Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0204176

Homotopical variations and high-dimensional Zariski-van Kampen theorems

D. Chéniot·C. Eyral
View original source ↗Read source document ↗Export metadata ↓

Abstract

In 1933, van Kampen described the fundamental groups of the complements of plane complex projective algebraic curves. Recently, Chéniot-Libgober proved an analogue of this result for higher homotopy groups of the complements of complex projective hypersurfaces with isolated singularities. Their description is in terms of some "homotopical variation operators". We generalize here the notion of "homotopical variation" to (singular) quasi-projective varieties. This is a first step for further generalizations of van Kampen's theorem. A conjecture, with a first approach, is stated in the special case of non-singular quasi-projective varieties.

Explore related subjects

math.AGmath.ATmath.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

D. Chéniot, C. Eyral. 2002-04-13. Homotopical variations and high-dimensional Zariski-van Kampen theorems. https://arxiv.org/abs/math/0204176

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Nodal degeneration of chiral algebras I: Global structure and gluing formula

We define a natural extension of a universal factorization algebra $\mathcal{A}$ to families of stable punctured curves, by integrating over all semistable modifications. We prove that the resulting sheaf of factorization homology satisfies a natural gluing formula, by tensoring over a certain derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, generalizing the Verlinde formula for gluing of conformal blocks.

math.AG↗