约化共振概型与陈秩

Chen ranks and resonance

The Chen groups of a group $G$ are the lower central series quotients of the maximal metabelian quotient of $G$. Under certain conditions, we relate the ranks of the Chen groups to the first resonance variety of $G$, a jump locus for the cohomology of $G$. In the case where $G$ is the fundamental group of the complement of a complex hyperplane arrangement, our results positively resolve Suciu's Chen ranks conjecture. We obtain explicit formulas for the Chen ranks of a number of groups of broad interest, including pure Artin groups associated to Coxeter groups, and the group of basis-conjugatin…

有效的陈秩猜想

The effective Chen ranks conjecture

本文证明了陈秩猜想的有效版本，并将其应用于超平面排列，同时提出了Koszul模的精确通用消失猜想。

Chen ranks and resonance varieties of the upper McCool groups

The group of basis-conjugating automorphisms of the free group of rank $n$, also known as the McCool group or the welded braid group $PΣ_n$, contains a much-studied subgroup, called the upper McCool group $PΣ_n^+$. Starting from the cohomology ring of $PΣ_n^+$, we find, by means of a Gröbner basis computation, a simple presentation for the infinitesimal Alexander invariant of this group, from which we determine the resonance varieties and the Chen ranks of the upper McCool groups. These computations reveal that, unlike for the pure braid group $P_n$ and the full McCool group $PΣ_n$, the Chen r…

Chen-Ruan cohomology of some moduli spaces, II

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξa holomorphic line bundle on it such that r is not a divisor of degree(ξ). Let {\mathcal M}_ξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γwe will denote the group of line bundles L over X such that $L^{\otimes r}$ is trivial. This group Γacts on {\mathcal M}_ξ(r). We compute the Chen-Ruan cohomology of the corresponding orbifold.

Flat connections and resonance varieties: from rank one to higher ranks

Given a finitely-generated group $π$ and a linear algebraic group $G$, the representation variety Hom$(π,G)$ has a natural filtration by the characteristic varieties associated to a rational representation of $G$. Its algebraic counterpart, the space of $\mathfrak{g}$-valued flat connections on a commutative, differential graded algebra $(A,d)$ admits a filtration by the resonance varieties associated to a representation of $\mathfrak{g}$. We establish here a number of results concerning the structure and qualitative properties of these embedded resonance varieties, with particular attention t…

二阶向量丛在椭圆曲线上的共振

Resonance of rank-two vector bundles over elliptic curves

研究二阶向量丛在椭圆曲线上的共振体，通过分析共振的平坦分层结构，探讨Grassmann流形线性截面的性质。