Atiyah and Todd classes arising from integrable distributions

Chen, Zhuo; Xiang, Maosong; Xu, Ping

doi:10.1016/j.geomphys.2018.10.011

Cited by 10 publications

(14 citation statements)

References 11 publications

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“…Via this contraction, we give an explicit description of the Atiyah class of the dg manifold (A [1], d A ) (see Theorem 4.1). In particular, we obtain the Atiyah class of a bundle of Lie algebras (see Proposition 4.4), and rediscover the fact in [4] that the Atiyah class of the dg manifold arising from an integrable distribution F ⊆ T M is isomorphic to that of the Lie pair (T M , F ) introduced in [3] (see Proposition 4.6). Since scalar Atiyah classes and Todd classes are generated by Atiyah classes, we also prove that all scalar Atiyah classes and Todd classes of a regular Lie algebroid A respect the Atiyah sequence of A (see Proposition 5.5 and Proposition 5.8).…”

Section: Introductionmentioning

confidence: 89%

“…Remark 3.22. When the anchor ρ A is injective, i.e., A is identified with its characteristic distribution F = Im ρ A , the above contraction was explicitly constructed in [4] to compute Atiyah and Todd classes of integrable distributions.…”

Section: Combining With the Side Conditionsmentioning

confidence: 99%

“…[4]). Let F ⊆ T M be an integrable distribution, i.e., the tangent bundle of some regular foliation in M .…”

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confidence: 99%

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Atiyah and Todd classes of regular Lie algebroids

Xiang¹

2021

Preprint

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For any regular Lie algebroid A, the kernel K and the image F of its anchor map ρA, together with A itself fit into a short exact sequence, called Atiyah sequence, of Lie algebroids. We prove that Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids respect the Atiyah sequence, that is, the Atiyah and Todd classes of A restrict to the Atiyah and Todd classes of the bundle K of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution F ⊆ TM on the other hand. CONTENTS 1. Introduction 2. Atiyah and Todd classes of dg vector bundles 2.1. Dg manifolds and dg vector bundles 2.2. Atiyah and Todd classes 2.3. Invariance under contractions 3. Cohomology of regular Lie algebroids 3.1. The graded geometry of regular Lie algebroids 3.2. Contractions of tangent dg vector bundles 4. Atiyah classes of regular Lie algebroids 4.1. Atiyah classes 4.2. Examples 4.3. Functoriality with respect to Atiyah sequence 5. Scalar Atiyah and Todd classes 5.1. Scalar Atiyah classes 5.2. Todd classes 5.3. Application to locally splittable cases References

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Section: Introductionmentioning

confidence: 89%

Section: Combining With the Side Conditionsmentioning

confidence: 99%

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Atiyah and Todd classes of regular Lie algebroids

Xiang¹

2021

Preprint

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“…Atiyah classes of dg Lie algebroids and Lie pairs. In this section, we briefly recall Atiyah classes of dg vector bundles with respect to a dg Lie algebroid defined in [23] and Atiyah classes of Lie pairs defined in [8] (see [9] for the equivalence between the two types of Atiyah classes arising from integrable distributions), and show that both of them can be viewed as twisted Atiyah classes.…”

Section: Since the Mapδ ⊗mentioning

confidence: 99%

Kapranov’s construction of $\operatorname{sh}$ Leibniz algebras

Chen

Liu

Xiang

2020

Homology, Homotopy and Applications

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Motivated by Kapranov's discovery of an sh Lie algebra structure on the tangent complex of a Kähler manifold and Chen-Stiénon-Xu's construction of sh Leibniz algebras associated with a Lie pair, we find a general method to construct sh Leibniz algebras. Let A be a commutative dg algebra. Given a derivation of A valued in a dg module Ω, we show that there exist sh Leibniz algebra structures on the dual module of Ω. Moreover, we prove that this process establishes a functor from the category of dg module valued derivations to the category of sh Leibniz algebras over A .

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“…For polyvector fields, indeed it was already proved in [4,14] that there exists an isomorphism of Gerstenhaber algebras…”

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confidence: 95%

Hochschild cohomology of dg manifolds associated to integrable distributions

Chen,

Xiang,

2021

Preprint

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For the field K = R or C, and an integrable distribution F ⊆ T K M = TM ⊗ R K on a smooth manifold M , we study the Hochschild cohomology of the dg manifold (F [1], dF ) and establish a canonical isomorphism with the Hochschild cohomology of the transversal polydifferential operators of F . In particular, for the dg manifold (T 0,1 X [1], ∂) associated with a complex manifold X, we prove that it is canonically isomorphic to the Hochschild cohomology HH • (X) of the complex manifold. As an application, we show that the Kontsevich-Duflo type theorem for the dg manifold (T 0,1 X [1], ∂) implies the Kontsevich-Duflo theorem for complex manifolds.

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Atiyah and Todd classes arising from integrable distributions

Cited by 10 publications

References 11 publications

Atiyah and Todd classes of regular Lie algebroids

Atiyah and Todd classes of regular Lie algebroids

Kapranov’s construction of $\operatorname{sh}$ Leibniz algebras

Hochschild cohomology of dg manifolds associated to integrable distributions

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