Valuative Invariants with Higher Moments

In this article we introduce a family of valuative invariants defined in terms the p-th moment expected vanishing order. These lie between $$\alpha $$ and $$\delta -invariants. They vary continuously big cone semi-continuously families. Most importantly, they give sufficient conditions for K-stability Fano varieties, which generalizes -criterions literature. are also related to $$d_p$$ -geometry maximal geodesic rays.