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

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 $f_{1,n}$.