K-stability for Fano manifolds with torus action of complexity 1

Abstract

We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension $1$. Using a recent result of Datar and Székelyhidi, we effectively determine the existence of Kähler–Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kähler–Einstein Fano threefolds and Fano threefolds admitting a nontrivial Kähler–Ricci soliton.