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We extend Nadel’s results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal subvarieties on toric del Pezzo surfaces which do not admit Kähler–Einstein metrics. We also show that one can define multiplier ideal sheaves for Kähler–Ricci solitons and extend the result of Nadel using the holomorphic invariant defined by Tian and Zhu.

Multiplier ideal sheaves and integral invariants on toric Fano manifolds