Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs

By S. Boucksom, Tomoyuki Hisamoto, Mattias Jonsson. 2015

273 citations recorded by the source corpus.

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in K\"ahler geometry. These functionals are defined on the space of test configurations, and the Donaldson-Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and singularities of pairs, reproving and strengthening Y. Odaka's results in our formalism. This provides various examples of uniformly K-stable varieties.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a26a6d59bcb08d492e0da0c6ab247b506c0fc449

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A non-Archimedean approach to K-stability (2018, methodology): “When ϕ ∈ H ( L ), E T ( ϕ ) can be computed using intersection numbers, as in [BHJ17, § 7.4].”
The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties (2019, methodology): “In this case, the notion of uniform K-stability as defined in [13, 28] is the algebraic correspondent to the properness of Mabuchi energy and is a priori a strengthening of the K-stability condition.”
On positivity of the CM line bundle on K-moduli spaces (2019, methodology): “j). The result now follows by induction hypothesis after removing the component Di from both ∆ and D. 2.3. Filtrations. Using ﬁltrations to study K-stability questions was systematically developed in [BHJ17] and further extended in many later works, see e.g. [BJ17,BL18,BX19]. We recall some basic de”
On the K-stability of Fano varieties and anticanonical divisors (2016, methodology): “, thus in n-dimensional geometry, while the proof of [OS12] was via analysis of test conﬁgurations, thus essentially depends on (n+1)-dimensional geometry. Theorem 0.6 (cf., [Tia87], [OS12], [Der14], [BHJ15]). For an ndimensional Q-Fano variety X, if α(X) >(resp., ≥)n/(n+ 1), then (X,−KX) is unif”