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.

Source: arXiv.org:2009.07381

Volume: Volume 5

Published on: August 31, 2021

Accepted on: June 23, 2021

Submitted on: September 22, 2020

Keywords: Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology,Mathematics - K-Theory and Homology,14L30 (Primary) 14C05, 14F42, 55R80 (Secondary)

Funding:

- Source : OpenAIRE Graph
*Hodge Theory and Classifying Spaces*; Funder: National Science Foundation; Code: 1701237

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