Stable map

In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable map was proposed by Maxim Kontsevich around 1992 and published in .
Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself.
The moduli space of smooth pseudoholomorphic curves.
Fix a closed symplectic manifold formula_1 with symplectic form formula_2. Let formula_3 and formula_4 be natural numbers (including zero) and formula_5 a two-dimensional homology class in formula_1. Then one may consider the set of pseudoholomorphic curves
where formula_8 is a smooth, closed Riemann surface of genus formula_3 with formula_4 marked points formula_11, and
is a function satisfying, for some choice of formula_2-tame almost complex structure formula_14 and inhomogeneous term formula_15, the perturbed Cauchy–Riemann equation
Typically one admits only those formula_3 and formula_4 that make the punctured Euler characteristic formula_19 of formula_20 negative; then the domain is stable, meaning that there are only finitely many holomorphic automorphisms of formula_20 that preserve the marked points.
The operator formula_22 is elliptic and thus Fredholm. After significant analytical argument (completing in a suitable Sobolev norm, applying the implicit function theorem and Sard's theorem for Banach manifolds, and using elliptic regularity to recover smoothness) one can show that, for a generic choice of formula_2-tame formula_14 and perturbation formula_15, the set of formula_26-holomorphic curves of genus formula_3 with formula_4 marked points that represent the class formula_5 forms a smooth, oriented orbifold
of dimension given by the Atiyah-Singer index theorem,
The stable map compactification.
This moduli space of maps is not compact, because a sequence of curves can degenerate to a singular curve, which is not in the moduli space as we've defined it. This happens, for example, when the energy of formula_32 (meaning the "L"2-norm of the derivative) concentrates at some point on the domain. One can capture the energy by rescaling the map around the concentration point. The effect is to attach a sphere, called a bubble, to the original domain at the concentration point and to extend the map across the sphere. The rescaled map may still have energy concentrating at one or more points, so one must rescale iteratively, eventually attaching an entire bubble tree onto the original domain, with the map well-behaved on each smooth component of the new domain.
In order to make this precise, define a stable map to be a pseudoholomorphic map from a Riemann surface with at worst nodal singularities, such that there are only finitely many automorphisms of the map. Concretely, this means the following. A smooth component of a nodal Riemann surface is said to be stable if there are at most finitely many automorphisms preserving its marked and nodal points. Then a stable map is a pseudoholomorphic map with at least one stable domain component, such that for each of the other domain components
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, called the stabilization formula_33 of the domain formula_20.
The set of all stable maps from Riemann surfaces of genus formula_3 with formula_4 marked points forms a moduli space
The topology is defined by declaring that a sequence of stable maps converges if and only if
The moduli space of stable maps is compact; that is, any sequence of stable maps converges to a stable map. To show this, one iteratively rescales the sequence of maps. At each iteration there is a new limit domain, possibly singular, with less energy concentration than in the previous iteration. At this step the symplectic form formula_2 enters in a crucial way. The energy of any smooth map representing the homology class formula_40 is bounded below by the symplectic area formula_41,
with equality if and only if the map is pseudoholomorphic. This bounds the energy captured in each iteration of the rescaling and thus implies that only finitely many rescalings are needed to capture all of the energy. In the end, the limit map on the new limit domain is stable.
The compactified space is again a smooth, oriented orbifold. Maps with nontrivial automorphisms correspond to points with isotropy in the orbifold.
The Gromov–Witten pseudocycle.
To construct Gromov–Witten invariants, one pushes the moduli space of stable maps forward under the evaluation map
to obtain, under suitable conditions, a rational homology class
Rational coefficients are necessary because the moduli space is an orbifold. The homology class defined by the evaluation map is independent of the choice of generic formula_2-tame formula_14 and perturbation formula_15. It is called the Gromov–Witten (GW) invariant of formula_1 for the given data formula_3, formula_4, and formula_5. A cobordism argument can be used to show that this homology class is independent of the choice of formula_2, up to isotopy. Thus Gromov–Witten invariants are invariants of symplectic isotopy classes of symplectic manifolds.
The "suitable conditions" are rather subtle, primarily because multiply covered maps (maps that factor through a branched covering of the domain) can form moduli spaces of larger dimension than expected.
The simplest way to handle this is to assume that the target manifold formula_1 is "semipositive" or "Fano" in a certain sense. This assumption is chosen exactly so that the moduli space of multiply covered maps has codimension at least two in the space of non-multiply-covered maps. Then the image of the evaluation map forms a pseudocycle, which induces a well-defined homology class of the expected dimension.
Defining Gromov–Witten invariants without assuming some kind of semipositivity requires a difficult, technical construction known as the virtual moduli cycle.