Vanishing Theorems on Toric Varieties in Positive Characteristic

TL;DR

This paper proves key vanishing theorems for toric varieties in positive characteristic by leveraging liftability properties, extending classical results to more general divisors beyond torus-invariant cases.

Contribution

It generalizes vanishing theorems for toric varieties in positive characteristic to include Weil divisors that are not necessarily torus invariant, using liftability techniques.

Findings

Proves Bott vanishing theorem in positive characteristic.

Shows degeneration of Hodge to de Rham spectral sequence.

Establishes Kawamata-Viehweg vanishing for log pairs.

Abstract

We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch-Thomsen-Lauritzen-Mehta, Mustata and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.