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

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If X is an algebraic variety, then a deformation of X is a (larger) variety   containing X, together with a map (called the projection)   (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  .

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  , 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

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Let N and P be manifolds, and f0 be a map (-germ) from N to P. A deformation of f0 is a map   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  .)

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

 .

for   Deformations of map-germs are often called unfoldings, especially in Singularity theory.

Group actions

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

 

such that for each  , the map   is a group action.

For example, the action of   on   given by   has as deformation (with  )

 

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).

Algebras

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References

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Category:Algebra Category:Geometry