Mathieu Stienon: Formal exponential maps and the Atiyah class of dg manifolds

Abstract

Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to Poincaré-Birkhoff-Witt isomorphisms and the complete symbols of differential operators. It turns out that these formal exponential maps can be extended to the context of graded manifolds. For dg manifolds, the formal exponential maps need not be compatible with the homological vector field and the incompatibility is captured by a cohomology class reminiscent of the Atiyah class of holomorphic vector bundles. Indeed, the space of vector fields on a dg manifold carries a natural \(L_\infty\) algebra structure whose binary bracket is a cocycle representative of the Atiyah class of the dg manifold. In particular, the de Rham complex associated with a foliation carries an \(L_\infty\) algebra structure akin to the \(L_\infty\) algebra structure on the Dolbeault complex of a Kähler manifold discovered by Kapranov in his work on Rozansky-Witten invariants.