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

Abstract

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.

Keywords

• mathematics - algebraic geometry
• mathematics - algebraic topology
• mathematics - k-theory and homology
• 14l30 (primary) 14c05, 14f42, 55r80 (secondary)