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Liftability and vanishing theorems for Fano threefolds in positive characteristic II

Tatsuro Kawakami, Hiromu Tanaka

TL;DR

This work extends the liftability and vanishing framework for smooth Fano threefolds in positive characteristic beyond earlier Lift1 results. It introduces and leverages quasi-$F$-splitting, showing that all smooth Fano threefolds with $ ho(X)>1$ or $r_X>1$ are quasi-$F$-split, which in turn yields Akizuki–Nakano vanishing, $E_1$-degeneration of the Hodge-to-de Rham spectral sequence, and torsion-freeness of crystalline cohomology; for many cases with $p>5$, actual $F$-splitting is established. The strategy interweaves inversion of adjunction for $F$-splitting and quasi-$F$-splitting, Cartier-operator criteria, Bott-type vanishing in ambient fourfolds, and analysis of exceptional No.s (notably 2-$2$, 2-$6$, 2-$8$, 3-$10$) via specialized Cartier techniques. The results yield liftability to $W(k)$ and invariance of Hodge numbers under lift, with a detailed classification by Picard number and Fano index that clarifies when pathological characteristic phenomena arise. The paper also provides concrete examples highlighting when $F$-split or quasi-$F$-split fails, guiding future investigations into the geometry of Fano threefolds in positive characteristic.

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 except when $|-K_X|$ is very ample and the Picard group is generated by $ω_X$. To this end, we show that an arbitrary smooth Fano threefold is quasi-$F$-split when the Picard number or the Fano index is larger than one.

Liftability and vanishing theorems for Fano threefolds in positive characteristic II

TL;DR

This work extends the liftability and vanishing framework for smooth Fano threefolds in positive characteristic beyond earlier Lift1 results. It introduces and leverages quasi--splitting, showing that all smooth Fano threefolds with or are quasi--split, which in turn yields Akizuki–Nakano vanishing, -degeneration of the Hodge-to-de Rham spectral sequence, and torsion-freeness of crystalline cohomology; for many cases with , actual -splitting is established. The strategy interweaves inversion of adjunction for -splitting and quasi--splitting, Cartier-operator criteria, Bott-type vanishing in ambient fourfolds, and analysis of exceptional No.s (notably 2-, 2-, 2-, 3-) via specialized Cartier techniques. The results yield liftability to and invariance of Hodge numbers under lift, with a detailed classification by Picard number and Fano index that clarifies when pathological characteristic phenomena arise. The paper also provides concrete examples highlighting when -split or quasi--split fails, guiding future investigations into the geometry of Fano threefolds in positive characteristic.

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 except when is very ample and the Picard group is generated by . To this end, we show that an arbitrary smooth Fano threefold is quasi--split when the Picard number or the Fano index is larger than one.
Paper Structure (38 sections, 66 theorems, 223 equations)

This paper contains 38 sections, 66 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Quasi-$F$-splitting
  3. $F$-splitting
  4. Strategy of the proof of Theorem \ref{['Introthm:quasi-F-split']}
  5. Preliminaries
  6. Notation
  7. Cartier operators
  8. Generic members
  9. $F$-splitting criteria
  10. Quasi-$F$-splitting criteria
  11. Weak del Pezzo surfaces
  12. Akizuki-Nakano vanishing for hhypersurfaces
  13. Fano threefolds with $\rho \geq 4$
  14. Fano threefolds with $\rho =3$ (except for 3-10)
  15. Fano threefolds with $\rho =2$ (except for 2-2, 2-6, 2-8)
  16. ...and 23 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 (167)

  • Theorem 1: Kawakami-Tanaka(Lift1)
  • Theorem 2: Kawakami-Tanaka(Lift1)
  • Theorem 3: Kawakami-Tanaka(Lift1)
  • Theorem 4: Kawakami-Tanaka(Lift1)
  • Theorem 5: =Theorem \ref{['thm:quasi-F-split']}
  • Theorem 6
  • Theorem 7
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • ...and 157 more