Basis divisors and balanced metrics

Abstract

Using log canonical thresholds and basis divisors Fujita--Odaka introduced purely algebro-geometric 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 $m$-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 ${\mathbb{P}}^n$. 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.

arXiv commentfinal version, to appear in J. Reine Angew. Math

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Cited by in corpus (3)

• The existence of the Kähler-Ricci soliton degeneration
• A quantization proof of the uniform Yau-Tian-Donaldson conjecture
• Asymptotics of quantized barycenters of lattice polytopes with applications to algebraic geometry