Uniform K-stability of 𝐺-varieties of complexity 1

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">k</m:mi> </m:math> k be an algebraically closed field of characteristic 0 and 𝐺 a connected, reductive, linear algebraic group of simply connected type over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">k</m:mi> </m:math> k . Let 𝑋 be a projective 𝐺-variety of complexity 1. We classify 𝐺-equivariant normal test configurations of 𝑋 with integral central fibre via the combinatorial data. We also give a formula of anti-canonical divisors on 𝑋. Based on this formula, when 𝑋 is ℚ-Fano, we give an expression of the Futaki invariant, and derive a criterion of uniform K-stability in terms of the combinatorial data.