Basis divisors and balanced metrics

Abstract

Abstract Using log canonical thresholds and basis divisors Fujita–Odaka introduced purely algebro-geometric invariants <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>δ</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> {\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>δ</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> {\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>δ</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> {\delta_{m}} is the coercivity threshold of a quantized Ding functional on the m th 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℙ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {{\mathbb{P}}^{n}} . Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kähler–Ricci solitons (and the more general g -solitons of Berman–Witt Nyström), 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.