Pseudoholomorphic curves

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1 Introduction

In this article, $(X,J)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(X,J)$ will denote an almost complex manifold of dimension $2n$$2n$.

Definition 1.1 ($J$$J$-holomorphic curve). Let $(\Sigma,j)$$(\Sigma,j)$ be a Riemann surface with complex structure $j$$j$. A $(j,J)$$(j,J)$-holomorphic curve

$\displaystyle u\co\Sigma\rightarrow X$ $$\displaystyle u\co\Sigma\rightarrow X$$

is a smooth map satisfying

$\displaystyle du\circ j=J\circ du$ $$\displaystyle du\circ j=J\circ du$$

or equivalently

$\displaystyle \bar{\partial}_Ju\co =\frac{1}{2}(du+J\circ du\circ j)=0$ $$\displaystyle \bar{\partial}_Ju\co =\frac{1}{2}(du+J\circ du\circ j)=0$$

A $J$$J$-holomorphic map is called simple if it cannot be factored as $u=v\circ\phi$$u=v\circ\phi$ where $\phi$$\phi$ is a $(j',j)$$(j',j)$-holomorphic branched cover $\phi\co\Sigma'\rightarrow\Sigma$$\phi\co\Sigma'\rightarrow\Sigma$ of degree strictly greater than 1. We will usually omit $j$$j$ from the notation and speak of $J$$J$-holomorphic curves. The term pseudoholomorphic curve will be used to describe a $J$$J$-holomorphic curve when we do not want to specify $J$$J$.

Pseudoholomorphic curves provide a useful tool for studying symplectic manifolds.

2 Taming J

Pseudoholomorphic curves in a general almost complex manifold can be quite wild (EXAMPLE?). Better behaviour can be ensured by the existence of a symplectic $\omega$$\omega$ taming $J$$J$, that is the quadratic form

$\displaystyle g(v,w)=\omega(v,Jw)$ $$\displaystyle g(v,w)=\omega(v,Jw)$$

is positive-definite. This gives us topological control on the energy of a $J$$J$-holomorphic curve

$\displaystyle E(u)=\int_{\Sigma}|du|^2d\mathrm{vol}$ $$\displaystyle E(u)=\int_{\Sigma}|du|^2d\mathrm{vol}$$

(here the norm and volume form are taken with respect to the metric $g$$g$) thanks to the identity

Lemma 2.1 (Energy identity). If $u$$u$ is a $J$$J$-holomorphic curve and $J$$J$ is tamed by $\omega$$\omega$ then

$\displaystyle E(u)=\int_{\Sigma}u^*\omega$ $$\displaystyle E(u)=\int_{\Sigma}u^*\omega$$

If moreover we require the metric $g$$g$ to be $J$$J$-invariant ($g(Jv,Jw)=g(v,w)$$g(Jv,Jw)=g(v,w)$) then we say that $J$$J$ is $\omega$$\omega$-compatible and we have the identity

$\displaystyle E(u)=\int_{\Sigma}|\partial_Ju|^2d\mathrm{vol}+\int_{\Sigma}u^*\omega$ $$\displaystyle E(u)=\int_{\Sigma}|\partial_Ju|^2d\mathrm{vol}+\int_{\Sigma}u^*\omega$$

so that the $J$$J$-holomorphic curves are the absolute minima of the energy functional on the space of maps $\Sigma\rightarrow X$$\Sigma\rightarrow X$.

3 Linearisation

One can think of $\bar{\partial}_J$$\bar{\partial}_J$ as a section of a Banach bundle. Explicitly, let $\mathcal{B}$$\mathcal{B}$ denote the $W^{1,p}$$W^{1,p}$-completion of the space of maps $\Sigma\to X$$\Sigma\to X$ and let $\mathcal{J}$$\mathcal{J}$ denote the space of $\omega$$\omega$-compatible almost complex structures. Define the Banach bundle $\mathcal{E}$$\mathcal{E}$ over $\mathcal{B}\times\mathcal{J}$$\mathcal{B}\times\mathcal{J}$ whose fibre over $(u,J)$$(u,J)$ is the $L^p$$L^p$-completion of the space

$\displaystyle \Omega^{0,1}(\Sigma,u^*TX)$ $$\displaystyle \Omega^{0,1}(\Sigma,u^*TX)$$

of $(0,1)$$(0,1)$-forms on $\Sigma$$\Sigma$ with values in $u^*TX$$u^*TX$. Here $(0,1)$$(0,1)$ refers to the complex structures $j$$j$ on $T\Sigma$$T\Sigma$ and $J$$J$ on $u^*TX$$u^*TX$. By definition, $\bar{\partial}\colon(u,J)\to\bar{\partial}_J(u)$$\bar{\partial}\colon(u,J)\to\bar{\partial}_J(u)$ is a section of this bundle over the smooth locus and it extends naturally to the Sobolev completions.

Definition 3.1 Linearised Cauchy-Riemann operator. The linearisation of $\bar{\partial}$$\bar{\partial}$ at a $J$$J$-holomorphic curve $u$$u$ is the operator

$\displaystyle D\bar{\partial}\colon W^{1,p}(\Sigma,u^*TX)\to L^p\Omega^{0,1}(\Sigma,u^*TX)$ $$\displaystyle D\bar{\partial}\colon W^{1,p}(\Sigma,u^*TX)\to L^p\Omega^{0,1}(\Sigma,u^*TX)$$

given (in local complex coordinates $a+ib$$a+ib$ on $\Sigma$$\Sigma$) by

$\displaystyle D\bar{\partial}(\xi)(\partial_a)=\partial_a+J\partial_b+\frac{\partial J}{\partial\xi}du(\partial_a)$ $$\displaystyle D\bar{\partial}(\xi)(\partial_a)=\partial_a+J\partial_b+\frac{\partial J}{\partial\xi}du(\partial_a)$$

Here we think of $J$$J$ as a section of $\mathrm{End}(TX)$$\mathrm{End}(TX)$ and $dJ$$dJ$ as a map $TX\to T\mathrm{End}(TX)$$TX\to T\mathrm{End}(TX)$ so $\frac{\partial J}{\partial\xi}$$\frac{\partial J}{\partial\xi}$ means the pushforward of $\xi$$\xi$ along $dJ$$dJ$. The linearisation measures to first order the change in $\bar{\partial}_J(u)$$\bar{\partial}_J(u)$ when $u$$u$ is deformed along a vector field $\xi$$\xi$. One can also allow $j$$j$ to vary in the space of complex structures on $\Sigma$$\Sigma$ or $J$$J$ to vary in a family of almost complex structures by adding corresponding terms to the linearisation.

Theorem 3.2 Ellipticity. The linearised Cauchy-Riemann operator of a holomorphic curve is a Fredholm operator.

In the case when $J$$J$ is an integrable complex structure the kernel and cokernel of the linearised Cauchy-Riemann operator agree with the usual Dolbeault cohomology groups.

4 Moduli spaces

Simple $J$$J$-holomorphic curves form nice moduli spaces

Theorem 4.1 Transversality for simple curves. Fix a homology class $A\in H_2(X;\mathbb{Z})$$A\in H_2(X;\mathbb{Z})$ and a Riemann surface $(\Sigma,j)$$(\Sigma,j)$. There is a subset $\mathcal{J}_{\mathrm{reg}}\subset\mathcal{J}$$\mathcal{J}_{\mathrm{reg}}\subset\mathcal{J}$ of the second-category such that the space

$\displaystyle \mathcal{M}^*(\Sigma,X;A,J)\co = \{u\co\Sigma\rightarrow X|du\circ j=J\circ du\}$ $$\displaystyle \mathcal{M}^*(\Sigma,X;A,J)\co = \{u\co\Sigma\rightarrow X|du\circ j=J\circ du\}$$

is a finite-dimensional manifold of dimension

$\displaystyle n(2-2g)+2c_1(X,J)[A]$ $$\displaystyle n(2-2g)+2c_1(X,J)[A]$$

If one allows $j$$j$ to vary then this dimension formula gains an extra $+2$$+2$ if $g=1$$g=1$ or $+(6g-6)$$+(6g-6)$ if $g>1$$g>1$.

These almost complex structures are the regular almost complex structures, for which the linearised problem has vanishing cokernel. Transversality is achieved by making small perturbations of the almost complex structure in regions through which the pseudoholomorphic curve has to pass.

Usually one defines a moduli space by dividing out the space of parametrised maps by the group of holomorphic reparametrisations. This group is:

• $\mathbf{P}\mathrm{SL}(2,\mathbf{C})$$\mathbf{P}\mathrm{SL}(2,\mathbf{C})$, the 6-dimensional group of M\"{o}bius transformations when $g=0$$g=0$,
• $\mathbf{R}^2/\mathbf{Z}^2$$\mathbf{R}^2/\mathbf{Z}^2$, acting by translations when $g=1$$g=1$,
• finite when $g>1$$g>1$

Definition 4.2 Expected dimension. The expected dimension of the moduli space is the number

$\displaystyle n(2-2g)+2c_1(X,J)[A]+6g-6$ $$\displaystyle n(2-2g)+2c_1(X,J)[A]+6g-6$$

and this coincides with the actual dimension of a regular moduli space of curves after dividing out by reparametrisation.

It is harder to achieve transversality for curves which are multiple covers. This is best seen in a simple example.

Example 4.3 Isolated spheres in Calabi-Yau 3-folds. Consider $X$$X$, the total space of the bundle $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$$\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ over $\mathbf{CP}^1$$\mathbf{CP}^1$. The inclusion of the zero-section $u\colon S^2\to X$$u\colon S^2\to X$ is the only closed holomorphic curve in its homology class (up to reparametrisation). This sphere is regular; the expected dimension is zero since $c_1=0$$c_1=0$. If $f\colon S^2\to S^2$$f\colon S^2\to S^2$ is a generic holomorphic branched cover of degree $d$$d$ then $u\circ f$$u\circ f$ lives in a moduli space of dimension $4(d-1)$$4(d-1)$ (corresponding to the configuration space of the $2(d-1)$$2(d-1)$ branch points). The expected dimension is still zero, so the linearised operator must have a nontrivial cokernel.

In this case one needs to introduce more general perturbations to achieve transversality.

Theorem 4.4 Transversality in the semipositive case. When $(X,\omega)$$(X,\omega)$ is semipositive one can achieve transversality even for multiply-covered curves by using a domain-dependent $J$$J$.

5 Compactness

While the $L^2$$L^2$-control on the derivatives of $u$$u$ given by the energy identity 2.1 is not enough to ensure a priori compactness of moduli spaces, there is a natural compactification by adding in strata of stable maps.

Definition 5.1 (Stable map). Let $(\Sigma,j)$$(\Sigma,j)$ be a nodal Riemann surface (i.e. a connected, compact reduced complex curve with at worst ordinary double points) and $x_1,\ldots,x_k$$x_1,\ldots,x_k$ a collection of distinct non-nodal marked points on $\Sigma$$\Sigma$. A stable map $u\co\Sigma\rightarrow X$$u\co\Sigma\rightarrow X$ is a $J$$J$-holomorphic map such that any irreducible component of $\Sigma$$\Sigma$ which is mapped down to a point $x\in X$$x\in X$ has either

• arithmetic genus 0 and at least three points which are either marked or nodal,
• arithmetic genus 1 and at least one point which is either marked or nodal,
• arithmetic genus 2 or more.

This is equivalent to the requirement that the group of holomorphic automorphisms $\phi$$\phi$ of $\Sigma$$\Sigma$ fixing the marked points and satisfying $u\circ\phi=u$$u\circ\phi=u$ is finite. A reparametrisation of a stable map is a holomorphic automorphism of the domain which does not necessarily leave $u$$u$ invariant and we usually only consider stable maps up to reparametrisation.

There is a notion of convergence for stable maps up to reparametrisation, called Gromov convergence, which allows us to define a topology of the moduli space of stable maps.

Theorem 5.2 (Gromov compactness). The moduli space of stable maps (modulo reparametrisations) with the topology of Gromov convergence is both compact and Hausdorff.