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Deformations and Lifts of Calabi-Yau Varieties in Characteristic $p$

Lukas Brantner, Lenny Taelman

Abstract

We study deformations of Calabi-Yau varieties in characteristic $p$ using techniques from derived algebraic geometry. We prove a mixed characteristic analogue of the Bogomolov-Tian-Todorov theorem (which states that Calabi-Yau varieties in characteristic $0$ are unobstructed), and we show that ordinary Calabi-Yau varieties admit canonical lifts to characteristic $0$, generalising the Serre-Tate theorem on ordinary abelian varieties.

Deformations and Lifts of Calabi-Yau Varieties in Characteristic $p$

Abstract

We study deformations of Calabi-Yau varieties in characteristic using techniques from derived algebraic geometry. We prove a mixed characteristic analogue of the Bogomolov-Tian-Todorov theorem (which states that Calabi-Yau varieties in characteristic are unobstructed), and we show that ordinary Calabi-Yau varieties admit canonical lifts to characteristic , generalising the Serre-Tate theorem on ordinary abelian varieties.
Paper Structure (48 sections, 115 theorems, 265 equations)

This paper contains 48 sections, 115 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Outline of proofs
  4. Acknowledgements
  5. Preliminaries
  6. Adjoining sifted colimits
  7. Animated commutative rings
  8. Modules over animated rings
  9. Cotangent complex
  10. Sheaves with values in an $\infty$-category
  11. Sheaves of modules
  12. Cotangent complex for sheaves of animated rings
  13. Cotangent complex for derived schemes
  14. Formal moduli problems
  15. Artinian animated rings
  16. ...and 33 more sections

Key Result

Theorem A

Let $k$ be a perfect field of characteristic $p$ with ring of Witt vectors $W(k)$. Let $X$ be a smooth, proper, and geometrically irreducible scheme over $k$ with trivial canonical bundle. Assume that Then the mixed characteristic formal deformations of $X$ are unobstructed.

Theorems & Definitions (291)

  • Theorem A: Unobstructedness
  • Remark 1.1
  • Definition 1.2: Bloch--Kato ordinary
  • Remark 1.3
  • Remark 1.4
  • Theorem B: Serre--Tate coordinates
  • Remark 1.5
  • Remark 1.6
  • Corollary 1.7: Canonical lifts
  • Theorem C: Algebraisation
  • ...and 281 more