Pseudoholomorphic curves

Revision as of 13:04, 7 March 2011

1 Introduction

In this article, $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; In this article, $(X,J)$ will denote an [http://en.wikipedia.org/wiki/Almost_complex_manifold almost complex manifold] of dimension n$. {{beginthm|Definition|($J$-holomorphic curve)}} Let $(\Sigma,j)$ be a Riemann surface with complex structure $j$. A '''$(j,J)$-holomorphic curve''' $$u\co\Sigma\rightarrow X$$ is a smooth map satisfying $du\circ j=J\circ du$. A $J$-holomorphic map is called '''simple''' if it cannot be factored as $u=v\circ\phi$ where $\phi$ is a $(j',j)$-holomorphic branched cover $\phi\co\Sigma'\rightarrow\Sigma$ of degree strictly greater than 1. We will usually omit $j$ from the notation and speak of $J$-holomorphic curves. The term '''pseudoholomorphic curve''' will be used to describe a $J$-holomorphic curve when we do not want to specify $J$. {{endthm}} Pseudoholomorphic curves provide a useful tool for studying [[symplectic manifolds]]. </wikitex> == References == {{#RefList:}} [[Category:Theory]](X,J)$ will denote an almost complex manifold of dimension $2n$.

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

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

is a smooth map satisfying

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

or equivalently

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

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

Pseudoholomorphic curves provide a useful tool for studying symplectic manifolds.

1 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 $\omega$ taming $J$, that is the quadratic form

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

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

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

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

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

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

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

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

2 Moduli spaces

Simple $J$-holomorphic curves form nice moduli spaces

Theorem 5.1 Transversality for simple curves. Fix a homology class $A\in H_2(X;\mathbb{Z})$ and a Riemann surface $(\Sigma,j)$. There is a subset $\mathcal{J}_{\mathrm{reg}}\subset\mathcal{J}$ of the second-category such that the space

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

is a finite-dimensional manifold of dimension

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

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.

3 Compactness

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

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

2 References