The formal Kuranishi parameterization via the universal homological perturbation theory solution of the deformation equation

Johannes Huebschmann

doi:10.1515/gmj-2018-0054

Published by De Gruyter October 5, 2018

The formal Kuranishi parameterization via the universal homological perturbation theory solution of the deformation equation

Johannes Huebschmann

From the journal Georgian Mathematical Journal

https://doi.org/10.1515/gmj-2018-0054

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Abstract

Using homological perturbation theory, we develop a formal version of the miniversal deformation associated with a deformation problem controlled by a differential graded Lie algebra over a field of characteristic zero. Our approach includes a formal version of the Kuranishi method in the theory of deformations of complex manifolds.

Keywords: Deformation theory; deformation controlled by a differential graded Lie algebra; miniversal deformation; formal deformation; Kuranishi method

MSC 2010: 14D15; 13D10; 14B07; 14B12; 16S80; 32G05; 32G08; 32S60; 58K60

Dedicated to Tornike Kadeishvili

Funding source: Labex

Award Identifier / Grant number: ANR-11-LABX-0007-01

Funding statement: I gratefully acknowledge support by the CNRS and by the Labex CEMPI (ANR-11-LABX-0007-01).

A Appendix

Let U be an 𝔽-vector space.

A.1 Symmetric coalgebra

For j≥2, the j-th symmetric copower Sjc⁢[U]⊆U⊗j of U is the linear 𝔽-subspace of invariants under the canonical action, on U⊗j, of the symmetric group on j letters.

The graded symmetric 𝔽-coalgebra cogenerated by U is the graded coaugmented subcoalgebra of the graded 𝔽-tensor coalgebra {U⊗j}j∈ℕ cogenerated by U having 𝔽 as its homogeneous degree 0 and U as its homogeneous degree 1 constituents and, for j≥2, as homogeneous degree j constituent the j-th symmetric copower Sjc⁢[U] of U. The 𝔽-vector space U being concentrated in degree 1, the canonical projection from {Sjc⁢[U]}j∈ℕ to U yields the requisite cogenerating morphism of graded 𝔽-vector spaces. Totalization yields the symmetric 𝔽-coalgebra Sc⁢[U] cogenerated by U. For ℓ≥0, the coaugmentation filtration degree ℓ constituent S≤ℓc⁢[U]⊆Sc⁢[U] of Sc⁢[U] has, for j≤ℓ, as its homogeneous constituents the homogeneous constituents Sjc⁢[U] of Sc⁢[U].

Recall that the canonical diagonal map U→U⊕U induces an 𝔽-bialgebra structure on the symmetric 𝔽-algebra S⁢[U] and that multiplication by -1 on U yields an antipode such that S⁢[U] acquires an 𝔽-Hopf algebra structure. The canonical projection πU:S⁢[U]→U induces, via the universal property of Sc⁢[U], the canonical morphism S⁢[U]→Sc⁢[U], cf. (3.1), of graded 𝔽-coalgebras. This canonical morphism, viewed as an 𝔽-linear map, coincides with the 𝔽-linear map that underlies the canonical morphism S⁢[U]→Sc⁢[U] of graded 𝔽-algebras associated with the canonical injection U→Sc⁢[U] and, since the ground field 𝔽 has characteristic zero, this morphism is an isomorphism of graded 𝔽-Hopf algebras.

As in Subsection 3.1 above, choose an 𝔽-basis B of U. The canonical morphism Γ⁢[B]→Sc⁢[U] of 𝔽-coalgebras is an isomorphism, and the canonical map can:𝔽⁢[B]→Γ⁢[B] renders the diagram

commutative. The inverse of can:𝔽⁢[B]→Γ⁢[B] sends γj⁢(b) to 1j!⁢bj (j≥1) as b ranges over B and thereby introduces divided powers in S⁢[U].

Remark A.1.

The obvious forgetful functor from the category of general cocommutative 𝔽-coalgebras, i.e., cocommutative 𝔽-coalgebras not necessarily endowed with a coaugmentation, to that of 𝔽-vector spaces admits likewise a right adjoint [47, p. 129, Theorem 6.4.3]. Write this functor as Σc. Consider an 𝔽-vector space U. The 𝔽-coalgebra Σc⁢[U] comes with a canonical coaugmentation map, and Sc⁢[U] is the largest cocomplete 𝔽-subcoalgebra of Σc⁢[U] cogenerated by U, but the two coalgebras do not coincide unless U is the trivial vector space, cf. [4, 20]. For example, when U has dimension 1 and when z denotes the standard coordinate function on U, the symmetric 𝔽-coalgebra Sc⁢[U] amounts to the coalgebra that underlies the polynomial algebra 𝔽⁢[z] (as noted before), whereas Σc⁢[U] comes down to the 𝔽-coalgebra that underlies the algebra of rational functions in the variable z which are regular at the origin. For general U, in [47], see also [41, p. 26, p. 51], [42, Section 3, note on p. 720], the 𝔽-coalgebra Σc⁢[U] is referred to as the symmetric coalgebra on U.

A.2 Formal tangent space

Let C be an 𝔽-coalgebra, and let Coder⁡(C) denote the 𝔽-vector space of coderivations of C, endowed with the Lie bracket arising from the commutator of 𝔽-linear endomorphisms of C. More generally, for a C-bicomodule B, we denote by Coder⁡(B,C) the 𝔽-vector space of coderivations from B to C.

A morphism C→Sc⁢[U] of 𝔽-coalgebras induces an Sc⁢[U]-bicomodule structure on C. Recall that, in view of the universal property of the symmetric 𝔽-coalgebra Sc⁢[U] on U, the cogenerating surjection Sc⁢[U]→U induces an isomorphism Coder⁡(C,Sc⁢[U])→Hom⁡(C,U) of 𝔽-vector spaces. In particular, the cogenerating surjection Sc⁢[U]→U of 𝔽-vector spaces induces an 𝔽-vector space isomorphism

Coder⁡(Sc⁢[U])⟶Hom⁡(Sc⁢[U],U).

Now, the coaugmentation map η:𝔽→Sc⁢[U] turns the ground field 𝔽 into an Sc⁢[U]-bicomodule, and we use the notation 𝔽η for this Sc⁢[U]-bicomodule. The above observation entails that the cogenerating surjection Sc⁢[U]→U induces an isomorphism

Coder⁡(𝔽η,Sc⁢[U])⟶Hom⁡(𝔽,U)≅U

of 𝔽-vector spaces. Thus the 𝔽-vector space Coder⁡(𝔽η,Sc⁢[U]) is canonically isomorphic to the 𝔽-vector space of primitives in Sc⁢[U] relative to the canonical Hopf algebra structure of Sc⁢[U]. We refer to the space of primitives in Sc⁢[U] as the formal tangent space to U at 0. When 𝔽 is the field of real or that of complex numbers, this formal tangent space comes of course down to the ordinary tangent space to U at 0.

Let 𝔽ε denote the base field 𝔽, endowed with the Hom⁡(Sc⁢[U],𝔽)-module structure induced by the augmentation map ε:Hom⁡(Sc⁢[U],𝔽)→𝔽. The canonical map

(A.1)Coder⁡(𝔽η,Sc⁢[U])⟶Der⁡(Hom⁡(Sc⁢[U],𝔽),𝔽ε)

is an isomorphism. Under (A.1), a basis element of U goes to the associated operation of partial derivative. This justifies the terminology “formal tangent space” to U at 0.

Acknowledgements

I am indebted to Jim Stasheff for a number of most valuable comments on a draft of the paper.

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Received: 2018-04-03

Accepted: 2018-06-09

Published Online: 2018-10-05

Published in Print: 2018-12-01

The formal Kuranishi parameterization via the universal homological perturbation theory solution of the deformation equation

Abstract

A Appendix

A.1 Symmetric coalgebra

Remark A.1.

A.2 Formal tangent space

Acknowledgements

References

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