Torus actions, Morse homology, and the Hilbert scheme of points on affine space
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-homo...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Association Epiga
2021-08-01
|
| Series: | Épijournal de Géométrie Algébrique |
| Subjects: | |
| Online Access: | https://epiga.episciences.org/6792/pdf |
| _version_ | 1798029679142109184 |
|---|---|
| author | Burt Totaro |
| author_facet | Burt Totaro |
| author_sort | Burt Totaro |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T19:29:06Z |
| format | Article |
| id | doaj.art-e860757cfc954122aca4b47df1849033 |
| institution | Directory Open Access Journal |
| issn | 2491-6765 |
| language | English |
| last_indexed | 2024-04-11T19:29:06Z |
| publishDate | 2021-08-01 |
| publisher | Association Epiga |
| record_format | Article |
| series | Épijournal de Géométrie Algébrique |
| spelling | doaj.art-e860757cfc954122aca4b47df18490332022-12-22T04:07:04ZengAssociation EpigaÉpijournal de Géométrie Algébrique2491-67652021-08-01Volume 510.46298/epiga.2021.67926792Torus actions, Morse homology, and the Hilbert scheme of points on affine spaceBurt Totarohttps://orcid.org/0000-0002-5573-4808We 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.https://epiga.episciences.org/6792/pdfmathematics - algebraic geometrymathematics - algebraic topologymathematics - k-theory and homology14l30 (primary) 14c05, 14f42, 55r80 (secondary) |
| spellingShingle | Burt Totaro Torus actions, Morse homology, and the Hilbert scheme of points on affine space Épijournal de Géométrie Algébrique mathematics - algebraic geometry mathematics - algebraic topology mathematics - k-theory and homology 14l30 (primary) 14c05, 14f42, 55r80 (secondary) |
| title | Torus actions, Morse homology, and the Hilbert scheme of points on affine space |
| title_full | Torus actions, Morse homology, and the Hilbert scheme of points on affine space |
| title_fullStr | Torus actions, Morse homology, and the Hilbert scheme of points on affine space |
| title_full_unstemmed | Torus actions, Morse homology, and the Hilbert scheme of points on affine space |
| title_short | Torus actions, Morse homology, and the Hilbert scheme of points on affine space |
| title_sort | torus actions morse homology and the hilbert scheme of points on affine space |
| topic | mathematics - algebraic geometry mathematics - algebraic topology mathematics - k-theory and homology 14l30 (primary) 14c05, 14f42, 55r80 (secondary) |
| url | https://epiga.episciences.org/6792/pdf |
| work_keys_str_mv | AT burttotaro torusactionsmorsehomologyandthehilbertschemeofpointsonaffinespace |