Geodesic
Torus actions, Morse homology, and the Hilbert scheme of points on affine space
Épijournal de Géométrie Algébrique, Tome 5 (2021)

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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. 
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     author = {Totaro, Burt},
     title = {Torus actions, {Morse} homology, and the {Hilbert} scheme of points on affine space},
     journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
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     volume = {5},
     year = {2021},
     doi = {10.46298/epiga.2021.6792},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2021.6792/}
}
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Totaro, Burt. Torus actions, Morse homology, and the Hilbert scheme of points on affine space. Épijournal de Géométrie Algébrique, Tome 5 (2021). doi: 10.46298/epiga.2021.6792

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