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Liftability and vanishing theorems for Fano threefolds in positive characteristic I

Tatsuro Kawakami, Hiromu Tanaka

TL;DR

The paper studies liftability to $W(k)$ and vanishing properties for smooth Fano threefolds in positive characteristic, focusing on the case where $|-K_X|$ is very ample and $ ext{Pic}(X)$ is generated by $oldsymbol{ hicksim_X}$. It embeds $X$ into projective space via $|-K_X|$, uses a two-ray game to analyze its birational geometry, proves the existence of conics, and applies a lifting criterion to obtain a $W(k)$-lift, together with Akizuki-Nakano vanishing and $E_1$-degeneration of the Hodge-to-de Rham spectral sequence. For genus $6$ Fano threefolds, the Gushel–Mukai description and separability of the double cover are treated explicitly, with vanishing assertions verified through detailed cohomological and toric arguments. Overall, the work extends liftability and cohomological degeneration results to positive characteristic in a concrete, genus-stratified setting, aligning positive-characteristic behavior with characteristic-zero expectations and enabling further Hodge-theoretic applications. The results form part of a broader program to understand Fano varieties in positive characteristic and their liftings to characteristic zero.

Abstract

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results for the case when $|-K_X|$ is very ample and the Picard group is generated by $ω_X$.

Liftability and vanishing theorems for Fano threefolds in positive characteristic I

TL;DR

The paper studies liftability to and vanishing properties for smooth Fano threefolds in positive characteristic, focusing on the case where is very ample and is generated by . It embeds into projective space via , uses a two-ray game to analyze its birational geometry, proves the existence of conics, and applies a lifting criterion to obtain a -lift, together with Akizuki-Nakano vanishing and -degeneration of the Hodge-to-de Rham spectral sequence. For genus Fano threefolds, the Gushel–Mukai description and separability of the double cover are treated explicitly, with vanishing assertions verified through detailed cohomological and toric arguments. Overall, the work extends liftability and cohomological degeneration results to positive characteristic in a concrete, genus-stratified setting, aligning positive-characteristic behavior with characteristic-zero expectations and enabling further Hodge-theoretic applications. The results form part of a broader program to understand Fano varieties in positive characteristic and their liftings to characteristic zero.

Abstract

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, -degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results for the case when is very ample and the Picard group is generated by .

Paper Structure

This paper contains 28 sections, 66 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. Overview of proofs
  3. Overview of contents
  4. Preliminaries
  5. Notation
  6. Fano threefolds
  7. Two-ray game
  8. K3 surfaces
  9. Complete intersections
  10. Existence of conics ($g \geq 7$)
  11. Existence of non-superspecial members ($g \geq 7$)
  12. Liftability ($g \geq 7$)
  13. Lifting criterion
  14. Case $(p, g) \neq (2, 9)$
  15. Case $(p, g) = (2, 9)$
  16. ...and 13 more sections

Key Result

Theorem 1

Let $X$ be a smooth Fano threefold over an algebraically closed field $k$ of positive characteristic. Then $X$ lifts to $W(k)$.

Theorems & Definitions (166)

  • Theorem 1: Liftability to $W(k)$
  • Theorem 2: Akizuki-Nakano vanishing
  • Theorem 3
  • Remark 1.1
  • Theorem 4: Hodge numbers
  • Proposition 2.1
  • proof
  • Lemma 2.2: cf. IP99*Remark 4.1.10
  • proof
  • Definition 2.3
  • ...and 156 more