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 one. Using a recent result of Datar and Szekelyhidi, 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 non-trivial Kähler-Ricci soliton