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

Abstract

The Artin stack [Formula: see text] of zero-dimensional sheaves of length [Formula: see text] on [Formula: see text] carries two natural d-critical structures in the sense of Joyce. One comes from its description as a quotient stack [Formula: see text], 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 [Formula: see text], which is a global critical locus for every [Formula: see text], and also carries a derived-in-flavor d-critical structure besides the one induced by the potential [Formula: see text]. Again, we show these two d-critical structures agree. Moreover, we prove that they locally model the d-critical structure on [Formula: see text], where [Formula: see text] is a locally free sheaf of rank [Formula: see text] on a projective Calabi–Yau [Formula: see text]-fold [Formula: see text]. Finally, we prove that the perfect obstruction theory on [Formula: see text] induced by the Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the Hessian of the potential [Formula: see text].