During the past five years, minimal model program for 3-folds in char p >5 has been established by Hacon, Xu and Birkar and etal. Along this direction, the main remaining problem is abundance, which has been proved for 3-folds with q(X) >0 last year in char p >5. We attempt to prove abundance for 3-folds with q=0, and recently we can prove non-vanishing, which is an important step.
In this talk, we will explain the strategy to prove nonvanishing for 3-folds in char p>5. Analogous result in char 0 was proved by Miyaoka and Kawamta 30 years ago. In char p, to carry out the approach in char 0, we meet some strange stimulating phenomena, which cause some difficulties. This is joint work with Chenyang Xu. We will also discuss several related questions.
