Stable map

In mathematics, specifically in symplectic geometry and algebraic geometry, the moduli spaces of stable maps extend the notion of moduli spaces of curves, allowing the consideration of curves (or equivalently, Riemann surfaces) embedded in an ambient space (a smooth projective variety or a closed symplectic manifold), rather than just studying their intrinsic geometry. This is done by considering the different ways of embedding curves into a fixed ambient space via a certain kind of map, called a stable map. The "stability" condition, like in the case of stable curves, means these maps have only a finite number of automorphisms, which is necessary for the construction of a good moduli space.

By requiring a certain number of "marked points" on the domain curves, and considering where these are mapped to in the ambient space, we can calculate 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 Kontsevich (1995).

Definition

Fix a smooth projective variety over ${\displaystyle \mathbb {C} }$ (closed symplectic manifold) called ${\displaystyle X}$, and two nonnegative integers ${\displaystyle g,n}$. Consider a nodal algebraic curve ${\displaystyle C}$ over ${\displaystyle \mathbb {C} }$ (nodal Riemann surface) of genus ${\displaystyle g}$ and with ${\displaystyle n}$ distinct marked smooth points ${\displaystyle p_{1},\ldots ,p_{n}\in C}$. We say a point is a special point if it is either a marked point or a node.

A morphism (pseudoholomorphic map) ${\displaystyle f:C\to X}$ is a stable map if every component of ${\displaystyle C}$ which is contracted by ${\displaystyle f}$ (that is, ${\displaystyle f}$ is constant on that component) is itself a stable curve. This is equivalent to saying that contracted genus 0 components must have 3 or more special points, and contracted genus 1 components must have at least 1 special point. We typically write ${\displaystyle (C,p_{1},\ldots ,p_{n},f)}$ for such a map.

We say that two stable maps ${\displaystyle (C,p_{1},\ldots ,p_{n},f)}$ and ${\displaystyle ({\tilde {C}},q_{1},\ldots ,q_{n},{\tilde {f}})}$ are isomorphic if there is an isomorphism of curves ${\displaystyle \tau :C\to {\tilde {C}}}$ such that ${\displaystyle \tau (p_{i})=q_{i}}$ for all indices ${\displaystyle i}$, and such that ${\displaystyle {\tilde {f}}\circ \tau =f}$. The above stability condition is then equivalent to saying that the group of automorphisms of ${\displaystyle f}$ is finite.

With this in mind, we may construct moduli spaces of stable maps. Let ${\displaystyle \beta \in H_{2}(X)}$ be a curve class. The corresponding (compact) moduli space ${\displaystyle {\overline {M}}_{g,n}(X,\beta )}$ consists of stable maps ${\displaystyle (C,p_{1}\ldots ,p_{n},f)}$ such that ${\displaystyle f_{*}[C]=\beta }$. The (possibly empty) open subset $${\displaystyle M_{g,n}(X,\beta )=\{(C,p_{1},\ldots ,p_{n},f)\in {\overline {M}}_{g,n}(X,\beta ):C{\text{ smooth}}\}}$$also 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.

Examples

• For any positive integer ${\displaystyle m}$, the space ${\displaystyle {\overline {M}}_{0,0}(\mathbb {P} ^{m},1)}$ consists of degree 1 maps ${\displaystyle \mathbb {P} ^{1}\to \mathbb {P} ^{m}}$, 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 ${\displaystyle \mathbf {Gr} (\mathbb {P} ^{1},\mathbb {P} ^{m})}$ parametrizing all lines in ${\displaystyle \mathbb {P} ^{m}}$.
• For any nonnegative integer ${\displaystyle d}$, the space ${\displaystyle {\overline {M}}_{0,0}(\mathbb {P} ^{2},d)}$ is the space of degree ${\displaystyle d}$ plane curves, and is of dimension ${\displaystyle 3d-1}$. Spaces like this can be used to answer questions in enumerative geometry such as: how many degree ${\displaystyle d}$ plane curves pass through ${\displaystyle 3d-1}$ general points?

In symplectic geometry

Fix a closed symplectic manifold ${\displaystyle X}$ with symplectic form ${\displaystyle \omega }$. Let ${\displaystyle g}$ and ${\displaystyle n}$ be nonnegative integers and ${\displaystyle \beta }$ a two-dimensional homology class in ${\displaystyle X}$. Then one may consider the set of pseudoholomorphic curves

${\displaystyle ((C,j),f,(p_{1},\ldots ,p_{n}))\,}$

where ${\displaystyle (C,j)}$ is a smooth, closed Riemann surface of genus ${\displaystyle g}$ with ${\displaystyle n}$ marked points ${\displaystyle p_{1},\ldots ,p_{n}}$, and

${\displaystyle f:C\to X\,}$

is a function satisfying, for some choice of ${\displaystyle \omega }$-tame almost complex structure ${\displaystyle J}$ and inhomogeneous term ${\displaystyle \nu }$, the perturbed Cauchy–Riemann equation

${\displaystyle {\bar {\partial }}_{j,J}f:={\frac {1}{2}}(df+J\circ df\circ j)=\nu .}$

Typically one admits only those ${\displaystyle g}$ and ${\displaystyle n}$ that make the punctured Euler characteristic ${\displaystyle 2-2g-n}$ of ${\displaystyle C}$ negative. Then the domain is stable, meaning that there are only finitely many holomorphic automorphisms of ${\displaystyle C}$ that preserve the marked points.

The operator ${\displaystyle {\bar {\partial }}_{j,J}}$ 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 ${\displaystyle \omega }$-tame ${\displaystyle J}$ and perturbation ${\displaystyle \nu }$, the set of ${\displaystyle (j,J,\nu )}$-holomorphic curves of genus ${\displaystyle g}$ with ${\displaystyle n}$ marked points that represent the class ${\displaystyle A}$ forms a smooth, oriented orbifold

${\displaystyle M_{g,n}^{J,\nu }(X,\beta )}$

(or just ${\displaystyle M_{g,n}(X,\beta )}$) of dimension given by the Atiyah-Singer index theorem,

${\displaystyle d:=\dim _{\mathbb {R} }M_{g,n}(X,\beta )=2c_{1}^{X}(\beta )+(\dim _{\mathbb {R} }X-6)(1-g)+2n.}$

Compactifying the moduli space

The moduli space ${\displaystyle M_{g,n}(X,\beta )}$ is not compact, because a family of smooth curves can have a singular curve as its limit. This happens, for example, when two marked points converge to the same point in the limit, or when a marked point converges to a node; markings are required to be distinct and smooth, so the result in either case is not a valid domain curve for a stable map.

Such a limit can be stabilized through a process known as bubbling. The effect is to attach a sphere (copy of ${\displaystyle \mathbb {P} ^{1}}$), called a bubble, to the original domain at the problematic point and to extend the map across this new component. The new map may still have problematic points, but the process can be applied iteratively, eventually attaching an entire "tree" of bubbles onto the original domain, and obtaining a map which is well-behaved on each smooth component of the new domain.

The topology of the moduli space ${\displaystyle {\overline {M}}_{g,n}(X,\beta )}$ is defined by declaring that a family of stable maps converges if and only if

• the (stabilized) domains converge in the Deligne–Mumford moduli space of curves ${\displaystyle {\overline {M}}_{g,n}}$,
• they converge uniformly in all derivatives on compact subsets away from the nodes, and
• the energy (the L² norm of the derivative) concentrating at any point equals the energy in the bubble tree attached at that point in the limit map.

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 ${\displaystyle \omega }$ enters in a crucial way. The energy of any smooth map representing the homology class ${\displaystyle B}$ is bounded below by the symplectic area ${\displaystyle \omega (B)}$,

${\displaystyle \omega (B)\leq {\frac {1}{2}}\int |df|^{2},}$

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.

Gromov–Witten pseudocycle

To construct Gromov–Witten invariants, one pushes the moduli space of stable maps forward under the evaluation map

${\displaystyle M_{g,n}(X,\beta )\to {\overline {M}}_{g,n}\times X^{n},}$

${\displaystyle ((C,j),f,(p_{1},\ldots ,p_{n}))\mapsto (\mathrm {st} (C,j),f(p_{1}),\ldots ,f(p_{n}))}$

to obtain, under suitable conditions, a rational homology class

${\displaystyle GW_{g,n}^{X,\beta }\in H_{d}({\overline {M}}_{g,n}\times X^{n},\mathbb {Q} ).}$

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 ${\displaystyle \omega }$-tame ${\displaystyle J}$ and perturbation ${\displaystyle \nu }$. It is called the Gromov–Witten (GW) invariant of ${\displaystyle X}$ for the given data ${\displaystyle g}$, ${\displaystyle n}$, and ${\displaystyle \beta }$. A cobordism argument can be used to show that this homology class is independent of the choice of ${\displaystyle \omega }$, 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 ${\displaystyle X}$ 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.

References

• Dusa McDuff and Dietmar Salamon, J-Holomorphic Curves and Symplectic Topology, American Mathematical Society colloquium publications, 2004. ISBN 0-8218-3485-1.
• Kontsevich, Maxim (1995). "Enumeration of rational curves via torus actions". Progr. Math. 129: 335–368. MR 1363062.
• Hori, Kentaro (2003). Mirror symmetry / Kentaro Hori ... [et al.] Providence, R.I.: American Mathematical Society for the Clay Mathematics Institute. pp. 501–512. ISBN 0821829556.