[Paper Review] A^1-homotopy groups, excision, and solvable quotients

Aravind Asok, Brent Doran|ArXiv.org|Feb 10, 2009

Homotopy and Cohomology in Algebraic Topology1 references3 citations

TL;DR

This paper develops tools to compute A¹-homotopy groups of smooth toric varieties using combinatorial data from fans, establishing excision theorems and showing that quotients by free actions of split solvable groups yield A¹-covering spaces. Key results include combinatorial criteria for vanishing of low-degree A¹-homotopy groups and explicit computations of the next non-vanishing group, demonstrating that A¹-homotopy theory is tractable for large classes of algebraic varieties despite its algebraic-geometric complexity.

ABSTRACT

We study some properties of A^1-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of A^1-homotopy theory. These concepts and results are well-suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate A^1-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree A^1-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the "next" non-vanishing A^1-homotopy group (beyond π_1^{A^1}) of a smooth toric variety. From this point of view, A^1-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost "as tractable" (in low degrees) as ordinary homotopy for large classes of interesting varieties.

Motivation & Objective

• To develop computable invariants for A¹-homotopy types of smooth algebraic varieties using geometric and combinatorial data.
• To understand how algebraic group actions—especially free actions of split solvable groups—affect A¹-homotopy groups.
• To establish excision results for A¹-homotopy groups under connectivity assumptions, enabling comparison between schemes and open subschemes.
• To provide explicit, combinatorial criteria for the vanishing of low-degree A¹-homotopy groups in smooth toric varieties.
• To compute the first non-vanishing A¹-homotopy group beyond π₁^A¹ for certain smooth toric varieties.

Proposed method

• Utilizes Morel-Voevodsky A¹-homotopy theory and A¹-covering space theory to interpret geometric quotients as covering spaces in the A¹-category.
• Applies excision theorems (Theorem 4.1) to relate A¹-homotopy groups of a scheme and its open subchemes when the complement has high codimension.
• Employs the fact that smooth toric varieties arise as quotients of open subsets of affine space by free torus actions, with the fan encoding the quotient data.
• Uses the homogeneous coordinate ring construction to describe the total space of the quotient and relate it to the fan structure.
• Applies Morel’s theory of A¹-fundamental groups and the sheaf of Milnor-Witt K-theory to compute extensions in π₁^A¹.
• Applies Kleinschmidt’s classification of smooth proper toric varieties with few generators to analyze specific cases and compute A¹-homotopy groups.

Experimental results

Research questions

• RQ1 Under what combinatorial conditions on a fan do the low-degree A¹-homotopy groups of a smooth toric variety vanish?
• RQ2 How do A¹-homotopy groups behave under open immersions with high-codimension complements, and can excision be established?
• RQ3 Can geometric quotients by free actions of split solvable groups be interpreted as A¹-covering spaces, and how does this simplify computation of A¹-homotopy invariants?
• RQ4 What is the structure of the A¹-fundamental group of a smooth toric variety, and how does it depend on the fan and Picard group?
• RQ5 Can the first non-vanishing A¹-homotopy group beyond π₁^A¹ be explicitly computed for smooth toric varieties?

Key findings

• For smooth proper toric varieties with fans satisfying certain combinatorial conditions (e.g., all pairs of primitive vectors lie in a common cone), the A¹-homotopy groups π₁^A¹ and π₂^A¹ vanish.
• When the fan has at most d+2 generators, the A¹-fundamental group π₁^A¹ is isomorphic to G_m × G_m if r ≥ 2 and s ≥ 3, where r and s are parameters from Kleinschmidt’s classification.
• If r = 1 or s = 2 (but not both), π₁^A¹ is an extension of G_m × G_m by the sheaf KMW₂, and if r = 1 and s = 2, it is an extension by KMW₂⊕KMW₂.
• For Hirzebruch surfaces F_a, the A¹-fundamental group fits into a short exact sequence 1 → KMW₂⊕KMW₂ → π₁^A¹(F_a) → G_m⊕G_m → 1, with the extension depending on a mod 2.
• In the case of smooth projective toric 3-folds with rk(Pic) = 4 and non-projective structure, π₁^A¹ surjects onto the dual torus but the kernel remains undetermined due to combinatorial complexity.
• The paper constructs an example where the kernel of the surjection π₁^A¹(Bl_x,x' P^n) → G_m^×3 is non-trivial and difficult to describe, indicating limitations in kernel control even when the image is known.