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

Understand

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.