Pseudoholomorphic curves

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This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

In this article, (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

Lemma 3.1 (Energy 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 3.1 is not enough to ensure a priori compactness of moduli spaces, there is a natural compactification by adding in strata of stable maps.

Definition 7.1 (Stable map). Let (\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 a collection of distinct non-nodal marked points on \Sigma. A stable map u\co\Sigma\rightarrow X is a J-holomorphic map such that any irreducible component of \Sigma which is mapped down to a point 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 of \Sigma fixing the marked points aznd satisfying u\circ\phi=u is finite.

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.

Theorem 7.2 (Gromov compactness). The space of stable maps with the topology of Gromov convergence is both compact and Hausdorff.

2 References

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