Basis divisors and balanced metrics

Using log canonical thresholds and basis divisors Fujita-Odaka introduced purely algcbro-gcometric invariants delta(m) whose limit in m is now known to characterize uniform K-stability on a Fano variety. As shown by Blum-Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these delta(m)-invariants characterizes uniform Ding stability. A basic question since Fujita-Odaka's work has been to find an analytic interpretation of these invariants. We show that each delta(m) is the coercivity threshold of a quantized Ding functional on the mth Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for P-n. Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kahler-Ricci solitons (and the more general g-solitons of Berman-Witt Nystrom), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.