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Andreotti-Frankel-Hamm theorem for morphisms of algebraic varieties

Dmitry Kerner

TL;DR

The paper proves a relative version of the Andreotti-Frankel-Hamm theorem for morphisms $f:X\to B$ with $B$ a complex affine variety of $\dim_\mathbb{C} B=n$, by constructing a deformation-retraction of $B$ that lifts to a deformation-retraction of the morphism and keeps $\dim_\mathbb{R}$ of the total space contraction bounded by $n$. The method is an iterative, discriminant-driven strategy that compactifies $f$ to a proper morphism and applies the AFKH theorem on the base while attaching low-dimensional cells to extend the deformation to the total space. Because the morphism is not assumed to be locally trivial, the horizontal version is nontrivial and the authors focus on a base deformation that lifts coherently to $X$. The results have immediate applications to families in Algebraic Geometry and Singularity Theory, including insights into Milnor fibres and vanishing cycles, though Stein morphisms are not treated due to potential pathologies.

Abstract

The classical Andreotti-Frankel-Hamm theorem reads: a complex affine algebraic variety B, of dim_\C B=n, has homotopy type of dim_\R\le n. We prove the relative version for morphisms X\to B.

Andreotti-Frankel-Hamm theorem for morphisms of algebraic varieties

TL;DR

The paper proves a relative version of the Andreotti-Frankel-Hamm theorem for morphisms with a complex affine variety of , by constructing a deformation-retraction of that lifts to a deformation-retraction of the morphism and keeps of the total space contraction bounded by . The method is an iterative, discriminant-driven strategy that compactifies to a proper morphism and applies the AFKH theorem on the base while attaching low-dimensional cells to extend the deformation to the total space. Because the morphism is not assumed to be locally trivial, the horizontal version is nontrivial and the authors focus on a base deformation that lifts coherently to . The results have immediate applications to families in Algebraic Geometry and Singularity Theory, including insights into Milnor fibres and vanishing cycles, though Stein morphisms are not treated due to potential pathologies.

Abstract

The classical Andreotti-Frankel-Hamm theorem reads: a complex affine algebraic variety B, of dim_\C B=n, has homotopy type of dim_\R\le n. We prove the relative version for morphisms X\to B.
Paper Structure (4 sections, 1 theorem, 2 equations)

This paper contains 4 sections, 1 theorem, 2 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Several general (topological) facts
  4. The proof of the theorem

Key Result

Theorem 1.1

There exists a deformation-retraction $\{\Phi^B_t\}$ that lifts to $X,$ making the commutative diagram. The lifted version satisfies: $\Phi_0=Id_X,\space dim_\mathbb{R} \Phi^B_1(B)\le n,\space \Phi_t|_{\Phi_1(X)}=Id.$

Theorems & Definitions (1)

  • Theorem 1.1