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On Proper Descent of Smooth Affine Surfaces with Finite Homotopy Rank-Sum

Buddhadev Hajra

TL;DR

This work investigates when EM-space structure and finite homotopy rank-sum properties descend under finite surjective morphisms between smooth complex affine surfaces. It shows EM descent fails in general and introduces the finite rank-sum property as a weaker alternative, proving descent for the latter under finite maps when the source has nonpositive logarithmic Kodaira dimension. The authors leverage a recent classification of non-general-type affine surfaces to establish descent results, and they derive classifications for surfaces properly dominated by $\mathbb{C}^*\times\mathbb{C}^*$, clarifying Furushima’s remarks. An explicit counterexample demonstrates EM-descent can fail, highlighting the nuanced interplay between topological and algebro-geometric properties in this setting.

Abstract

We study the descent behaviour of homotopy-theoretic properties of smooth complex affine surfaces under finite surjective morphisms. We first examine the Eilenberg-MacLane property and show, by means of an explicit counterexample, that it does not descend under proper morphisms in general. This negative result motivates the introduction of a weaker notion, the finite homotopy rank-sum property. Our main theorem establishes that this property does descend under proper morphisms between smooth affine surfaces of logarithmic Kodaira dimension at most zero. The proof relies essentially on the recent classification of smooth complex affine surfaces of log non-general type characterized by these two properties. As a further application, we classify smooth affine surfaces properly dominated by the complex algebraic 2-torus, thereby clarifying an earlier remark of M. Furushima.

On Proper Descent of Smooth Affine Surfaces with Finite Homotopy Rank-Sum

TL;DR

This work investigates when EM-space structure and finite homotopy rank-sum properties descend under finite surjective morphisms between smooth complex affine surfaces. It shows EM descent fails in general and introduces the finite rank-sum property as a weaker alternative, proving descent for the latter under finite maps when the source has nonpositive logarithmic Kodaira dimension. The authors leverage a recent classification of non-general-type affine surfaces to establish descent results, and they derive classifications for surfaces properly dominated by , clarifying Furushima’s remarks. An explicit counterexample demonstrates EM-descent can fail, highlighting the nuanced interplay between topological and algebro-geometric properties in this setting.

Abstract

We study the descent behaviour of homotopy-theoretic properties of smooth complex affine surfaces under finite surjective morphisms. We first examine the Eilenberg-MacLane property and show, by means of an explicit counterexample, that it does not descend under proper morphisms in general. This negative result motivates the introduction of a weaker notion, the finite homotopy rank-sum property. Our main theorem establishes that this property does descend under proper morphisms between smooth affine surfaces of logarithmic Kodaira dimension at most zero. The proof relies essentially on the recent classification of smooth complex affine surfaces of log non-general type characterized by these two properties. As a further application, we classify smooth affine surfaces properly dominated by the complex algebraic 2-torus, thereby clarifying an earlier remark of M. Furushima.
Paper Structure (11 sections, 33 theorems, 30 equations)

This paper contains 11 sections, 33 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations & Conventions
  4. Basics about Eilenberg-MacLane smooth complex algebraic varieties
  5. A formula of M. Suzuki
  6. Nori's lemma and Xiao Gang's generalization
  7. Ramified Covering Trick
  8. Proof of Main Results
  9. On the descent of Eilenberg--MacLane affine surfaces when $\bar{\kappa}(X)=-\infty$
  10. On the descent of finite homotopy rank-sum if $\bar{\kappa}(X)=0$.
  11. Examples

Key Result

Theorem 1.3

Let $X$ be a non-contractible smooth complex affine $K(G,1)$-surface. Then the following hold:

Theorems & Definitions (53)

  • Theorem 1.3: cf. GGH2023
  • Theorem 1.4: cf. BH2025
  • Theorem A: Theorem \ref{['Thm: Descent of A^1 bundle over affine curve']}
  • Theorem B: Theorem \ref{['Thm: Affine surfaces with kappa=0 properly dominated by 2-torus']}, \ref{['Thm: Affine surfaces with kappa negative properly dominated by 2-torus']}
  • Theorem C: Theorem \ref{['Main Theorem - kappa bar negative case']}, \ref{['Main Theorem - kappa bar = 0 case']}
  • Lemma 2.1: J.P. Serre, cf. Ser1959
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4: Suzuki's Formula; cf. Suz1977
  • Theorem 2.5: Nori's lemma; Nor1983
  • ...and 43 more