In mathematics, specifically in symplectic geometry and algebraic geometry, the moduli spaces of stable maps generalise the moduli spaces of curves, via a special kind of function called a stable map. The word "stable", like in the case of stable curves, means that these maps have only a finite number of automorphisms" - that is, a moduli space.

, we can calculate the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable maps was proposed by Maxim Kontsevich around 1992 and published in .

There are two competing points of view: those of algebraic and symplectic geometry. This article aims to treat both; the word "curve" refers both to (complex) algebraic curves and to Riemann surfaces can be taken either as a smooth projective variety or as a closed symplectic manifold (equipped with a symplectic form and an almost complex structure-tameness, defined below).

are nonnegative integers.

Algebraic definition

Consider a nodal is required to be connected, but may be made up of several irreducible components joined to one another at the nodes. We say a point is a special point if it is either a marked point or a node.

A morphism is a stable map for such a map.

are isomorphic if there is an isomorphism is finite.

. The (possibly empty) open subsetalso has a moduli space structure, but is not compact.

It is significant that the domain of a stable map need not be a stable curve. However, one can contract its unstable components (iteratively) to produce a stable curve.

In symplectic geometry

-holomorphic curves:

is -tame has this property.

is called a -holomorphic curvefor some conjugate-linear-holomorphic curve.

, such that the fundamental class of the surface is pushed forward. One way to do this is by considering the space (no stable curves exist otherwise).

is semipositive has a natural compactification, called the Gromov-Uhlenbeck compactification. A stable map in the sense of symplectic geometry can then be considered an element of this compactified space.

Examples

• For any positive integer, where the domain has no marked points. Such a map contracts no components of the domain (of which there is only one), and so is stable. This space is the Grassmannian.

• general points?

Properties

• .

• is convex

• does not have a well-defined dimension, we may define its virtual dimension. The virtual fundamental class, which is important for the definition of algebraic Gromov-Witten invariants, lives in the homology vector space.

• .

• . These are also used in the definition of Gromov-Witten invariants, as they allow the pulling back.

Stabilisation and bubbling

is compact, as there are several problems that can occur when taking the limit of a family in this space. Possible issues are:

• Two marked points converge to the same point

• A marked point converges to a node

• Two nodes converge to the same point, creating a triple point

• The map becomes constant on an unstable component of the domain

These can all be dealt with through a two-step process. The first step is known as bubbling (a sphere), called a bubble, to the domain curve at the problematic point. If the problem involves a node, this bubble separates the two components that were joined by that node. Any markings that were present at the problem point are moved onto the bubble - this ensures that it is stable, so we may extend the original map to this new curve by defining it to be constant on the bubble. The second step consists of contracting any unstable-but-contracted components of the domain curve.

.

Notes

References