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Strictification and gluing of Lagrangian distributions on derived schemes with shifted symplectic forms

Dennis Borisov, Ludmil Katzarkov, Artan Sheshmani, Shing-Tung Yau

Abstract

A strictification result is proved for isotropic distributions on derived schemes equipped with negatively shifted homotopically closed $2$-forms. It is shown that any derived scheme over $\mathbb{C}$ equipped with a $-2$-shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally defined Lagrangian distribution as a dg $\mathbb{C}^{\infty}$-manifold.

Strictification and gluing of Lagrangian distributions on derived schemes with shifted symplectic forms

Abstract

A strictification result is proved for isotropic distributions on derived schemes equipped with negatively shifted homotopically closed -forms. It is shown that any derived scheme over equipped with a -shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally defined Lagrangian distribution as a dg -manifold.

Paper Structure

This paper contains 10 sections, 24 theorems, 61 equations.

Table of Contents

  1. Integrable distributions
  2. Well presented dg algebras and ${C^\infty}$-rings
  3. Graded mixed algebras and integrable distributions
  4. Pull-backs and derived foliations
  5. Symplectic and Lagrangian structures
  6. Homotopically closed and symplectic differential forms
  7. Isotropic and Lagrangian distributions
  8. Sheaves of symplectic and isotropic structures
  9. Strictification of Lagrangian distributions
  10. Global Lagrangian distributions over $\mathbb R$

Key Result

Proposition 1

Let ${{A}^\bullet}$ be a dg algebra and let ${\boldsymbol\omega}$ be a homotopically closed $2$-form on ${{\rm Spec}({{A}^\bullet})}$. Let ${\alpha}\colon({\mathcal{L}},{\boldsymbol\lambda})\longrightarrow({\mathbb T_{{{{A}^\bullet}}}},{\boldsymbol\omega})$ be a purely derived foliation with an isot Then locally on ${{\rm Spec}({{A}^\bullet})}$ there is an equivalent distribution with an isotropic

Theorems & Definitions (76)

  • Proposition
  • Theorem
  • Proposition 1
  • proof
  • Definition 1
  • Remark 1
  • Definition 2
  • Proposition 2
  • Definition 3
  • Remark 2
  • ...and 66 more