$\mathbb A^1$-connected components of affine quadrics

Problem Statement
The paper addresses the computation of the sheaf of $\mathbb{A}^1$-connected components, denoted $\pi^{\mathbb{A}^1}_0(X)$, for smooth affine quadratic hypersurfaces $X$ in $\mathbb{A}^n_k$ over a field $k$ of characteristic 0. While the sheaf $\pi^{\mathbb{A}^1}_0(F)$ is a fundamental invariant in the Morel–Voevodsky unstable $\mathbb{A}^1$-homotopy category, explicitly computing it is generally difficult. A standard approach involves the functor of naive $\mathbb{A}^1$-connected components, $S(F)$, and its iterations $S^n(F)$. The colimit $L(F) = \varinjlim S^n(F)$ is the universal $\mathbb{A}^1$-invariant quotient. Although Morel conjectured that $\pi^{\mathbb{A}^1}_0(F)$ is always $\mathbb{A}^1$-invariant (implying $\pi^{\mathbb{A}^1}_0(F) \cong L(F)$), counter-examples exist for general sheaves, though the conjecture remains open for representable sheaves.

A specific open question is whether, for a given variety $X$, there exists a finite integer $n_X$ such that the sequence of field-valued sections stabilizes, i.e., $\pi^{\mathbb{A}^1}_0(X)(K) = S^{n_X}(X)(K)$ for all finitely generated field extensions $K/k$. Previous results indicate that for proper schemes, stabilization occurs at $n=1$, but for non-proper varieties, no uniform finite bound was known. This paper seeks to determine the stabilization point and the explicit structure of $\pi^{\mathbb{A}^1}_0(X)(K)$ for smooth affine quadrics.

Methodology
The authors employ a combination of $\mathbb{A}^1$-homotopy theory and the classical theory of quadratic forms.

1. Reduction to Standard Forms: Using Theorem 3.2, the authors show that any smooth affine quadratic hypersurface is isomorphic to a product of an affine space and a hypersurface $Q_\psi$ defined by $\psi(x_1, \dots, x_m) - 1 = 0$, where $\psi$ is a regular quadratic form.
2. Iterative Functor Analysis: The study focuses on the sequence of sheaves $S^n(Q_\psi)$. The authors utilize the concept of "ghost homotopies" (Definition 2.7) to characterize when two points are identified in $S^n(X)$. Specifically, they analyze $n$-ghost homotopies to determine when the sequence stabilizes.
3. Quadratic Form Invariants: The classification relies heavily on the isotropy properties of the quadratic form $\psi$ and its associated forms. Key invariants include:
   • The Witt index $i_0(\theta)$ (dimension of the maximal isotropic subspace).
   • The first Witt index $i_1(\theta)$ (the Witt index of $\theta$ over its own function field).
   • The set of represented values $D(\theta)$ and the subgroup $\langle D(\theta) \rangle$.
4. Geometric Construction:
   • Isotropic Case: When $\psi$ is isotropic, the authors construct explicit $\mathbb{A}^1$-homotopies and use the projection of the quadric to $\mathbb{A}^1$ to relate the connected components to the group $F^*/\langle D(\phi_F) \rangle$, where $\phi$ is a related form of dimension $n-1$.
   • Anisotropic Case: When $\psi$ is anisotropic, the authors introduce "good rational curves" (Definition 4.12). They prove that the stabilization of the sequence depends on whether rational curves in the quadric satisfy specific valuation conditions related to the quadratic form. They utilize blow-ups and Nisnevich coverings to resolve indeterminacies and establish the equivalence of points under 2-ghost homotopies.

Key Contributions and Results

• Stabilization at $n=2$: The primary result (Theorem 1.1) establishes that for any smooth affine quadratic hypersurface $Q_\psi$ and any finitely generated field extension $F/k$, the sequence of naive $\mathbb{A}^1$-connected components stabilizes at $n=2$. Specifically:

  • If $\psi$ is isotropic, $\pi^{\mathbb{A}^1}_0(Q_\psi)(F) \cong S(Q_\psi)(F) \cong F^*/\langle D(\phi_F) \rangle$. Here, stabilization occurs at $n=1$.
  • If $\psi$ is anisotropic, $\pi^{\mathbb{A}^1}_0(Q_\psi)(F) \cong S^2(Q_\psi)(F)$. In this case, $S(Q_\psi)(F) \neq S^2(Q_\psi)(F)$, meaning the sequence requires exactly two iterations to stabilize.
  • This provides a finite, uniform bound ($n=2$) for the stabilization of field-valued sections for this class of non-proper varieties.

• Characterization of $\mathbb{A}^1$-Connectedness: Combining the stabilization result with Morel's characterization (a space is $\mathbb{A}^1$-connected iff its field-valued sections of $\pi^{\mathbb{A}^1}_0$ are trivial), the authors provide a complete characterization of when an affine quadric is $\mathbb{A}^1$-connected (Theorem 1.2 and Theorem 5.1). Let $\kappa = \psi \perp \langle -1 \rangle$. The quadric $Q_\psi$ is $\mathbb{A}^1$-connected if and only if:

  1. $i_0(\kappa) \ge 2$; or
  2. $i_0(\kappa) = 1$, $i_0(\psi) = 0$, and $i_1(\psi) \ge 2$.

• Explicit Isomorphisms: The paper constructs explicit isomorphisms for the set of connected components in the isotropic case, identifying them with the quotient of the multiplicative group of the field by the subgroup generated by values represented by a specific subform.

Significance
The paper claims to resolve the question of finite stabilization for the specific and important class of smooth affine quadratic hypersurfaces. By proving that the sequence stabilizes at $n=2$, the authors demonstrate that the "infinitely many iterations" required in the general definition of the universal $\mathbb{A}^1$-invariant quotient are not necessary for these varieties.

Furthermore, the work bridges the gap between $\mathbb{A}^1$-homotopy theory and the classical arithmetic of quadratic forms. It translates the topological property of $\mathbb{A}^1$-connectedness into concrete algebraic conditions involving the Witt index and the first Witt index of quadratic forms. As noted in the introduction, this extends known results for algebraic groups and proper varieties to the non-proper affine setting, providing a complete classification of $\mathbb{A}^1$-connected affine quadrics in terms of classical invariants. The results imply that over algebraically closed or quadratically closed fields, any smooth affine quadratic hypersurface of dimension $n \ge 2$ is either isomorphic to $\mathbb{G}_m \times \mathbb{A}^{n-2}_k$ or is $\mathbb{A}^1$-connected.