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 $L_\infty[1]$ 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 $L_{\mathrm{\infty }}[1]$ 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 $(T_X^{0,1}[1],\bar{\mathrm{\partial }})$ arising from a complex manifold $X$, we prove that this $L_{\mathrm{\infty }}[1]$ algebra structure is quasi-isomorphic to the standard $L_{\mathrm{\infty }}[1]$ algebra structure on the Dolbeault complex ${\mathrm{\Omega }}^{0,\bullet }(T_X^{1,0})$.