This talk is about a close relationship between two seemingly different topics: complex geometry and commutative algebra in characteristic p. This relationship is facilitated by certain invariants of singularities, namely the log canonical threshold and the F-pure threshold. We will present an application of this idea to the study of Fano varieties by introducing a characteristic p analog of Tian's alpha-invariant. Tian introduced the alpha-invariant in 1987 to detect the existence of Kähler–Einstein metrics on Fano varieties.
This invariant has played a central role in the study of complex Fano varieties and their K-stability. We will discuss the many similarities, and some surprising differences between the Frobenius-alpha invariant and its complex counterpart.