Pseudoholomorphic curves
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== Introduction == | == Introduction == |
Revision as of 13:19, 7 March 2011
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
In this article, will denote an almost complex manifold of dimension .
Definition 1.1 (-holomorphic curve). Let be a Riemann surface with complex structure . A -holomorphic curve
is a smooth map satisfying
or equivalently
A -holomorphic map is called simple if it cannot be factored as where is a -holomorphic branched cover of degree strictly greater than 1. We will usually omit from the notation and speak of -holomorphic curves. The term pseudoholomorphic curve will be used to describe a -holomorphic curve when we do not want to specify .
Pseudoholomorphic curves provide a useful tool for studying symplectic manifolds.
1 Taming
Pseudoholomorphic curves in a general almost complex manifold can be quite wild (EXAMPLE?). Better behaviour can be ensured by the existence of a symplectic taming , that is the quadratic form
is positive-definite. This gives us topological control on the energy of a -holomorphic curve
(here the norm and volume form are taken with respect to the metric ) thanks to the identity
Lemma 3.1 (Energy identity).
If moreover we require the metric to be -invariant () then we say that is -compatible and we have the identity
so that the -holomorphic curves are the absolute minima of the energy functional on the space of maps .
2 Moduli spaces
Simple -holomorphic curves form nice moduli spaces
Theorem 5.1 Transversality for simple curves. Fix a homology class and a Riemann surface . There is a subset of the second-category such that the space
is a finite-dimensional manifold of dimension
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 -control on the derivatives of 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 be a nodal Riemann surface (i.e. a connected, compact reduced complex curve with at worst ordinary double points) and a collection of distinct non-nodal marked points on . A stable map is a -holomorphic map such that any irreducible component of which is mapped down to a point 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 of fixing the marked points aznd satisfying 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.