Derived intersections and free dg-Lie algebroids

Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid

We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra $\mathcal O $ and homotopy equivalence classes of negatively graded Lie $\infty $-algebroids over their resolutions (=acyclic Lie $\infty$-algebroids). This extends to a purely algebraic setting the construction of the universal $Q$-manifold of a locally real analytic singular foliation of Lavau-C.L.-Strobl. In particular, it makes sense for the universal Lie $\infty$-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algeb…

Behavior of the Torsion of the Differential Module of an Algebroid Curve under Quadratic Transformations

We study the difference between the lengths of the torsion of the differential modules of the local ring R of an algebroid curve and its first quadratic transform. The conjecture is that this difference should be positive if R is not regular. This is shown to be true in some cases, especially for stable complete intersections and for semigroup rings which are complete intersections.

Invariants of Lie algebroids

Several new invariants for Lie algebroids have been discovered recently. We give an overview of these invariants and establish several relationships between them.

关于抛物向量束 admits 抛物李代数束连接的准则

A criterion for parabolic vector bundles to admit a parabolic Lie algebroid connection

本文研究了抛物向量束在给定抛物结构下是否能配备抛物李代数束连接的条件，通过分析给出了必要且充分的判断标准。

Nonabelian holomorphic Lie algebroid extensions

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild-Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure.

Lie algebroids as $L_\infty$ spaces

In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid $L$-and the natural generalization to dg Lie algebroids-provides an (essentially unique) $L_\infty$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_\infty$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_\infty$ space. Indeed, this functor sends the …

全纯李代数丛联络的一个判据

A criterion for holomorphic Lie algebroid connections

该研究针对紧连通黎曼曲面上的全纯李代数丛，给出全纯向量丛容许其全纯联络的充要条件，区分非分裂与分裂情形分别得到对应结论。

Families and unfoldings of singular holomorphic Lie Algebroids

In this paper, we investigate families of singular holomorphic Lie algebroids on complex analytic spaces. We introduce and study a special type of deformation called unfoldings of Lie algebroids, which generalizes the theory of singular holomorphic foliations developed by T. Suwa. We show that a one to one correspondence between transversal unfoldings and holomorphic flat connections on a natural Lie algebroid on the bases exists.