Relative Calabi-Yau completions

Abstract

We generalize Keller's construction [38] of deformed n-Calabi-Yau completions to the relative contexts. This gives a universal construction which extends any given DG functor F : A → B to a DG functor F : Ã → B, together with a family of deformations of F parametrized by relative negative cyclic homology classes [η] ∈ HC − n−2 (B, A). We show that, under a finiteness condition, these extensions have canonical relative n-Calabi-Yau structures in the sense of [8].This is applied to give a construction which associates a DG category A (N, M ; Φ) to a pair (N, M ) consisting of a manifold N and an embedded submanifold M of codimension ≥ 2, together with a trivialization Φ of the unit normal bundle of M in N . In the case when (N, M ) is the pair consisting of a set of n-points in the interior of the 2-dimensional disk, A (N, M ; Φ) is the multiplicative preprojective algebra [16] with non-central parameters. In the case when (N, M ) is the pair consisting of a link L in R 3 , A (N, M ; Φ) is the link DG category [5] that extends the Lengendrian DG algebra [45,46,24] of the unit conormal bundle ST * L (R 3 ) ⊂ ST * (R 3 ). We show that, when M ⊂ N has codimension 2, then the category of finite dimensional modules over the 0-th homology H 0 ( A (N, M ; Φ) ) of this DG category is equivalent to the category of perverse sheaves on N with singularities at most along M .

Exact Hochschild extensions and deformed Calabi-Yau completions

“…So we obtain a bijection Φ : T † → Π, which is a graded K-algebra morphism and compatible with differentials, i.e., a dg K-algebra isomorphism. By Theorem 5, i.e., [46,Theorem 3.17], Π n (A † , α † ), and thus T n (A, α) † , is an almost exact n-Calabi-Yau dg algebras.…”

“…More general, for a Hochschild class [α] ∈ HH n−2 (A), he introduced its deformed Calabi-Yau completion Π n (A, α) called derived preprojective algebra as well [25]. If [α] ∈ HH n−2 (A) is an almost exact Hochschild homology class, i.e., it is the image of a negative cyclic homology class, then Π n (A, α) is an almost exact Calabi-Yau algebra [46]. Therefore, Ginzburg dg algebras associated to quivers with potential are Calabi-Yau dg algebras.…”

Relative cluster categories and Higgs categories

“…Notice that the morphism µ ∨ G → Σ n−1 ν G is invertible if and only if its associated morphism of triangles (3) is invertible. We point out that condition b) is not imposed by Brav-Dyckerhoff [8] but is imposed by Yeung [49].…”

“…Let G : B → A be a dg functor between finitely cellular type dg categories. By [49,Remark 4.19], we can assume that B and A are finitely cellular and G : B → A is a semi-free extension, i.e. there is a finite graded quiver Q and a subquiver F ⊆ Q such that the underlying graded k-category of B and A are isomorphic to T kF 0 (kF 1 ) and T kQ 0 (kQ 1 ), respectively.…”

“…In particular, if the dg algebra B is zero, then A is n-Calabi-Yau as a bimodule. A canonical way to produce relative left Calabi-Yau structures is the deformed relative Calabi-Yau completion which was introduced by Wai-kit Yeung (see [49]). This generalized Keller's construction [34] of deformed n-Calabi-Yau completions to the relative context.…”