A relative Yau-Tian-Donaldson conjecture and stability thresholds

Generalizing Fujita-Odaka invariant, we define a function δ˜ on a set of generalized b-divisors over a smooth Fano variety. This allows us to provide a new characterization of uniform K-stability. A key role is played by a new Riemann-Zariski formalism for K-stability. For any generalized b-divisor D, we introduce a (uniform) D-log K-stability notion. We prove that the existence of a unique Kähler-Einstein metric with prescribed singularities implies this new K-stability notion when the prescribed singularities are given by the generalized b-divisor D. We connect the existence of a unique Kähler-Einstein metric with prescribed singularities to a uniform D-log Ding-stability notion which we introduce. We show that these conditions are satisfied exactly when δ˜(D)>1, extending to the D-log setting the δ-valuative criterion of Fujita-Odaka and Blum-Jonsson. Finally we prove the strong openness of the uniform D-log Ding stability as a consequence of the strong continuity of δ˜.