[Paper Review] Vanishing theorems of Kodaira type for Witt canonical sheaves

Hiromu Tanaka|arXiv (Cornell University)|Jul 13, 2017

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper establishes positive-characteristic analogues of Kodaira and Kawamata–Viehweg vanishing theorems for Witt canonical sheaves using the de Rham–Witt complex. It proves that higher cohomology groups vanish for tensor products of the Witt canonical sheaf with Teichmüller lifts of ample line bundles after tensoring with ℚ, resolving a key failure of classical Kodaira vanishing in positive characteristic.

ABSTRACT

Given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmuller lift of an ample invertible sheaf. We also give a generalisation of this vanishing theorem to one of Kawamata-Viehweg type.

Motivation & Objective

• To establish a positive-characteristic analogue of the Kodaira vanishing theorem using the de Rham–Witt complex.
• To address the failure of classical Kodaira vanishing in positive characteristic by introducing Witt canonical sheaves and Teichmüller lifts.
• To generalize the result to a Kawamata–Viehweg-type vanishing theorem in positive characteristic.
• To clarify the necessity of tensoring with ℚ in cohomological vanishing statements by constructing counterexamples over ℤ.
• To investigate the infinite-dimensional nature of cohomology of Teichmüller lifts over Q.

Proposed method

• Utilizes the de Rham–Witt complex to define Witt canonical sheaves and their tensor products with Teichmüller lifts of ample invertible sheaves.
• Applies Frobenius twisting and Verschiebung maps to induce isomorphisms on cohomology after tensoring with ℚ.
• Employs the Mittag–Leffler condition on inverse systems of cohomology groups to ensure compatibility with limits.
• Uses Serre duality and the exact sequence involving Verschiebung to analyze torsion-freeness and injectivity in cohomology.
• Applies the $p$-adic valuation and $p$-torsion freeness to prove linear independence over $W(k)$ and hence over $\mathbb{Q}$.
• Constructs explicit counterexamples using the exact sequence $0 \to (F_X)_*\underline{L}^{-p} \xrightarrow{V} \underline{L}^{-1} \to L^{-1} \to 0$ to show non-vanishing over $\mathbb{Z}$.

Experimental results

Research questions

• RQ1 Does the higher cohomology of the tensor product of the Witt canonical sheaf and the Teichmüller lift of an ample line bundle vanish in positive characteristic?
• RQ2 Can the Kodaira vanishing theorem be generalized to a Kawamata–Viehweg-type vanishing theorem in positive characteristic using Witt canonical sheaves?
• RQ3 Why is tensoring with $\mathbb{Q}$ necessary in the vanishing theorems, and what happens to cohomology groups over $\mathbb{Z}$?
• RQ4 Are the cohomology groups of Teichmüller lifts of ample line bundles infinite-dimensional over $\mathbb{Q}$?
• RQ5 Can the vanishing results be extended to nef and big line bundles or for higher-degree forms?

Key findings

• For a smooth projective variety $X$ of dimension $N$ over a perfect field of characteristic $p>0$, $H^i(X, W\Omega_X^N \otimes_{W\mathcal{O}_X} \underline{A}) = 0$ for all $i > 0$, where $\underline{A}$ is the Teichmüller lift of an ample invertible sheaf $A$.
• The cohomology $H^j(X, \underline{A}^{-1}) \otimes_{\mathbb{Z}} \mathbb{Q} = 0$ for all $j < N$, demonstrating a positive-characteristic Kodaira vanishing analogue.
• The cohomology $H^0(X, \underline{A}) \otimes_{\mathbb{Z}} \mathbb{Q}$ is infinite-dimensional over $\mathbb{Q}$, showing that the Teichmüller lift does not preserve finite-dimensionality.
• For smooth $X$, $H^N(X, \underline{A}^{-1}) \otimes_{\mathbb{Z}} \mathbb{Q}$ is also infinite-dimensional over $\mathbb{Q}$, indicating non-trivial cohomological behavior.
• Over $\mathbb{Z}$, $H^1(X, \underline{A}^{-1})$ and $H^2(X, \underline{A}^{-1})$ can be non-zero, proving that $\otimes \mathbb{Q}$ is necessary in the vanishing theorems.
• The relative Kawamata–Viehweg vanishing holds after $\otimes \mathbb{Q}$: $R^i f_*(\mathcal{H}om(W\mathcal{O}_X(-A), W\Omega_X^N)) \otimes_{\mathbb{Z}} \mathbb{Q} = 0$ for $i > 0$, under suitable ampleness and normal crossing assumptions.