arXiv subjects

Seokbong Seol

Publications and source records attributed to Seokbong Seol.

6 recordsLinked to original sources

arXiv2026

Linearisation, splitting property and homotopy algebras

We study the linearisation of formal vector fields and homotopy algebras using graded coalgebras. We prove that a formal vector field is linearisable if and only if it satisfies a splitting property, and we give an explicit recursive construction of a linearising coordinate transformation. This provides a direct coalgebraic proof of Bandiera's characterisation of linearisable, equivalently homotopy abelian, $L_{\infty}[1]$ algebras. We establish an associative analogue, characterising linearisability of $A_{\infty}[1]$ algebras by a splitting property for one-sided Hochschild cochains. We also examine the corresponding splitting property for standard Hochschild cochains and show that it is equivalent to $A_{\infty}[1]$ commutativity in the sense of Briggs and Gélinas. Finally, we establish a corresponding criterion for formal endomorphisms and give sufficient conditions for splitting that recover Basto-Gonçalves' linearisation theorem for admissible formal vector fields.

math.DG↗

arXiv2026

The Atiyah class of DG manifolds of amplitude $+1$

A DG manifold of amplitude $+1$ encodes the derived intersection of a section $s$ and the zero section of a vector bundle $E$. In this paper, we compute the Atiyah class of DG manifolds of amplitude $+1$. In particular, we show that the Atiyah class vanishes if and only if the intersection of $s$ with the zero section is a clean intersection. As an application, we study the Atiyah class of DG manifolds that encodes the derived intersection of two smooth manifolds.

math.DG↗

arXiv2025

Atiyah class of DG manifolds of positive amplitude

Behrend, Liao, and Xu showed that differential graded (DG) manifolds of positive amplitude forms a category of fibrant objects. In particular, this ensures that notion of derived intersection -- more generally, homotopy fibre product -- is well-defined up to weak equivalences. We prove that the Atiyah and Todd classes of DG manifolds of positive amplitude are invariant under the weak equivalences. As an application, we study Hochschild cohomology of DG manifolds of positive amplitude defined using poly-differential operators, which is compatible with Kontsevich formality theorem and Duflo--Kontsevich-type theorem established by Liao, Stiénon and Xu. We prove that this Hochschild cohomology is invariant under weak equivalences.

math.DG↗

arXiv2025

Kapranov $L_{\infty}[1]$ algebras

Given any Kähler manifold $X$, Kapranov discovered an $L_\infty[1]$ algebra structure on $Ω^{0,\bullet}_X(T^{1,0}_X)$. Motivated by this result, we introduce, as a generalization of $L_\infty[1]$ algebras, a notion of $L_\infty[1]$ $\mathfrak{R}$-algebra, where $\mathfrak{R}$ 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 $(\mathcal{L},Q_{\mathcal{L}})$ over a DG manifold $(\mathcal{M},Q)$, there exists an induced $L_\infty[1]$ $\mathfrak{R}$-algebra structure on $Γ(\mathcal{L})$, where $\mathfrak{R}$ is the DG commutative algebra $(C^\infty(\mathcal{M}),Q)$ -- its unary bracket is $Q_{\mathcal{L}}$ while its binary bracket is a cocycle representative of the Atiyah class of the DG Lie algebroid. This $L_\infty[1]$ $\mathfrak{R}$-algebra $Γ(\mathcal{L})$ is linearizable if and only if the Atiyah class of the DG Lie algebroid vanishes. However, the $L_\infty[1]$ ($\mathbb{K}$-)algebra $Γ(\mathcal{L})$ induced by this $L_\infty[1]$ $\mathfrak{R}$-algebra is necessarily homotopy abelian. As a special case, we prove that, given any complex manifold $X$, the Kapranov $L_\infty[1]$ $\mathfrak{R}$-algebra $Ω^{0,\bullet}_X(T^{1,0}_X)$, where $\mathfrak{R}$ is the DG commutative algebra $(Ω^{0,\bullet}_X,\bar{\partial})$, is linearizable if and only if the Atiyah class of the holomorphic tangent bundle $T_X$ vanishes. Nevertheless, the induced $L_\infty[1]$ $\mathbb{C}$-algebra structure on $Ω^{0,\bullet}_X(T^{1,0}_X)$ is necessarily homotopy abelian.

math.DG↗

arXiv2023

Keller admissible triples and Duflo theorem

The present paper is devoted to the study of Keller admissible triples. We prove that a Keller admissible triple induces an isomorphism of Gerstenhaber algebras between the Hochschild cohomologies of direct-sum type of the pair of differential graded algebras bound to one another by the admissible triple. As an application, we give a new concrete proof of the Duflo--Kontsevich theorem for finite-dimensional Lie algebras.

math.QA↗

arXiv2021

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

math.DG↗