TY - JOUR

T1 - Positivity in Kähler-Einstein theory

AU - Di Cerbo, Gabriele

AU - Di Cerbo, Luca F.

N1 - Publisher Copyright:
© Cambridge Philosophical Society 2015.

PY - 2015/9/4

Y1 - 2015/9/4

N2 - 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.

AB - 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.

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U2 - 10.1017/S0305004115000377

DO - 10.1017/S0305004115000377

M3 - Article

AN - SCOPUS:84938743075

SN - 0305-0041

VL - 159

SP - 321

EP - 338

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 2

ER -