The d-critical structure on the Quot scheme of points of a Calabi-Yau 3-fold

TL;DR

This paper establishes the equivalence of two natural d-critical structures on the Artin stack of 0-dimensional sheaves and Quot schemes of points on affine 3-space, and relates these to obstruction theories on Hilbert schemes of points.

Contribution

It proves the agreement of different d-critical structures on Quot schemes of points on affine Calabi-Yau 3-folds and relates them to known obstruction theories.

Findings

The d-critical structures from quotient descriptions and derived deformation theory agree.

The Quot scheme of points admits a global critical locus description for all ranks r.

The obstruction theory from the Atiyah class matches the critical obstruction theory on Hilbert schemes.

Abstract

The Artin stack $\mathcal M_n$ of $0$-dimensional sheaves of length $n$ on $\mathbb A^3$ carries two natural d-critical structures in the sense of Joyce. One comes from its description as a quotient stack $[\textrm{crit}(f_n)/\textrm{GL}_n]$, another comes from derived deformation theory of sheaves. We show that these d-critical structures agree. We use this result to prove the analogous statement for the Quot scheme of points $\textrm{Quot}_{\mathbb A^3}(\mathscr O^{\oplus r},n) = \textrm{crit}(f_{r,n})$, which is a global critical locus for every $r>0$, and also carries a derived-in-flavour d-critical structure besides the one induced by the potential $f_{r,n}$. Again, we show these two d-critical structures agree. Moreover, we prove that they locally model the d-critical structure on $\textrm{Quot}_X(F,n)$, where $F$ is a locally free sheaf of rank $r$ on a projective Calabi-Yau $3$-fold $X$. Finally, we prove that the perfect obstruction theory on $\textrm{Hilb}^n\mathbb A^3=\textrm{crit}(f_{1,n})$ induced by the Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the Hessian of the potential $f_{1,n}$.