ \phi = u$. So a “$J$-holomorphic curve” is formally an equivalence class of triples $(\Sigma,j,u)$ satisfying the above conditions.

Note that if $u:(\Sigma,j)\to(X,J)$ is an embedding, then the equivalence class of the $J$-holomorphic curve $u$ is determined by its image $C=u(\Sigma)$ in $X$. Indeed, an embedded $J$-holomorphic curve is the same thing as a closed two-dimensional submanifold $C\subset X$ such that $J(TC)=TC$.

More generally, a holomorphic curve $u:\Sigma\to X$ is called somewhere injective if there exists $z\in\Sigma$ such that $u^{-1}(u(z))=\{z\}$ and $du_z:T_z\Sigma\to T_{u(z)}X$ is injective. One can show that in this case $u$ is an embedding on the complement of a countable (finite in the case of interest where $n=2$) subset of $\Sigma$, and the equivalence class of $u$ is still determined by its image in $X$.  Detailed proofs of this fact and the facts that I am about to state can be found in McDuff-Salamon, “J-holomorphic curves and symplectic topology”. On the other hand, $u$ is called multiply covered if there exists a branched cover $\phi:(\Sigma,j)\to(\Sigma',j')$ of degree $d>1$ and a holomorphic map $u':(\Sigma',j')\to(X,J)$ such that $u=u'\circ \phi$. Another basic fact is that every holomorphic curve is either somewhere injective or multiply covered. In particular, every holomorphic curve is the composition of a somewhere injective holomorphic curve with a branched cover of degree $d\ge 1$. When $d>1$, the holomorphic curve is not determined just by its image in $X$; it depends also on $d$, the images of the branch points in $X$, and the combinatorics of the branched cover.

A transversality argument shows that if $J$ is generic, then the space of somewhere injective $J$-holomorphic curves with connected domain of genus $g$ representing the homology class $A\in H_2(X)$ is a smooth manifold of dimension

$dim = (n-3)(2-2g) + 2\langle c_1(TX),A\rangle.$

Here $c_1(TX)$ denotes the first Chern class of $TX$, regarded as a complex vector bundle using the almost complex structure $J$. (This complex vector bundle depends only on the symplectic structure and not on the compatible almost complex structure.) Unfortunately, this transversality usually does not work for multiply covered curves. Buckets of ink have been spilled, and forests have been leveled, to deal with this problem; and it also causes some trouble for ECH as we will see later.

Special properties in four dimensions

In four dimensional symplectic manifolds, holomorphic curves have two (related) special properties which will play an important role in our story.

The first special property is intersection positivity. One way to state it is as follows: Let $C_1$ and $C_2$ be distinct irreducible (i.e. their domains are connected) somewhere injective $J$-holomorphic curves in $(X^4,\omega)$. Then $C_1$ and $C_2$ have only finitely many intersection points, and for each intersection point $p$, the intersection multiplicity is positive, and equals $1$ if and only if $C_1$ and $C_2$ are embedded near $p$ and intersect transversely at $p$. (It is easy to see that if $C_1$ and $C_2$ are embedded near $p$ and intersect transversely at $p$, so that the intersection multiplicity is $\pm1$ then the intersection multiplicity is in fact $+1$, basically because the complex  vector space $T_pX$ has a canonical orientation. The rest of the theorem is a lot harder.) In particular, the algebraic intersection number $[C_1]\cdot[C_2]\ge 0$, with equality if and only if $C_1$ and $C_2$ are disjoint. Note that a holomorphic curve $C$ can have $[C]\cdot [C]<0$, for example the exceptional divisor in a blowup is a holomorphic sphere $C$ of self-intersection $-1$. What intersection positivity implies here is that $C$ is the unique holomorphic curve in its homology class.

The second special property of holomorphic curves in four dimensions is the adjunction formula. One way to say it is that if $C$ is a somewhere injective $J$-holomorphic curve in $(X^4,\omega)$, then

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

Here $\chi(C)$ denotes the Euler characteristic of the domain of $C$. In addition, $\delta(C)$ is a count of the singularities of $C$ (i.e. points in $X$ where $C$ is not locally embedded) with positive integer weights. A singularity $p$ has weight $1$ if and only if $p$ is a node, meaning a transverse self-intersection of $C$. In particular, $\delta(C)\ge 0$, with equality if and only if $C$ is embedded. It is an exercise to prove the adjunction formula in the special case when the only singularities of $C$ are nodes.

Taubes’s Gromov invariant

We now have enough background in place to define Taubes’s Gromov invariant. While the definition is still a bit complicated, bear with me because soon we will be able to compute examples.

Let $(X^4,\omega)$ be a closed connected symplectic four-manifold, and let $A\in H_2(X)$. We define the Gromov invariant $Gr(X,\omega,A)\in{\mathbb Z}$ as follows.

To start, define an integer

$I(A) = \langle c_1(TX),A\rangle + A\cdot A.$

In fact one can show that $I(A)$ is always even (maybe not so easy exercise). The integer $I(A)$  is the closed four-manifold version of the ECH index, a crucial notion which we will introduce later. For now, the significance of this integer $I(A)$ is the following.

Choose a generic $\omega$-compatible almost complex structure $J$ so that all somewhere injective $J$-holomorphic curves are cut out transversely as above. If $C$ is a somewhere injective $J$-holomorphic curve, define the “index” of $C$ by

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

By the general dimension formula which we stated above, $ind(C)$ is the dimension of the component of the moduli space of somewhere injective holomorphic curves containing $C$. Now by the adjunction formula, we have

$ind(C) = I([C]) - 2\delta(C).$

That is, if we fix the homology class $[C]$, then the maximum possible value of $ind(C)$ is $I([C])$, which is attained exactly when $C$ is embedded.

Now the idea of the definition of the Gromov invariant is that we will count holomorphic curves in moduli spaces for which $ind(C)$ is maximized. But we also allow disconnected curves.

The precise definition is as follows. If $I(A)<0$, then we define $Gr(X,\omega,A)=0$. If $I(A)\ge 0$, choose $I(A)/2$ generic points $x_1,\ldots,x_{I(A)/2}\in X$. Then $Gr(X,A)$ is a count of holomorphic curves $C$ representing the homology class $A$ that pass through all of the points $x_1,\ldots,x_{I(A)/2}$. The curves $C$ are allowed to be disconnected and/or multiply covered; however they are not allowed to contain multiple covers of spheres of negative self-intersection. An important aspect of Taubes’s approach is that multiply covered holomorphic curves are regarded as currents; that is, we only keep track of the image in $X$ and the covering multiplicity of each component, and aside from that we do not care about the details of the covering map. So what we are counting more precisely are finite sets $C= \{(C_i,d_i)\}$ where the $C_i$‘s are distinct somewhere injective irreducible holomorphic curves, the $d_i$‘s are positive integers, $\sum_id_i[C_i]=A$, and $d_i=1$ whenever $C_i$ is a sphere and $[C_i]\cdot[C_i]<0$. Each such curve $C$ is counted with a certain weight $w(C)\in{\mathbb Z}$ which I will define below. For simplicity let us restrict attention to the case $I(A)=0$.

First let us analyze what one of these curves $C=\{(C_i,d_i)\}$ can look like. It follows directly from the definition of $I$ that if $B_1,B_2\in H_2(X)$ then $I(B_1+B_2)=I(B_1)+I(B_2)+2B_1\cdot B_2$. Applying this to $A=\sum_id_i[C_i]$ gives

$I(A) = \sum_id_iI([C_i]) + \sum_i(d_i^2-d_i)[C_i]\cdot[C_i] + \sum_{i\neq j}[C_i]\cdot [C_j].$

Now the terms on the right hand side are all nonnegative. In particular, we must have $ind(C_i)\ge 0$ (or else transversality would be violated), so $I(C_i)\ge 0$, with equality only if $C_i$ is embedded. In addition, if we combine the inequality $ind(C_i)\ge 0$ with the adjunction formula for $C_i$, we find that

$\chi(C_i) + 2[C_i]\cdot[C_i]\ge 0$

with equality only if $C_i$ is embedded. In particular, the only way that $[C_i]\cdot[C_i]$ can be negative is if $C_i$ is an embedded sphere of self-intersection number $-1$; and in this case we have required that $d_i=1$. Finally, we know by intersection positivity that $[C_i]\cdot[C_j]\ge 0$ with equality if and only if $C_i$ and $C_j$ are disjoint. Since $I(A)=0$, we conclude that the curves $C_i$ are embedded and disjoint, $ind(C_i)=I(C_i)=0$, and $d_i>1$ only if $C_i$ is a torus of self-intersection number zero. (A priori $[C_i]\cdot[C_i]=0$ is possible if $C_i$ is a sphere, but this would require $I(C_i)= 2$ and so cannot happen here.)

Now here is the definition of the weight $w(C)$ when $I(A)=0$. First, $w(C)$ is the product over $i$ of a weight $w(C_i,m_i)\in{\mathbb Z}$. If $m_i=1$, then $w(C_i,m_i)\in\{\pm1\}$ is the sign of the determinant of the deformation operator of $C_i$, which we denote by $\epsilon(C_i)\in\{\pm1\}$. We now outline what this means (full details would require a lot of analysis).

Let $C$ be an embedded holomorphic curve in $(X^4,\omega)$. Consider differentiating the equation for $C$ to be holomorphic under a deformation of $C$. The derivative of a deformation of $C$ can be regarded as a section of the normal bundle $N_{C}$ of $C$ (there is no problem with this point of view since $C$ is embedded here), and the derivative of the holomorphic curve equation defines a real linear operator

$D_{C}: L^2_1(N_{C})\to L^2(T^{0,1}C\otimes N_{C}).$

This is what we call the “deformation operator” of $C$. This operator is Fredholm and has index $ind(C)$. We can write this operator more explicitly, but that is not necessary for the present discussion.

Now suppose that $ind(C)=0$ and that transversality holds so that $D_C$ is invertible.  What do we mean by the “sign of the determinant” of $D_{C}$? The answer is that this is the sign of the spectral flow from $D_{C}$ to a complex linear operator. What that means is the following: one can show that there exists a suitably smooth 1-parameter family of self-adjoint operators $\{D_t\}_{t\in[0,1]}$ such that $D_0=D_{C}$, $D_1$ is complex linear, and there are only finitely many $t$ such that $D_t$ is not invertible. Moreover for each such $t$, the operator $D_t$ has one-dimensional kernel, and the derivative of $D_t$ defines an isomorphism from the kernel of $D_t$ to the cokernel of $D_t$. Then $\epsilon(D_C)$ is simply $-1$ to the number of such $t$. One can show that this is well-defined. We will compute an example or two later.

That defines the weight $w(C_i,m_i)$ when $m_i=1$. It remains to define the weights $w(C_i,m_i)$ when $m_i>1$ and $C_i$ is a torus of self-intersection zero. In this case there are three connected unbranched double covers of $C_i$, classified by nonzero elements of $H^1(C_i;{\mathbb Z}/2)$ and we can and do choose $J$ sufficiently generic so that these are also cut out transversely. Each of these has a sign above. The weight $w(C_i,m_i)$ depends only on $m_i$, the sign of $C_i$, and the number of double covers with each sign. We denote this number by $f_{\pm,k}(m_i)$, where $\pm$ indicates the sign of $C_i$, and $k\in\{0,1,2,3\}$ is the number of double covers whose sign disagrees with that of $C_i$. The numbers $f_{\pm,k}(m)$ are defined as follows. Combine these into a generating function $f_{\pm,k}=1+\sum_{m\ge 1}f_{\pm,k}(m)t^m$. Then

$f_{+,0} = \frac{1}{1-t},$

$f_{+,1}=1+t,$

$f_{+,2} = \frac{1+t}{1+t^2},$

$f_{+,3} = \frac{(1+t)(1-t^2)}{1+t^2},$

$f_{-,k}=\frac{1}{f_{+,k}}.$

That completes the definition of $Gr(X,\omega,A)$ when $I(A)=0$, and the general case is a slight extension of this. In case the above generating functions look strange, note that if you want to get an invariant of $(X,\omega)$, then the generating functions $f_{\pm,k}$ must satisfy certain relations because of “torus doubling” bifurcations that can happen as one deforms $J$ (we will see some examples of this). These relations do not completely determine the $f_{\pm,k}$, and the above choice is needed to agree with Seiberg-Witten theory. Anyway Taubes’s paper “Counting pseudoholomorphic submanifolds in dimension four” gives a detailed proof that $Gr(X,\omega,A)$ does not depend on $J$ and is invariant under deformation of $\omega$ through symplectic forms. Ionel and Parker have given an alternate proof of this.

Taubes’s SW=Gr theorem

For our closed connected symplectic four-manifold $(X,\omega)$, we now state Taubes’s theorem relating the Gromov invariants $Gr(X,\omega,A)$ to the Seiberg-Witten invariants of $X$.

We first have to clarify the relation between the parametrization of the invariants. Recall that the Seiberg-Witten invariants depend on a choice of spin-c structure on $X$, and the set of spin-c structures on a general closed oriented four-manifold $X$ is an affine space over $H^2(X;{\mathbb Z})=H_2(X)$. So there is a bijection between the set of spin-c structures and $H_2(X)$, but this bijection is not canonical. However for a symplectic four-manifold $(X,\omega)$, the symplectic structure determines a distinguished bijection between the set of spin-c structures and $H_2(X)$. To describe this, let a spin-c structure be given. Choose an $\omega$-compatible almost complex structure $J$. This defines a Riemannian metric as explained above, and for this metric, $\omega$ is self-dual and $|\omega|=\sqrt{2}$ (exercise). It then follows that Clifford multiplication $cl(\omega): S_+\to S_+$ has eigenvalues $\pm 2i$. Define $E$ to be the $-2i$ eigenspace; this is a complex line bundle on $X$, and as such it is classified by $c_1(E) \in H^2(X;{\mathbb Z})=H_2(X)$. So the bijection from the set of spin-c structures to $H_2(X)$ sends a spin-c structure to the Poincare dual of $c_1(E)$. One can also define the bijection in the other direction by starting with a complex line bundle $E$ and defining a spin-c structure by fixing a compatible almost complex structure and setting $S_+ = T^{0,even}\otimes E$ and $S_-=T^{0,odd}\otimes E$ with Clifford multiplication given by a certain combination of interior and exterior products.  In particular, $S_+ = E \oplus K^{-1}E$ where $K^{-1}E$ is the $+2i$ eigenspace of Clifford multiplication by $\omega$.  Anyway, let us denote the distinguished spin-c structure corresponding to $o\in H_2(X)$ by ${\mathfrak s}_\omega$.

Taubes’s SW=Gr theorem now says the following: If $(X,\omega)$ is a closed connected symplectic four-manifold, then $X$ has a canonical homology orientation (recall that this fixes the signs in the Seiberg-Witten invariants) such that for every $A\in H_2(X)$,

$SW(X,{\mathfrak s}_\omega + A) = Gr(X,\omega,A).$

That is, the Seiberg-Witten invariant for the spin-c structure ${\mathfrak s}_\omega+A$ is a certain count of holomorphic curves in the class $A$. One can now use this to start computing examples and obtaining applications. For example $SW(X,{\mathfrak s}_\omega)=1$ (since there is exactly one holomorphic curve in the class $0\in H_2(X)$, namely the empty set, as every nonempty holomorphic curve has positive symplectic area). That fact can be proved more easily (see for example “An introduction to the Seiberg-Witten equations on symplectic manifolds”, which are my notes on Taubes’s lectures), but we will see more nontrivial examples shortly.

The detailed proof of SW=Gr takes about 400 pages, so any attempt to describe the proof will necessarily be very superficial. If we accept that, then here is a one-paragraph summary: Remember that the Seiberg-Witten equations have a parameter which is a self-dual 2-form $\eta$. Taubes simply takes $\eta = -r\omega$ where $r$ is a large real number. For a fixed spin-c structure, if $r$ is large, then a Seiberg-Witten solution has the property that the zero set of the $E$ component of the spinor is close (as a current) to a holomorphic curve in the class $A$. This is how one goes from Seiberg-Witten solutions to holomorphic curves, and one goes in the other direction by starting with a holomorphic curve and gluing transverse solutions to the vortex equations (a two-dimensional analogue of the Seiberg-Witten equations).

Anyway, next time we will compute an important family of examples of Gromov invariants, which will tell us a lot about how ECH should be defined. Meanwhile, it seems that I am writing a lot more here than I can possibly say in my minicourses (especially since in those courses I want to quickly get to the definition of ECH), so please regard my postings here as optional supplemental reading.