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Shifted Lagrangian thickenings of shifted Poisson derived schemes

Nikola Tomić

Abstract

We prove that the space of shifted Poisson structures on a derived scheme $X$ locally of finite presentation is equivalent to the space of shifted Lagrangian thickenings out $X$, solving a conjecture in shifted Poisson geometry. As a corollary, we show that for $M$ a compact oriented $d$-dimensional manifold and an $n$-shifted Poisson structure on $X$, the mapping stack $\mathrm{Map}(M,X)$ has an $(n-d)$-shifted Poisson structure. It extends a known theorem for shifted symplectic structures to shifted Poisson structures.

Shifted Lagrangian thickenings of shifted Poisson derived schemes

Abstract

We prove that the space of shifted Poisson structures on a derived scheme locally of finite presentation is equivalent to the space of shifted Lagrangian thickenings out , solving a conjecture in shifted Poisson geometry. As a corollary, we show that for a compact oriented -dimensional manifold and an -shifted Poisson structure on , the mapping stack has an -shifted Poisson structure. It extends a known theorem for shifted symplectic structures to shifted Poisson structures.

Paper Structure

This paper contains 30 sections, 54 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Preliminaries
  5. Graded, filtered, and complete filtered objects
  6. Adic and de Rham filtrations
  7. The de Rham functor
  8. Model categorical settings
  9. Complete filtered operads and cooperads
  10. Convolution algebras
  11. Forms, formal geometry, and forms over formal prestacks
  12. Prestacks, forms, and symplectic structures
  13. Finiteness properties on algebras
  14. Lie algebroids and derived foliations
  15. Formal geometry and formal integration of derived foliations
  16. ...and 15 more sections

Key Result

Theorem 1

(thm:main) Let $X$ be a locally of finite presentation derived scheme, denote by $\mathrm{Pois}(X,n)$ the space of $n$-shifted Poisson structures on $X$, and by $\mathrm{LagThick}(X,n+1)$ the space of $(n+1)$-shifted Lagrangian thickenings. There is an equivalence:

Theorems & Definitions (137)

  • Theorem
  • Corollary
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 127 more