Cohomology of the Quot scheme of an infinite affine space
Paweł Pielasa
TL;DR
Addresses the cohomology of the Quot scheme of points on affine space and its colimit to the infinite ambient, aiming to understand stable invariants via loci whose complement has codimension diverging in $n$. The approach combines a detailed classification of maximal-dimensional commutative $r$-spanning subspaces of $ ext{End}(V)$ with a Białynicki-Birula type cohomology analysis, translating to explicit geometry of loci. It proves that these high-codimension loci have cohomology stable under the colimit and computes the cohomology for the case $d=2$, confirming a Pandharipande conjecture in this setting. Overall, the results advance the understanding of ind-schemes of Quot schemes and their cohomology, with implications for purity phenomena and motivic perspectives.
Abstract
We study the Quot scheme of points $\mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})$. We exhibit and compute the cohomology of explicit loci in $\mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})$, whose complement has codimension diverging to infinity as $n\rightarrow \infty$. In the case $1<r<\frac{d+1}{2}$ this loci is an irreducible component. The main ingredient in our proof are classification results on maximal-dimensional spaces of commutative matrices satisfying certain generating conditions. Our primary motivation is the study of the ind-scheme \[ \mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{\infty}}^{\oplus r}) := \underset{n\rightarrow \infty}{colim} \mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r}). \] Finally, we compute the cohomology (with integral coefficients) of the Quot scheme $\mathrm{Quot}_2(\mathcal{O}_{\mathbb{A}^n}^{\oplus r})$, confirming, in the case $d=2$, a conjecture of Pandharipande.