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An operadic proof of the BTT Theorem

Joana Cirici, Geoffroy Horel

Abstract

In this note, we explain an operadic proof of the BTT Theorem stating that the deformation theory of Calabi-Yau varieties is unobstructed. We also provide a short new proof of the non-commutative BTT for Calabi-Yau dg-categories. Finally, we observe that our proof also produces partial results in positive characteristic.

An operadic proof of the BTT Theorem

Abstract

In this note, we explain an operadic proof of the BTT Theorem stating that the deformation theory of Calabi-Yau varieties is unobstructed. We also provide a short new proof of the non-commutative BTT for Calabi-Yau dg-categories. Finally, we observe that our proof also produces partial results in positive characteristic.

Paper Structure

This paper contains 7 sections, 7 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. A bestiary of operads
  3. The degeneration condition
  4. Proof of the main results
  5. The classical BTT Theorem
  6. Some more BV-algebras of geometric origin
  7. The non-commutative BTT Theorem

Key Result

Theorem 1.1

Let $K$ be a field. Let $A$ be a dg-algebra over the framed little disks operad. Assume that $A$ satisfies the degeneration condition. Then

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Remark 3.2
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • proof : Proof of Theorem \ref{['maintheo']}
  • Remark 4.3
  • ...and 10 more