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Positivity in Kähler–Einstein theory

Published online by Cambridge University Press:  25 June 2015

GABRIELE DI CERBO  and
LUCA F. DI CERBO
GABRIELE DI CERBO
Affiliation:
Department of Mathematics, Princeton University, NJ 08544-1000, U.S.A. e-mail: gdi@math.princeton.edu
LUCA F. DI CERBO
Affiliation:
Department of Mathematics, Duke University, Durham NC 27708-0320, U.S.A. e-mail: luca@math.duke.edu
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Abstract

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 α∈ (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 α∈((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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 2 , September 2015 , pp. 321 - 338
Copyright
Copyright © Cambridge Philosophical Society 2015 

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