K-Stability for Fano Manifolds with Torus Action of Complexity One

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 Kahler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kahler-Einstein Fano threefolds, and Fano threefolds admitting a non-trivial Kahler-Ricci soliton.

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