Torus actions, Morse homology, and the Hilbert scheme of points on affine space
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913572082155520 |
|---|---|
| author | Totaro, Burt |
| author_facet | Totaro, Burt |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_07381 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Torus actions, Morse homology, and the Hilbert scheme of points on affine space Totaro, Burt Algebraic Geometry Algebraic Topology K-Theory and Homology 14L30 (Primary) 14C05, 14F42, 55R80 (Secondary) 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. |
| title | Torus actions, Morse homology, and the Hilbert scheme of points on affine space |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology 14L30 (Primary) 14C05, 14F42, 55R80 (Secondary) |
| url | https://arxiv.org/abs/2009.07381 |