The Zariski-Lefschetz principle for higher homotopy groups of nongeneric pencils

Connectivity via nongeneric pencils

We use slicing by nongeneric pencils of hypersurfaces and prove a new theorem of Lefschetz type for singular non compact spaces, at the homotopy level. As applications, we derive results on the topology of the fibres of polynomial functions or the topology of complements of hypersurfaces in $\mathbb C^n$.

勒夫谢茨束的特征簇上的单值群作用

Monodromy action on character varieties for Lefschetz pencils

该数学研究确定了来自勒夫谢茨束的单值群在黎曼曲面基本群的特征簇上的作用性质，解答了Katzarkov等人的相关猜想。

Zariski-van Kampen theorems for singular varieties --- an approach via the relative monodromy variation

The classical Zariski-van Kampen theorem gives a presentation of the fundamental group of the complement of a complex algebraic curve in $\mathbb{P}^2$. The first generalization of this theorem to singular (quasi-projective) varieties was given by the first author. In both cases, the relations are generated by the standard monodromy variation operators associated with the special members of a generic pencil of hyperplane sections. In the present paper, we give a new generalization in which the relations are generated by the relative monodromy variation operators introduced by D. Chéniot and th…

Birational geometry of Fano hypersurfaces of index two

We prove that every non-trivial structure of a rationally connected fibre space (and so every structure of a Mori-Fano fibre space) on a general (in the sense of Zariski topology) hypersurface of degree $M$ in the $(M+1)$-dimensional projective space for $M\geq 14$ is given by a pencil of hyperplane sections. In particular, the variety $V$ is non-rational and its group of birational self-maps coincide with the group of biregular automorphisms and is therefore trivial. The proof is based on the techniques of the method of maximal singularities and the inversion of adjunction.

一般纤维型曲线的补集的基本群

The fundamental group of the complement of a generic fiber-type curve

本文刻画了无分量线束中有限多条一般纤维并集补集的基本群，指出其曾出现在 Eyral 和 Oka 的纤维型曲线例子中，并由此得到处理 Zariski 交换性问题的一个准则。