TY - JOUR

T1 - Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

AU - Seol, Seokbong

AU - Stiénon, Mathieu

AU - Xu, Ping

N1 - Funding Information:
We would like to thank Ruggero Bandiera, Camille Laurent-Gengoux, Hsuan-Yi Liao, Rajan Mehta and Luca Vitagliano for fruitful discussions and useful comments. Seokbong Seol is grateful to the Korea Institute for Advanced Study for its hospitality and generous support.
Funding Information:
Research partially supported by NSF Grants DMS-1707545 and DMS-2001599.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/4

Y1 - 2022/4

N2 - This paper is devoted to the study of the relation between ‘formal exponential maps,’ the Atiyah class, and Kapranov L∞[1] algebras associated with dg manifolds in the C∞ context. We prove that, for a dg manifold, a ‘formal exponential map’ exists if and only if the Atiyah class vanishes. Inspired by Kapranov’s construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an L∞[1] algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative with respect to the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold (TX0,1[1],∂¯) arising from a complex manifold X, we prove that this L∞[1] algebra structure is quasi-isomorphic to the standard L∞[1] algebra structure on the Dolbeault complex Ω0,∙(TX1,0).

AB - This paper is devoted to the study of the relation between ‘formal exponential maps,’ the Atiyah class, and Kapranov L∞[1] algebras associated with dg manifolds in the C∞ context. We prove that, for a dg manifold, a ‘formal exponential map’ exists if and only if the Atiyah class vanishes. Inspired by Kapranov’s construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an L∞[1] algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative with respect to the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold (TX0,1[1],∂¯) arising from a complex manifold X, we prove that this L∞[1] algebra structure is quasi-isomorphic to the standard L∞[1] algebra structure on the Dolbeault complex Ω0,∙(TX1,0).

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U2 - 10.1007/s00220-021-04265-x

DO - 10.1007/s00220-021-04265-x

M3 - Article

AN - SCOPUS:85126255760

SN - 0010-3616

VL - 391

SP - 33

EP - 76

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -