Dg manifolds, formal exponential maps and homotopy Lie algebras

Abstract

This paper is devoted to the study of the relation between `formal exponential maps,' the Atiyah class, and Kapranov $L_{\infty }[1]$ algebras associated with dg manifolds in the $C^{\infty }$ context. Given a dg manifold, we prove that 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 $L_{\infty }[1]$ algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative w.r.t. 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 $(T_X^{0,1}[1],\bar{\partial })$ arising from a complex manifold $X$, we prove that this $L_{\infty }[1]$ algebra structure is quasi-isomorphic to the standard $L_{\infty }[1]$ algebra structure on the Dolbeault complex ${\Omega }^{0,\bullet }(T_X^{1,0})$.

arXiv commentMinor improvement; to appear in Communications in Mathematical Physics

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Cited by in corpus (4)

• Poincaré--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Basic curvature and the Atiyah cocycle in gauge theory
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs