We find Fano threefolds X admitting Kähler-Ricci solitons (KRS) with non-trivial moduli, which are \({\mathbb {T}}\) -varieties of complexity two. More precisely, we show that the weighted K-stability of \((X,\xi _0)\:(\) where \(\xi _0\) is the soliton candidate \()\) is equivalent to certain GIT-stability. In particular, this provides the first examples of strictly weighted K-semistable Fano varieties. On the other hand, we generalize Koiso’s theorem to the log Fano setting. Indeed, we show that the K-stability of a log Fano pair \((V,\Delta _V)\) is equivalent to the weighted K-stability of a cone \((Y, \Delta _Y, \xi _0)\) over it. This also leads to new examples of KRS Fano varieties with non-trivial moduli and small automorphism groups. To achieve these, we establish the weighted Abban-Zhuang estimate generalizing the work of [2], which gives a lower bound of the weighted stability threshold \(\delta ^g_{{\mathbb {T}}}(X,\Delta )\) . This is an effective way to check the weighted K-semistablity of a log Fano triple \((X,\Delta,\xi _0)\) . This estimate is also useful in testing (weighted) K-polystability based on the work of [9].
