On positivity and semistability of vector bundles in finite and mixed characteristics (Q2871003)

The author surveys ``results concerning behavior of positivity of line bundles and possible vanishing theorems in positive characteristic.''NEWLINENEWLINEFrom the Introduction: ``The motivating problems are the following:NEWLINENEWLINEWhat can we say about relation between nefness, semiampleness, effectivity and pseudoeffectivity for line bundles on varieties defined over finite field?NEWLINENEWLINEWhat vanish theorems can hold for suitably positive line bundles in positive characteristic \dots?NEWLINENEWLINEIs there any relation between nefness in characteristic zero and in positive characteristic?NEWLINENEWLINEWhat can we say about variation in families of positivity of line bundles and semistability of vector bundles?NEWLINENEWLINEThe known results do not answer any of these questions. In this paper we pose and study some conjectures that try to answer all of the above questions.''