Deformations of Affine Varieties and the Deligne Crossed Groupoid

Descent of Deligne groupoids

To any non-negatively graded dg Lie algebra $g$ over a field $k$ of characteristic zero we assign a functor $Σ_g: art/k \to Kan$ from the category of commutative local artinian $k$-algebras with the residue field $k$ to the category of Kan simplicial sets. There is a natural homotopy equivalence between $Σ_g$ and the Deligne groupoid corresponding to $g$. The main result of the paper claims that the functor $Σ$ commutes up to homotopy with the "total space" functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves…

Deligne Cohomology for Orbifolds, discrete torsion and B-fields

In this paper we introduce the concept of Deligne cohomology of an orbifold. We prove that the third Deligne cohomology group of a smooth étale groupoid classify gerbes with connection over the groupoid. We argue that the $B$-field and the discrete torsion in type II superstring theories are special kinds of gerbes with connection, and finally, for each one of them, using Deligne cohomology we construct a flat line bundle over the inertia groupoid, namely a Ruan inner local in the case of an orbifold.

Combinatorial Descent Data for Gerbes

We consider descent data in cosimplicial crossed groupoids. This is a combinatorial abstraction of the descent data for gerbes in algebraic geometry. The main result is this: a weak equivalence between cosimplicial crossed groupoids induces a bijection on gauge equivalence classes of descent data.

具有反演的拓扑群oid与实代数堆

Topological groupoids with involution and real algebraic stacks

本文研究具有反演的拓扑群oid的固定点子群oid，将其与实代数堆的实定位联系起来，提出Smith-Thom型猜想以推广拓扑空间的Smith-Thom不等式。

Curved L-infinity algebras and lifts of torsors

Consider an extension of finite dimensional nilpotent Lie algebras $0 \to \mathfrak{h} \to \tilde{\mathfrak{g}} \to \mathfrak{g} \to 0$ (over a field $k$ of characteristic zero) corresponding to an extension of unipotent algebraic groups $1 \to H \to \tilde{G} \to G \to 1$. For a $G$-torsor $P$ on an algebraic variety $X$ over $k$, we study the problem of lifting $P$ to $\widetilde{G}$-torsor $\widetilde{P}$. Fixing a trivialization of $P$ on open subsets of an affine cover, we give the Cech complex of $\mathfrak{h}$-valued functions the structure of a curved $L_\infty$-algebra and define a cu…

准射影层映射的变形

Deformations of morphisms of coherent sheaves

本文推广Hinich的Deligne群oid下降定理到dgLa不具有负余维数的情况，研究了相干层映射的微分变形，并利用Deligne函子和Thom-Whitney总化来控制这些变形。