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

TL;DR

This paper explores the impact of G_m actions on quasi-projective schemes within motivic homotopy theory, establishing conditions under which certain inclusions are homotopy equivalences, with applications to Hilbert schemes of points.

Contribution

It formulates a conjecture relating G_m actions to homotopy equivalences and proves partial results, including over complex numbers, using Morse theory analogs for singular varieties.

Findings

Inclusion of Y into U is a homotopy equivalence over complex numbers.

The Hilbert scheme of points on affine space is homotopy equivalent to schemes supported at the origin.

Partial results support the conjecture in motivic homotopy theory.

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.