Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

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).