no emie c. combe mpi mis wednesday 29/04 at 17:00...classical deformation theory (for algebraic...

Deformation theory and Higher Lie theory

Noemie C. CombeMPI MiS

Wednesday 29/04 at 17:00

Noemie C. Combe MPI MiS


A bit of deformation theory: heuristics

The aim of deformation theory is to study structures of agiven type, existing on an object up to some ’equivalence’.♣ We want to know if a given structure is:

1. rigid

2. if it can deform into another one. For the case 2,questions is:I how can we deform?I what is a deformation?I how much freedom do we have while deforming?



Typical situationA typical situation would be a flat morphism of schemesf : X → T .For varying t ∈ T , we regard the fibres Xt as a family ofschemes.Deformation theory is the infinitesimal study of the family inthe neighborhood of a special fibre X0.



Example about deformations

ExampleLet k be a field (char 0). Consider A = k[x , y ]/(xy). For anyvalue t 6= 0 we can have At = k[x , y ]/(xy − t).



The k-algebra A = k[x , y ]/(xy) can be deformed toAt = k[x , y ]/(xy − t) for any value t 6= 0 of the parameter t.The family {At : t ∈ k} of commutative k-algebras can beconsidered as a family of deformations of the commutativek-algebra A = A0.

♣ Classical deformation theory is concerned with deformationin a more restricted sense.



This example illustrates (exactly!) the notion of localdeformations.• Let A a commutative k-algebra A = k[x , y ]/(xy)• R = k[[t]] be the formal power series ring on one variable.Then, AR = k[x , y ][[t]]/(xy − t) is a local deformation of Aparametrized by R .

Indeed, since AR is R-flat with isomorphism k ⊗ Ar → Ainduced by the map AR → A given by t → 0.



Classical deformation theory (for Algebraic geometry) is basedon the work of Kodaira–Spencer Nirenberg, Kuranishi on smalldeformations of complex manifolds. It was formalised byGrothendieck in language of algebraic schemes.



Steps to follow

1. consider formal, local deformations parametrised bycomplete local commutative rings,

2. find formal deformation which is universal (orsemi-universal). Then find an ’algebraization’.

♣ Analytic setting: first one wants formal solutions and thenconsider convergence.

ExampleBack to At = k[x , y , t]/(xy − t) AR is a formal deformation ofA parametrized by R = k[[t]]. It is semi-universal. Analgebraization of R and the deformation AR is given by k[t]and the deformation is At = k[x , y , t]/(xy − t).



More details and rigour

ExampleComplex structures, on a smooth manifold, up todiffeomorphism.K. Kodaira and D. C. Spencer, On deformations of complexanalytic structures. I, II, Ann. of Math. (2) 67 (1958),328–466. 16



Other examples where deformation theory shows up

ExamplePoisson structures, on a smooth manifold, up todiffeomorphism.Maxim Kontsevich, Deformation quantization of Poissonmanifolds, Lett. Math. Phys. 66 (2003), no. 3, 157– 216. 16,18

ExampleAssociative algebras on a vector space up to isomorphism.Murray Gerstenhaber, On the deformation of rings andalgebras, Ann. of Math. (2) 79 (1964), 59–103. 16



Pierre Deligne formulated the following philosophy:

In char 0, any deformation problem is controlled by DGLA(Differentially Graded Lie Algebra).

Proven in the last years using homotopic algebra, and highercategories (categories of models and ∞-categories).



DGLA’s

In all examples cited above, there exists a graded differentialLie algebra (DGLA) given by (g, d , [, ]).It is a graded vector space {gn}n∈Z endowed with:

1. Lie bracket a bilinear map [, ] : gn ⊗ gm → gn+m andverifying the Jacobi relation

2. Differential a linear map d ∈ Hom(g, g) of degree -1,d2 = 0, compatible with the Lie bracket.



Maurer–Cartan equations

In any cases, the set of studied structures (deformations oforiginal structure) is in bijection with the set of solutions of thedeformation equation of DGLA (the Maurer–Cartan equation)

MC (g) = {α ∈ g−1|dα +1

2[α, α] = 0}.



The part in degree 0 forms a classical Lie algebra (g0, [, ]),which can be integrated (under some convergence hypothesis)using the Baker–Campell–Hausdorff

BCH(x , y) := ln(exp x exp y) = x + y +1

2[x , y ]+

1

12[x , [x , y ]] +

1

12[[x , y ], y ] + . . .

.♣ This formula is:

1. Associative: BCH(x ,BCH(y , z)) = BCH(BCH(x , y), z)

2. Unitary: BCH(x , 0) = BCH(0, x) = x .



This formula endows g0 with a group structureG := (g0,BCH , 0) called the Gauge group, acting on theelements of Maurer–Cartan.

♣ This action corresponds to the wanted ’equivalences’ !



Comment

It is good to have a differential graded Lie algebra to code adeformation problem.

The study of deformations (infinitesimal or formal) becomeseasy:

MC (g) is an algebraic variety (intersection of quadrics) andthe dimension of the tangent space at a given point α tells ushow much we can locally deform the corresponding structure.

Exercice: prove this ! (someone for next week?)(And this is what lead Deligne to his philosophy ... )



Example from Algebraic Geometry- after Drinfeld,

Kontsevitch, Hinich, Manetti, Pridham, Lurie,...

Deformation theory studies infinitesimal variations of objectscoming from algebraic-geometry.Notation• X be an algebraic object over k . ()• Set: category of sets (in a fixed universe).• Artk category of local Artinian k-algebras with residue field k(with morphisms the local homomorphisms). If A ∈ Artk , mA

is the maximal ideal.• ˆArtk its pro-category. It coincides with complete localNoetherian commutative k-algebras.



Definition (Deformation functor)An infinitesimal deformation of X is a local deformation of Xparametrized by an algebra R in the category Artk .

Let DefX (R) be the set of equivalence classes of localdeformations of X , parametrized by R .

The assignment R 7→ DefX (R) defines a functor:

DefX : Artk → Sets

called the classical deformation functor.

Remark: A formal deformation of X is a deformationparametrized by an algebra R in Artk .



How does this work?

Take X0 a smooth algebraic variety over k . We want todeform this.

The functorDefX0 : Art∗k → Set

maps an Artin (augmented) algebra R to a set of isomorphismclasses of pairs (X , u), where X is a scheme, flat overSpec(R), and u is an isomorphism of algebraic varietiesu : X ⊗R k ∼= X0.



The classical formalism (Grothendieck–Mumford–Schlessinger)of infinitesimal deformation theory is described by theprocedure:

Deformation problem→ Deformation functor/groupoid



Lost information!

The above picture is easy, but information is lost !

Deligne, Drinfeld, Quillen, Kontsevich, Schlessinger–Stasheff,Goldman–Millson and many others suggested the following:

Deformation problem→ DGLA→ Deformation functor/groupoid



How does it work?

• Let L be a DGLA.• R a local Artin algebra augmented of maximal ideal m.

We form the set MC (L⊗k m) of elements x of degree 1 inL⊗m verifying the Maurer–Cartan equation.

This gives the functor

XL : Art∗k → Sets,

the deformation problem associated to L.♣ When a problem can be encoded by XL the knowledge of Lgives a lot of information.



Example• For example, the tangent space is H1(L), and H2(L) is thespace of obstructions to smoothness of XL.

• If the differential of L is nul then the singularities of XL arequadratic and if the bracket is nul then XL is formally smooth.



Thanks !



no emie c. combe mpi mis wednesday 29/04 at 17:00...classical deformation theory (for algebraic...

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