Stable map

In mathematics, specifically in symplectic geometry and algebraic geometry, the moduli spaces of stable maps generalise the moduli spaces of curves, allowing the study of the geometry of curves with respect to their position in some larger space ${\displaystyle X}$. This is done by considering ways of embedding curves into ${\displaystyle X}$, 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, which is important for the construction of a "space of all curves (of a certain type) in ${\displaystyle X}$" - that is, a moduli space.

By "marking" a certain number of points on the embedded curves, and considering where these are positioned in the ambient space ${\displaystyle X}$, 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 Kontsevich (1995).

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, and the ambient space ${\displaystyle X}$ can be taken either as a smooth projective variety or as a closed symplectic manifold (equipped with a symplectic form ${\displaystyle \omega }$ and an almost complex structure ${\displaystyle J}$ satisfying a certain "compatibility condition" known as ${\displaystyle \omega }$-tameness, defined below).

Throughout this article ${\displaystyle X}$ denotes a fixed ambient space as above, and ${\displaystyle g,n}$ are nonnegative integers.