Some Failures of Vanishing Theorems

The minimal model program for 3-folds has been developed only in characteristics p greater than or equal to 5. A key difficulty at small primes is that the singularities occurring in the minimal model program need not be Cohen-Macaulay, as they are in characteristic zero. This is related to the fact that Kodaira vanishing fails even on Fano varieties. We present the latest work in this direction.