Positivity in Kähler-Einstein theory

Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle 2π(1-α) for αε (0, 1). In this paper we study how the existence of such Kähler-Einstein metrics depends on α. We show that in the negative scalar curvature case, if such Kähler-Einstein metrics exist for all small cone-angles then they exist for every αε((n+1)/(n+2), 1), where n is the dimension. We also give a characterisation of the pairs that admit negatively curved cone-edge Kähler-Einstein metrics with cone angle close to 2π. Again if these metrics exist for all cone-angles close to 2π, then they exist in a uniform interval of angles depending on the dimension only. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed.