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Vanishing and a counterexample for Witt divisorial sheaves

Niklas Lemcke

TL;DR

This work tackles vanishing phenomena for Witt divisorial sheaves on smooth projective varieties in positive characteristic. It refines the duality framework by removing a residual derived limit, obtaining the cleaner isomorphism $R\phi_* \mathop{\mathcal{H}\!om}_{W\mathscr{O}_X}(\omega(D), W\Omega_X^N) \cong R\mathop{\mathrm{Hom}}_\omega(R\phi_* \omega(D), \check{\omega}[-N])$ between Witt divisorial cohomology and derived Hom over the Cartier-Dieudonné ring, valid for divisors with SNC support. It then proves a Ramanujam-type vanishing for Witt divisorial sheaves on smooth surfaces when $D$ is nef and big, establishing vanishing of $H^1(X, W\mathscr{O}_X(-D))_\mathbb{Q}$ and of $H^1(X, \mathop{\mathcal{H}\!om}_{W\mathscr{O}_X}(W\mathscr{O}_X(-D), W\Omega_X^2))$. Finally, it constructs a two-dimensional counterexample, following Langer and Cascini--Tanaka, showing that Kawamata-Viehweg-type vanishing for Witt divisorial sheaves can fail in dimension two, thus delineating the limits of Witt-vector vanishing methods in positive characteristic. These results clarify when Witt-vector techniques yield vanishing and highlight subtle torsion phenomena in $V$-torsion vs $p$-torsion behavior.

Abstract

First we refine the duality theory for Witt divisorial sheaves on smooth projective varieties over a perfect field of positive characteristic. Building on previous work [Lem22], we remove the residual derived limit to obtain a cleaner isomorphism. As an application, we prove a Ramanujam-type vanishing theorem for Witt divisorial sheaves of nef and big divisors on surfaces. Finally, we show that a surface constructed by Langer [Lan16] with a divisor constructed by Cascini and Tanaka [CT18] gives a counterexample to Kawamata-Viehweg-type vanishing of Witt divisorial sheaves in dimension two.

Vanishing and a counterexample for Witt divisorial sheaves

TL;DR

This work tackles vanishing phenomena for Witt divisorial sheaves on smooth projective varieties in positive characteristic. It refines the duality framework by removing a residual derived limit, obtaining the cleaner isomorphism between Witt divisorial cohomology and derived Hom over the Cartier-Dieudonné ring, valid for divisors with SNC support. It then proves a Ramanujam-type vanishing for Witt divisorial sheaves on smooth surfaces when is nef and big, establishing vanishing of and of . Finally, it constructs a two-dimensional counterexample, following Langer and Cascini--Tanaka, showing that Kawamata-Viehweg-type vanishing for Witt divisorial sheaves can fail in dimension two, thus delineating the limits of Witt-vector vanishing methods in positive characteristic. These results clarify when Witt-vector techniques yield vanishing and highlight subtle torsion phenomena in -torsion vs -torsion behavior.

Abstract

First we refine the duality theory for Witt divisorial sheaves on smooth projective varieties over a perfect field of positive characteristic. Building on previous work [Lem22], we remove the residual derived limit to obtain a cleaner isomorphism. As an application, we prove a Ramanujam-type vanishing theorem for Witt divisorial sheaves of nef and big divisors on surfaces. Finally, we show that a surface constructed by Langer [Lan16] with a divisor constructed by Cascini and Tanaka [CT18] gives a counterexample to Kawamata-Viehweg-type vanishing of Witt divisorial sheaves in dimension two.
Paper Structure (4 sections, 14 theorems, 72 equations)

This paper contains 4 sections, 14 theorems, 72 equations.

Table of Contents

  1. Notation
  2. Duality
  3. Vanishing on Surfaces
  4. Counter example

Key Result

Theorem 1

Let $k$ be a perfect field of positive characteristic $p$, and $X \xrightarrow \phi \text{Spec }k$ be an $N$--dimensional smooth projective variety. If $A$ is an ample $\mathbb Q$--divisor with simple normal crossing support on $X$, then

Theorems & Definitions (26)

  • Theorem 1: Tanaka*Corollary 4.16, Theorem 4.17, Theorem 5.3 Step 5
  • Theorem 2: cf. Theorem \ref{['thm']}
  • Corollary 3: cf. Corollary \ref{['cor']}
  • Theorem 4: cf. Theorem \ref{['thmKVV']}
  • Theorem 5: cf. Theorem \ref{['example']}
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • ...and 16 more