Formality theorem for $ \mathfrak{g}$-manifolds

Hsuan-Yi Liao; Mathieu Stiénon; Ping Xu

doi:10.1016/j.crma.2017.03.008

Differential geometry

Formality theorem for

g

-manifolds
[Théorème de formalité pour les

g

-variétés]

Présenté par : Michèle Vergne

Hsuan-Yi Liao ¹ ; Mathieu Stiénon ¹ ; Ping Xu ¹

¹ Department of Mathematics, Pennsylvania State University, United States

Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 582-589.

Résumé
Abstract

À toute $g$ -variété M sont associées deux algèbres de Lie différentielles graduées $tot (Λ^{•} g^{\lor} \otimes_{k} T_{poly}^{•} (M))$ et $tot (Λ^{•} g^{\lor} \otimes_{k} D_{poly}^{•} (M))$ , dont les cohomologies $H_{CE}^{•} (g, T_{poly}^{•} (M) \overset{0}{\to} T_{poly}^{• + 1} (M))$ et $H_{CE}^{•} (g, D_{poly}^{•} (M) \overset{d_{H}}{\to} D_{poly}^{• + 1} (M))$ sont des algèbres de Gerstenhaber. Nous obtenons un théorème de formalité pour les $g$ -variétés : il existe un quasi-isomorphisme $Φ : tot (Λ^{•} g^{\lor} \otimes_{k} T_{poly}^{•} (M)) \to tot (Λ^{•} g^{\lor} \otimes_{k} D_{poly}^{•} (M))$ d'algèbres $L_{\infty}$ dont le premier « coefficient de Taylor » (1) est égal à l'application de Hochschild–Kostant–Rosenberg tordue par la racine carrée du cocycle de Todd de la $g$ -variété M et (2) induit un isomorphisme d'algèbre de Gerstenhaber au niveau des cohomologies. Par conséquent, l'application de Hochschild–Kostant–Rosenberg tordue par la racine carrée de la classe de Todd de la $g$ -variété M est un isomorphisme d'algèbres de Gerstenhaber de $H_{CE}^{•} (g, T_{poly}^{•} (M) \overset{0}{\to} T_{poly}^{• + 1} (M))$ sur $H_{CE}^{•} (g, D_{poly}^{•} (M) \overset{d_{H}}{\to} D_{poly}^{• + 1} (M))$ .

With any $g$ -manifold M are associated two dglas $tot (Λ^{•} g^{\lor} \otimes_{k} T_{poly}^{•} (M))$ and $tot (Λ^{•} g^{\lor} \otimes_{k} D_{poly}^{•} (M))$ , whose cohomologies $H_{CE}^{•} (g, T_{poly}^{•} (M) \overset{0}{\to} T_{poly}^{• + 1} (M))$ and $H_{CE}^{•} (g, D_{poly}^{•} (M) \overset{d_{H}}{\to} D_{poly}^{• + 1} (M))$ are Gerstenhaber algebras. We establish a formality theorem for $g$ -manifolds: there exists an $L_{\infty}$ quasi-isomorphism $Φ : tot (Λ^{•} g^{\lor} \otimes_{k} T_{poly}^{•} (M)) \to tot (Λ^{•} g^{\lor} \otimes_{k} D_{poly}^{•} (M))$ whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the $g$ -manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the $g$ -manifold M is an isomorphism of Gerstenhaber algebras from $H_{CE}^{•} (g, T_{poly}^{•} (M) \overset{0}{\to} T_{poly}^{• + 1} (M))$ to $H_{CE}^{•} (g, D_{poly}^{•} (M) \overset{d_{H}}{\to} D_{poly}^{• + 1} (M))$ .

Reçu le : 2017-01-20
Accepté le : 2017-03-13
Publié le : 2017-03-30

DOI : 10.1016/j.crma.2017.03.008

Affiliations des auteurs :

Hsuan-Yi Liao ¹ ; Mathieu Stiénon ¹ ; Ping Xu ¹

¹ Department of Mathematics, Pennsylvania State University, United States

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     title = {Formality theorem for $ \mathfrak{g}$-manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {582--589},
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     doi = {10.1016/j.crma.2017.03.008},
     language = {en},
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Hsuan-Yi Liao; Mathieu Stiénon; Ping Xu. Formality theorem for $ \mathfrak{g}$-manifolds. Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 582-589. doi : 10.1016/j.crma.2017.03.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.03.008/

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^☆ Research partially supported by NSF grants DMS-1406668 and DMS-1101827, and NSA grant H98230-14-1-0153.

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