We show the existence and uniqueness of a one-parameter family of smooth complete $U(1)$-invariant gradient steady Ricci solitons on the total space of any complex line bundle over a Fano K-Einstein base with first Chern class proportional to that of the base. These solitons are non-K except on the total space of the canonical bundle.
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Maxwell Stolarski (2024) studied this question.
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