Raynaud-Mukai construction and Calabi-Yau Threefolds in Positive Characteristic

TL;DR

This paper investigates the construction of Calabi-Yau threefolds in positive characteristic that serve as counter-examples to Kodaira vanishing, focusing on the Raynaud-Mukai method and its modifications.

Contribution

It analyzes and extends Mukai's construction method to produce Calabi-Yau threefolds that violate Kodaira vanishing in positive characteristic.

Findings

Identification of conditions for counter-examples to Kodaira vanishing.

Extension of Mukai's method for constructing Calabi-Yau threefolds.

Computation of cohomology groups related to these threefolds.

Abstract

In this article, we study the possibility of producing a Calabi-Yau threefold in positive characteristic which is a counter-example to Kodaira vanishing. The only known method to construct the counter-example is so called inductive method such as the Raynaud-Mukai construction or Russel construction. We consider Mukai's method and its modification. Finally, as an application of Shepherd-Barron vanishing theorem of Fano threefolds, we compute $H^1(X, H^{-1})$ for any ample line bundle $H$ on a Calabi-Yau threefold $X$ on which Kodaira vanishing fails.