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stable map ：ウィキペディア英語版

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

X

\omega

. Let

g

and

n

be natural numbers (including zero) and

A

a two-dimensional homology class in

X

. Then one may consider the set of pseudoholomorphic curves
:

((C, j), f, (x_1, \ldots, x_n))\,

where

(C, j)

is a smooth, closed Riemann surface of genus

g

with

n

marked points

x_1, \ldots, x_n

, and
:

f : C \to X\,

is a function satisfying, for some choice of

\omega

J

and inhomogeneous term

\nu

\bar \partial_ f := \frac(df + J \circ df \circ j) = \nu.

Typically one admits only those

g

and

n

that make the punctured Euler characteristic

2 - 2g - n

C

negative; then the domain is stable, meaning that there are only finitely many holomorphic automorphisms of

C

that preserve the marked points.
The operator

\bar \partial_

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

\omega

-tame

J

and perturbation

\nu

, the set of

(j, J, \nu)

-holomorphic curves of genus

g

with

n

marked points that represent the class

A

forms a smooth, oriented orbifold
:

M_^(X, A)

of dimension given by the Atiyah-Singer index theorem,
:

d := \dim_(X, A) = 2 c_1^X(A) + (\dim_{\mathbb{R}} X - 6)(1 - g) + 2 n.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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