Dolbeault dga and L∞-algebroid of the formal neighborhood

Abstract

We continue the study the Dolbeault dga of the formal neighborhood of an arbitary closed embedding of complex manifolds previously defined by the author in [Yua]. The special case of the diagonal embedding has been studied in [Yu15]. We describe the Dolbeault dga explicitly in terms of the formal differential geometry of the embedding. Moreover, we show that the Dolbeault dga is the completed Chevalley-Eilenberg dga an L ∞ -algebroid structure on the shifted normal bundle of the submanifold. This generlizes the result of Kapranov on the diagonal embedding and Atiyah class.

Formal moduli problems and formal derived stacks

“…Similar results had been previously proven by Bhargav Bhatt in a slightly different formulation (see [1]). We also refer to the work of Shilin Yu [33], who obtained parallel results in the complex analytic context. Note that this time, the derived scheme Ŷ doesn't live anymore over X (although it still lives under X).…”

Lie algebroids are curved Lie algebras

“…From the point of view of X, one can view Y as the quotient of X by the formal groupoid X × Y X ⇒ X. Consequently, Y should give rise to a Lie algebroid on X (see e.g. [CCT14,GR17,Yu17]). In the algebraic context, the third author proved [Nui19b] that dg-Lie algebroids are indeed equivalent to formal moduli problems under X = Spec(A) (see also [CG18]), for A a connective commutative differential graded algebra (cdga).…”

Derived intersections and free dg-Lie algebroids

“…Let us now focus on non-quantized cycles. This is a more difficult task that has been completed in the groundbreaking paper [4], and independently in [21]. The main idea in [4] is that the self-derived intersection of X in Y is an algebraic counterpart of the set of continuous paths in Y whose beginning and end points lie in X.…”