Stable figure

In the symplectic topology one can define the module space of stable mappings , of Riemann surfaces in a given symplectic manifold . This modular space is essential for the construction of the Gromov-Witten invariants , which are used in counting algebraic geometry and string theory .

The module space of pseudoholomorphic curves

${\displaystyle X}$be a closed symplectic manifold with the symplectic form . and let be natural numbers including zero and a two-dimensional homology class in . Then one can get the set of pseudoholomorphic curves ${\displaystyle \omega }$ ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle X}$

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

consider, where is a smooth, closed Riemann surface of the sex with marked (distinguished) points , and ${\displaystyle (C,j)}$ ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle x_{1},\ldots ,x_{n}}$

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

is a function that is necessary for the choice of a certain -tame (tame) almost complex structure and inhomogeneous term , the perturbed Cauchy-Riemann differential equations ${\displaystyle \omega }$ ${\displaystyle J}$ ${\displaystyle \nu }$

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

Fulfills. Typically, they are allowed only such and that the dotted Euler-Poincare characteristic of making negative. Then the area is stable , that is, there are only a finite number of automorphisms of that receive the marked points. ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle 2-2g-n}$ ${\displaystyle C}$ ${\displaystyle C}$

The operator is an elliptic operator and therefore of the Fredholm type. After considerable analytical work, it can be shown that for a generic choice of -tame and perturbation, the set of -holomorphic sex curves with marked points representing class have a smooth, oriented orbifold ${\displaystyle {\bar {\partial }}_{j,J}}$ ${\displaystyle \omega }$ ${\displaystyle J}$ ${\displaystyle \nu }$ ${\displaystyle (j,J,\nu )}$ ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle A}$

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

is, with one dimension given by the Atiyah-Singer index theorem :

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

The compactification of stable images

This module space is not compact, since a sequence of curves can degenerate into a singular curve which would then lie outside the module space previously defined. This happens e.g. B. if the energy of (what is meant is the L ² norm of the derivative) is concentrated in a point of the domain of the function (hereinafter referred to as the domain for short). The energy can be “captured” by rescaling the image around the point of concentration, whereby a sphere (called a bubble ) is attached to the area and the image is extended to this bubble. Additional concentration points can be created, so that you have to proceed iteratively and create a whole bubble tree. ${\displaystyle f}$

To formulate the whole thing more precisely, a stable mapping is defined as a pseudoholomorphic mapping of a Riemann surface with, in the worst case, node singularities, so that there are only finitely many automorphisms in the mapping. A smooth component of a Riemann surface with nodes is called stable if there are only finitely many automorphisms that contain the nodes and marked points. A stable mapping is a pseudoholomorphic mapping with at least one stable area component, so that for all other area components the mapping is non-constant or the component is stable.

The area of ​​a stable map does not have to be a stable curve, but one can iteratively contract its unstable components to create a stable curve called the stabilization of the area . ${\displaystyle \mathrm {st} (C)}$ ${\displaystyle C}$

The set of all stable images of Riemann surfaces of the sex with marked points forms a module space ${\displaystyle g}$ ${\displaystyle n}$

${\displaystyle {\bar {M}}_{g,n}^{J,\nu }(X,A).}$

The topology is defined by the fact that a sequence of stable mappings converges if and only if

• their (stabilized) areas converge in the Deligne-Mumford module space of the curves , ${\displaystyle {\bar {M}}_{g,n}}$
• they converge uniformly in all derivatives on compact subsets outside the nodes, and
• The energy concentration at each point is equal to the energy in each bubble tree associated with that point in the boundary diagram.

The module space of the stable images is compact: each sequence of stable images converges to a stable image. To prove this, one iteratively rescaled the sequence of mappings. Each time at the border crossing you get a new area (possibly singular) with less energy concentration than in the iteration step before. At this point, the presence of a symplectic shape is essential. The energy of every smooth map that represents the homology class is limited from below by the symplectic surface , ${\displaystyle \omega }$ ${\displaystyle B}$ ${\displaystyle \omega (B)}$

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

where the equal sign applies if and only if the mapping is pseudoholomorphic. This limits the energy that is captured in the repetition of the rescaling and ensures that only a finite number of rescalings are necessary to capture all of the energy. In the end, the border mapping is stable in the new border area.

The compacted space is again a smooth, oriented orbifold . Maps with non-trivial automorphisms correspond to points on the orbifold with isotropy .

The Gromov-Witten pseudocycle

In order to construct Gromov-Witten invariants , an evaluation map is carried out in the module space of the stable mappings

${\displaystyle M_{g,n}^{J,\nu }(X,A)\to {\bar {M}}_{g,n}\times X^{n},}$

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

to get a rational homology class under suitable conditions:

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

Rational coefficients are necessary because the module space is an orbic fold . The homology class defined by the execution figure is independent of the choice of the generic -tame and the disorder . It is the Gromov-Witten invariant (GW) of for given , and called. The independence (except for isotopy ) of the homology class from the choice of can be shown by a cobordism argument. The GW are thus invariants of symplectic isotope classes of symplectic manifolds. ${\displaystyle \omega }$ ${\displaystyle J}$ ${\displaystyle \nu }$ ${\displaystyle X}$ ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle \omega }$

The "suitable conditions" are quite technical, mainly because maps with multiple sheets (an overlay manifold) can form larger module spaces than expected. The simplest way is then to assume that the target multiplicity of illustration in a specific sense semi positive (semi-positive) or a Fano manifold is. The module space of multiple overlapping images then has at least co-dimension 2 in the space of single overlapping images. The image of the evaluation map then forms a pseudocycle, with which a well-defined homology class can be defined in the expected dimension. GW without a kind of semi-positivity requires a difficult technical construction, known as a virtual module cycle . ${\displaystyle X}$

literature

• Dusa McDuff , Dietmar Salamon : J-Holomorphic Curves and Symplectic Topology ( American Mathematical Society. Colloquium publications 52) American Mathematical Society, Providence RI 2004, ISBN 0-8218-3485-1 .

Individual evidence

1. ↑ Completion of a suitable Sobolew space , application of the theorem about implicit functions , Sard's theorem for Banach spaces , use of elliptic regularity to regain smoothness.