The $\mathcal{D}$-Geometric Hilbert Scheme -- Part II: Hilbert and Quot DG-Schemes

TL;DR

This paper develops a derived geometric framework for moduli spaces of differential ideal sheaves related to algebraic PDEs, enhancing the understanding of their deformation theory and representability.

Contribution

It constructs a derived enhancement of moduli spaces for differential ideal sheaves, proving representability by differential graded $ ext{D}$-manifolds and analyzing their properties.

Findings

Derived $ ext{D}$-Quot functor admits a global differential graded refinement.

Established foundations for a derived deformation theory of algebraic differential equations.

Proved finiteness, representability, and functoriality of the derived moduli spaces.

Abstract

This is the second in a series of two papers developing a moduli-theoretic framework for differential ideal sheaves associated with formally integrable, involutive systems of algebraic partial differential equations (PDEs). Building on earlier work, which established the existence of moduli stacks for such systems with prescribed regularity and stability conditions, we now construct a derived enhancement of these moduli spaces. We prove the derived $\mathcal{D}$-Quot functor admits a global differential graded refinement representable by a suitable differential graded $\mathcal{D}$-manifold. We further analyze the finiteness, representability, and functoriality properties of these derived moduli spaces, establishing foundations for a derived deformation theory of algebraic differential equations.