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

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.