Derived Quot schemes

TL;DR

This paper constructs a derived analog of Grothendieck's Quot scheme as a differential graded manifold, providing a smooth geometric object that encodes deformation theory of subsheaves in a coherent sheaf.

Contribution

It introduces a derived Quot scheme as a differential graded manifold, advancing the derived deformation theory framework for parametrizing subsheaves.

Findings

The derived Quot scheme is always smooth in an appropriate sense.

The tangent space at a point is a cochain complex quasi-isomorphic to RHom(K, F/K).

Provides a new geometric object for studying deformations of subsheaves.

Abstract

Realizing a part of the Derived Deformation Theory program, we construct a "derived" analog of the Grothendieck's Quot scheme parametrizing subsheaves in a given coherent sheaf F on a smooth projective variety X. This analog is a differential graded manifold RQuot_h(F) (so it is always smooth in an appropriate sense) whose tangent space at a point represented by a subsheaf K in F, is a cochain complex quasiisomorphic to RHom(K, F/K).