In this 90-min talk, I will first give a simple introduction to the rough idea of K-stability theory, and explain how to compute delta invariant, the key invariant in studying the K-stability of ℚ-Fano varieties, using compatible divisors. Then I will apply this technique to reprove the Blum-Jonsson’s miraculously simple formula for the delta invariant of toric varieties.
After a short break. I will explain what spherical varieties are and introduce a similar formula of delta invariant for spherical varieties. In short, there is a unique anticanonical ℚ-divisor computing the equivariant stability threshold, and thus this ℚ-divisor characterizes the K-stability of a Q-Fano spherical variety.