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Shifted symplectic structures on derived Quot-stacks II -- Derived Quot-schemes as dg manifolds

Dennis Borisov, Ludmil Katzarkov, Artan Sheshmani

TL;DR

The paper proves that derived Quot-stacks, built from graded submodules with a fixed Hilbert polynomial $h(\lambda)$, are representable by dg manifolds of finite type, advancing the program to study shifted symplectic structures on moduli of coherent sheaves on Calabi–Yau manifolds. It builds a tower of affine dg manifolds encoding $A_\infty$-structures and morphisms, then imposes open conditions to model submodules and develops three quotient strategies (partial, algebraic, geometric) to mod out symmetries, showing agreement with the classical derived Quot construction. The core result shows the homotopy limit in derived stacks is represented by a dg manifold of finite type, obtained by a colimit of almost-free dg algebras over $\mathbf{Gr}_{[a,b]}$ and refining with coherence of cohomology to a finite-type model. This yields a concrete dg-Quot-manifold that enables direct application of shifted-symplectic methods to derived moduli of coherent sheaves.

Abstract

It is proved that derived Quot-schemes, as defined by Ciocan-Fontanine and Kapranov, are represented by dg manifolds of finite type. This is the second part if a work aimed to analyze shifted symplectic structures on moduli spaces of coherent sheaves on Calabi--Yau manifolds. The first part related dg manifolds to derived schemes as defined by Toën and Vezzosi.

Shifted symplectic structures on derived Quot-stacks II -- Derived Quot-schemes as dg manifolds

TL;DR

The paper proves that derived Quot-stacks, built from graded submodules with a fixed Hilbert polynomial , are representable by dg manifolds of finite type, advancing the program to study shifted symplectic structures on moduli of coherent sheaves on Calabi–Yau manifolds. It builds a tower of affine dg manifolds encoding -structures and morphisms, then imposes open conditions to model submodules and develops three quotient strategies (partial, algebraic, geometric) to mod out symmetries, showing agreement with the classical derived Quot construction. The core result shows the homotopy limit in derived stacks is represented by a dg manifold of finite type, obtained by a colimit of almost-free dg algebras over and refining with coherence of cohomology to a finite-type model. This yields a concrete dg-Quot-manifold that enables direct application of shifted-symplectic methods to derived moduli of coherent sheaves.

Abstract

It is proved that derived Quot-schemes, as defined by Ciocan-Fontanine and Kapranov, are represented by dg manifolds of finite type. This is the second part if a work aimed to analyze shifted symplectic structures on moduli spaces of coherent sheaves on Calabi--Yau manifolds. The first part related dg manifolds to derived schemes as defined by Toën and Vezzosi.
Paper Structure (11 sections, 8 theorems)

This paper contains 11 sections, 8 theorems.

Table of Contents

  1. Introduction
  2. The question
  3. The answer
  4. Not taking symmetries into account
  5. Classifying all ${{\rm A}_\infty}$-structures
  6. Classifying ${{\rm A}_\infty}$-submodules
  7. Dividing by the symmetries
  8. Taking a partial quotient
  9. Taking an algebraic quotient
  10. Taking a geometric quotient
  11. Representation by a dg manifold

Key Result

Proposition 1

Let ${{\rm p}}\in{{\widetilde{\mathcal{H}}}_{{a}}}$ be a closed point, let ${N_{{a}}}\subseteq{M_{{a}}}$ be the corresponding $\mathbb C$-linear subspace, and let ${{\widetilde{N}}_{\geq{a}}}\subseteq{M_{\geq{a}}}$ be the ${R_{+}}$-submodule generated by ${N_{{a}}}$. Then $\forall {t}\geq{a}$$\dim_\

Theorems & Definitions (17)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 7 more