Motivic and cohomological stabilisation of the Quot scheme of points

TL;DR

This paper demonstrates the stabilization of motives and cohomology of Quot schemes of points on affine space as the dimension tends to infinity, providing explicit limits and confirming conjectures about their asymptotic behavior.

Contribution

It establishes the stabilization of motives and cohomology of Quot schemes and nested Hilbert schemes in high dimensions, with explicit computations of their limits.

Findings

Motives of punctual Quot schemes stabilize to a product involving infinite Grassmannians.

Poincaré polynomials of Quot schemes stabilize and their limits are computed.

Motives of nested Hilbert schemes stabilize to the motive of the infinite flag variety.

Abstract

We prove that the motive of the punctual Quot scheme $\mathrm{Quot}^d(\mathscr O^{\oplus r}_{\mathbb A^n})_0$ stabilises, when $n \to \infty$, to $[\mathrm{Gr}(d-1,\infty)]\cdot \sum_{i=0}^{r-1}\mathbb L^{di}$. We similarly show that the Poincar\'e polynomial of the Quot scheme $ \mathrm{Quot}^d(\mathscr O^{\oplus r}_{\mathbb A^n})$ stabilises and we compute the limit in terms of the infinite Grassmannian. Finally, we prove that the motive of the nested Hilbert scheme stabilises to the motive of the infinite flag variety and we compute the cohomology ring in the limit. These results provide affirmative evidence to a question of Pandharipande concerning the cohomology of Quot schemes on $\mathbb A^\infty$.