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

epiga:6792 - Épijournal de Géométrie Algébrique, August 31, 2021, Volume 5 - https://doi.org/10.46298/epiga.2021.6792
Torus actions, Morse homology, and the Hilbert scheme of points on affine space

Authors: Burt Totaro

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.


Volume: Volume 5
Published on: August 31, 2021
Submitted on: September 22, 2020
Keywords: Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology,Mathematics - K-Theory and Homology,14L30 (Primary) 14C05, 14F42, 55R80 (Secondary)


Share

Consultation statistics

This page has been seen 16 times.
This article's PDF has been downloaded 7 times.