User:Simplifix/Deformations

In mathematics a deformation of a mathematical object is either a single nearby object ('nearby' in some appropriate sense depending on the context), or a family of such nearby objects, and often an object is understood better by understanding its possible deformations. Examples are very numerous: deformations of group actions, of maps, of algebraic varieties, of algebras (eg deformation quantization), dynamical systems (bifurcation theory), complex structures (deformation theory).

At this very general level, it is not possible (?) to give a formal definition of deformation, so in this article we describe different definitions for different contexts.

In many cases, the possible deformations of a given object are classified by some cohomology construction.

A deformation is said to be trivial if every member of the family is equivalent to the original object (equivalent in an appropriate sense depending on the context).

Algebraic varieties

If X is an algebraic variety, then a deformation of X is a (larger) variety $\scriptstyle {\mathcal {X}}$ containing X, together with a map (called the projection) $\scriptstyle \pi :{\mathcal {X}}\to B$ (B is the base of the deformation), such that X=π^-1(0). Individual deformations of X are then obtained as π^-1(b) for $\scriptstyle b\in B$.

An example of a variety is the real cone X, given by equations x² + y² - z² = 0. A deformation of X can be given by $\scriptstyle {\mathcal {X}}=\mathbb {R} ^{3}$, with coordinates x, y, z, and projection

π(x,y,z) = x² + y² - z².

The individual deformations are then 1- or 2-sheeted hyperbolae, depending on the sign of b = x² + y² - z².

Map-germs

Let N and P be manifolds, and f₀ be a map (-germ) from N to P. A deformation of f₀ is a map $\scriptstyle F:N\times B\to P\times B$ of the form F(x,b) = (F₁(x,b),b), and such that F(x,0) = (f(x),0). (In practice the B factor in the target is often omitted, so one has $\scriptstyle F_{1}:N\times B\to P$.)

For example, the map f₀(x,y) = x²+y³ has a deformation,

$F(x,y;b)=(x^{2}+y^{3}-by,\;b)$.

for $\scriptstyle b\in \mathbb {R} .$ Deformations of map-germs are often called unfoldings, especially in Singularity theory.

Group actions

An action of a group G on a manifold M is a smooth map $\phi :G\times M\to M$ satisfying 2 conditions. A deformation (with base B) of such an action is a smooth map

$\Phi :G\times M\times B\longrightarrow M\times B\quad \mathrm {with} \quad \Phi (g,m,\,b)=(\phi _{b}(g,m),\,b)$

such that for each $\scriptstyle b\in B$, the map $\phi _{b}\,$ is a group action.

For example, the action of $\mathbb {R}$ on $\mathbb {R} ^{2}$ given by $\phi (s,\,(x,y))=\mathrm {e} ^{s}(x,y)$ has as deformation (with $\scriptstyle b\in \mathbb {R}$)

$\phi _{b}(s,(x,y))\,=\mathrm {e} ^{s}(x+bsy,y)$

The actions in this example for b zero and non-zero are not equivalent, so this deformation is non-trivial.

If M=V is a representation of G, then the deformations of a given representation ρ are classified by the group cohomology H¹(G,ρ).

References

Category:Algebra Category:Geometry