The deletion restriction method for plane curve germ complement

Local systems on analytic germ complements

We prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori. This is a generalization of the classical Monodromy Theorem stating that the eigenvalues of the monodromy on the cohomology of the Milnor fiber of a germ of a holomorphic function are roots of unity.

关于$E_6$型实焦散芽的补集

On the complement to a real caustic germ of type $E_6$

该研究证明了$E_6^\pm$型稳定拉格朗日映射芽的焦散补集的连通分支数量与拓扑性质，明确了不可缩分支原像的连通性及映射在各分支上的限制类型。

On homotopy types of complements of analytic sets and Milnor fibres

We prove that for any germ of complex analytic set in $\CC^n$ there exists a hypersurface singularity whose Milnor fibration has trivial geometric monodromy and fibre homotopic to the complement of the germ of complex analytic set. As an application we show an example of a quasi-homogeneous hypersurface singularity, with trivial geometric monodromy and simply connected and non-formal Milnor fibre.

Characteristic Cohomology I: Singularities of Given Type

For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of type $\mathcal{V}$, is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense so that $\mathcal{V}_0 = f_0^{-1}(\mathcal{V})$. For these singularities, we introduce "characteristic cohomology" to capture the contribution of the topology of $\mathcal{V}$ to that of $\mathcal{V}_0$, for the Milnor fiber (for $\mathcal{V}, 0$ a hypersurface), and complement and link of $\mathcal{V}_0$ (in the general case). The characteristic cohomology …

On the geometry and topology of partial configuration spaces of Riemann surfaces

We examine complements (inside products of a smooth projective complex curve of arbitrary genus) of unions of diagonals indexed by the edges of an arbitrary simple graph. We use Gysin models associated to these quasi-projective manifolds to compute pairs of analytic germs at the origin, both for rank 1 and 2 representation varieties of their fundamental groups, and for degree 1 topological Green--Lazarsfeld loci. As a corollary, we describe all regular surjections with connected generic fiber, defined on the above complements onto smooth complex curves of negative Euler characteristic. We show…

平面曲线通过其上函数环的Alexander多项式

The Alexander polynomial of a plane curve singularity via the ring of functions on it

证明了用解析函数芽环${\cal O}_{C}$表示平面曲线奇异点$C$的多元Alexander多项式$Δ^C$的两个公式。其中之一表示$Δ^C$的尺寸的一些因素对应于一个（多指标）过滤环${\cal O}_{C}$。另一个给出了Alexander多项式$Δ^C$的系数作为某些显式描述空间的欧拉示性数（对射影超平面排列的补充）。本文的最终版本将发表在杜克大学数学杂志上。