2025/09/30 by Ruggero Bandiera, Bandiera, Ruggero, Seokbong Seol +5
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2509.26341
openalex publication_date 2025/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given any Kähler manifold X, Kapranov discovered an L_∞[1] algebra structure on Ω0,\bulletX(T1,0X). Motivated by this result, we introduce, as a generalization of L_∞[1] algebras, a notion of L_∞[1] \mathfrakR-algebra, where \mathfrakR is a differential graded commutative algebra with unit. We show that standard notions (such as quasi-isomorphism and linearization) and results (including homotopy transfer theorems) can be extended to this context. For instance, we provide a linearization theorem. As an application, we prove that, given any DG Lie algebroid (L,QL) over a DG manifold (M,Q), there exists an induced L_∞[1] \mathfrakR-algebra structure on Γ(L), where \mathfrakR is the DG commutative algebra (C^∞(M),Q) -- its unary bracket is QL while its binary bracket is a cocycle representative of the Atiyah class of the DG Lie algebroid. This L_∞[1] \mathfrakR-algebra Γ(L) is linearizable if and only if the Atiyah class of the DG Lie algebroid vanishes. However, the L_∞[1] (\mathbbK-)algebra Γ(L) induced by this L_∞[1] \mathfrakR-algebra is necessarily homotopy abelian. As a special case, we prove that, given any complex manifold X, the Kapranov L_∞[1] \mathfrakR-algebra Ω0,\bulletX(T1,0X), where \mathfrakR is the DG commutative algebra (Ω0,\bulletX,∂), is linearizable if and only if the Atiyah class of the holomorphic tangent bundle TX vanishes. Nevertheless, the induced L_∞[1] ℂ-algebra structure on Ω0,\bulletX(T1,0X) is necessarily homotopy abelian.