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Torus actions, Morse homology, and the Hilbert scheme of points on affine space

Burt Totaro

Abstract

We formulate a conjecture on actions of the multiplicative group in motivic homotopy theory. In short, if the multiplicative group G_m acts on a quasi-projective scheme U such that U is attracted as t approaches 0 in G_m to a closed subset Y in U, then the inclusion from Y to U should be an A^1-homotopy equivalence. We prove several partial results. In particular, over the complex numbers, the inclusion is a homotopy equivalence on complex points. The proofs use an analog of Morse theory for singular varieties. Application: the Hilbert scheme of points on affine n-space is homotopy equivalent to the subspace consisting of schemes supported at the origin.

Torus actions, Morse homology, and the Hilbert scheme of points on affine space

Abstract

We formulate a conjecture on actions of the multiplicative group in motivic homotopy theory. In short, if the multiplicative group G_m acts on a quasi-projective scheme U such that U is attracted as t approaches 0 in G_m to a closed subset Y in U, then the inclusion from Y to U should be an A^1-homotopy equivalence. We prove several partial results. In particular, over the complex numbers, the inclusion is a homotopy equivalence on complex points. The proofs use an analog of Morse theory for singular varieties. Application: the Hilbert scheme of points on affine n-space is homotopy equivalent to the subspace consisting of schemes supported at the origin.

Paper Structure

This paper contains 9 sections, 12 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results, and a conjecture on $\mathbf{G}_m$-actions in motivic homotopy theory
  3. $\mathbf{G}_m$-actions and broken trajectories
  4. Openness
  5. Proof of Theorem \ref{['complex']}
  6. The real case
  7. $\mathbf{A}^1$-connectedness of the Hilbert scheme
  8. Conservativity for the motivic stable homotopy category
  9. $\mathbf{G}_m$-actions on smooth varieties and motivic homotopy theory

Key Result

Theorem 2.2

Under the assumptions of Conjecture conj:torus with base field $\text{\bf C}$, the inclusion $Y\to U$ is a homotopy equivalence $($in the classical topology$)$.

Figures (2)

  • Figure 1: Example of a $T$-action on $U\cong \text{\bf P}^1\times \mathbf{A}^1$, $t([x_0,x_1],y)=([x_0,tx_1],ty)$, with $Y=\text{\bf P}^1\times 0$ shown as the horizontal line ($T=\mathbf{G}_m$). The arrows point in the direction $t\to 0$. The fixed point set $Y^T$ consists of two points.
  • Figure 2: Newton polygon of the pairs $(a_i,\mathop{\mathrm{ord}}\nolimits_u(z_i))$

Theorems & Definitions (25)

  • Conjecture 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof : Proof of Corollary \ref{['move-to-origin']}
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • proof
  • ...and 15 more