arXiv research

In this paper we consider deformations of an algebroid stack on an etale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild …

The paper studies deformations of Filippov algebroids using cohomology and DGLA.

Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.

Formula decomposes multiplicative forms on Poisson groupoids into two parts.

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system V on it. We consider a differential graded Lie algebra (DGLA) of forms with holomorphic logarithmic singularities and vanishing residues. We construct a spectral sequence corresponding to the ant…

Study vector fields and derivations on differentiable stacks.

The derived bracket of a Maurer-Cartan element in a differential graded Lie algebra (DGLA) is well-known to define a differential graded Leibniz algebra. It is also well-known that a Lie infinity morphism between DGLAs maps a Maurer-Cartan element to a Maurer-Cartan element. Given a Lie-infinity morphism, a Maurer-elem…

The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differentia…

We identify the 2-groupoid of deformations of a gerbe on a smooth manifold with the Deligne 2-groupoid of a corresponding twist of the DGLA of local Hochschild cochains on infinite jets of smooth functions.

We investigate the formal deformation theory of (rank 1) branes on generalized complex (GC) manifolds. This generalizes, for example, the deformation theory of a complex submanifold in a fixed complex manifold. For each GC brane $\mathcal{B}$ on a GC manifold $(X,\mathbb{J})$, we construct a formal (pointed) groupoid $…

Paper studies symmetries in singular foliations using Lie $\infty$-morphisms.

Deformations of a Courant Algebroid E and its Dirac subbundle A have been widely considered under the assumption that the pseudo-Euclidean metric is fixed. In this paper, we attack the same problem in a setting that allows the pseudo-Euclidean metric to deform. Thanks to Roytenberg, a Courant algebroid is equivalent to…

Study on deforming complex manifolds and Higgs bundles.

We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids. When the action is strict, we show that it integrates to …

The associator of a non-associative algebra is the curvature of the Hochschild quasi-complex. The relationship ``curvature-associator'' is investigated. Based on this generic example, we extend the geometric language of vector fields to a purely algebraic setting, similar to the context of Gerstenhaber algebras. We int…

This research extends Lie algebra actions to singular foliations.

We show how to compute the spectral flow of the odd signature operator $\pm *d_{a_t}-d_{a_t}*$ along an analytic path of flat connections $a_t$ on a bundle over a closed odd-dimensional manifold in terms of Massey products in the DGLA of bundle-valued differential forms. To obtain this information, we set up a sequence…

This thesis studies deformations of VB-algebroids and VB-groupoids in Lie algebroid and groupoid categories.

$\newcommand{\poly}{_{\operatorname{poly}}^{\bullet}}\newcommand{\td}{(\operatorname{td}_{L/A}^{\nabla})^{\frac{1}{2}}}\newcommand{\cx}[1]{\operatorname{tot}\big(Γ(Λ^\bullet A^\vee)\otimes_R\mathcal{#1}\poly\big)}\newcommand{\cy}[1]{\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{#1}\poly)}$Kontsevich's formality the…

This article gives an exposition of the deformation theory for pairs $(X, E)$, where $X$ is a compact complex manifold and $E$ is a holomorphic vector bundle over $X$, adapting an analytic viewpoint à la Kodaira-Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer--Cartan equa…

The purpose of this paper is to investigate shifted $(+1)$ Poisson structures in context of differential geometry. The relevant notion is shifted $(+1)$ Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equivalence of quasi-Poisson groupoids. Thus isomorphism classes of $(+1)$…

To any $\mathfrak{g}$-manifold $M$ are associated two dglas $\operatorname{tot}\big(Λ^{\bullet} \mathfrak{g}^\vee \otimes_{\Bbbk} T_{\operatorname{poly}}^{\bullet} \big)$ and $\operatorname{tot} \big(Λ^{\bullet} \mathfrak{g}^\vee\otimes_{\Bbbk} D_{\operatorname{poly}}^{\bullet} \big)$, whose cohomologies $H_{\operatorn…