Ricolfi, Savvas — The D-Critical Structure On The Quot Scheme Of Points Of A Calabi-Yau 3-Fold (2023)

The Artin stack \(\mathcal M_n\) of $0$-dimensional sheaves of length \(n\) on \(\mathbb A\textasciicircum 3\) 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 \(\textrm{ Quot}_{\mathbb A\textasciicircum 3} (\mathscr O\textasciicircum{\oplus r},n) = \textrm{ crit} (f_{ r,n} )\), 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 \(\textrm{ Hilb}\textasciicircum n\mathbb A\textasciicircum 3=\textrm{ crit} (f_{ 1,n} )\) induced by the Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the Hessian of the potential \(f_{ 1,n}\).