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

Abstract

The purpose of this paper is to set up a formalism inspired by non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations for arbitrary polarized schemes, characterizing in particular almost trivial test configurations. Second, for any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study non-Archimedean analogues of certain classical functionals in Kähler 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. Contents 56 9. Uniform K-stability and singularities of pairs 58 Appendix A. Asymptotic Riemann-Roch on a normal variety 66 Appendix B. The equivariant Riemann-Roch theorem for schemes 67 References 69

Relative K-stability for Kähler manifolds

“…In particular, the example of [1] is not relatively K-stable in our Käh-ler sense. This is the first time this example has been ruled out (it would also be ruled out by relative analogues of other stronger notions of K-stability [12,22,51], however it appears to be a very challenging problem to prove that the existence of an extremal metric implies these notions). Although our stronger Kähler notion of relative K-stability rules out the phenomenon described in [1], it may perhaps be too optimistic to conjecture that it implies the existence of an extremal metric; we discuss this further in Remark 2.…”

“…Using this, we will also be able to characterise the trivial Kähler test configurations, clarifying the definition of K-stability given in [24] (by the pathological examples of Li-Xu [34], it is a rather subtle problem to understand what it means for a test configuration to be trivial, even in the projective case). The minimum norm [22] is also called the "non-Archimedean J-functional" [12]. It follows that uniform K-stability with respect to the L 1 -norm (in the sense of [47]) is equivalent to uniform K-stability with respect to the minimum norm (in the sense of [12,22]) in this general Kähler setting, extending the corresponding projective result [12] (the advantage of the minimum norm being that it is closely related to analytic functionals and intersection theory).…”

“…The minimum norm [22] is also called the "non-Archimedean J-functional" [12]. It follows that uniform K-stability with respect to the L 1 -norm (in the sense of [47]) is equivalent to uniform K-stability with respect to the minimum norm (in the sense of [12,22]) in this general Kähler setting, extending the corresponding projective result [12] (the advantage of the minimum norm being that it is closely related to analytic functionals and intersection theory). The result relating the norm of a test configuration to the norm of the associated geodesic is due to Hisamoto in the projective case (without any smoothness assumption), who proved this by relating both quantities to an associated Duistermaat-Heckman measure [30]; we give a more direct proof which applies in the Kähler setting.…”

K-Stability of Fano Manifolds With Not Small Alpha invariants

“…In this article, we will focus on the conditions K-stability and K-semistability; K-stability is stronger than K-polystability and K-polystability is stronger than K-semistability. Odaka and Sano [OS12, Theorem 1.4] (see also its generalizations [Der14,BHJ15,FO16]) proved a variant of Tian's theorem; if an ndimensional Q-Fano variety X satisfies that α(X) > n/(n + 1) (resp. α(X) ≥ n/(n + 1)), then X is K-stable (resp.…”

K-semistable Fano manifolds with the smallest alpha invariant

“…Tian [Tia87] proved that for a Fano manifold X, if α(X) > dim X/(dim X + 1), then X admits Kähler-Einstein metrics. Odaka and Sano [OS12, Theorem 1.4] (see also its generalizations [Der16,BHJ15,FO16,Fuj16c]) proved a variant of Tian's theorem: if a Q-Fano variety X satisfies that α(X) > dim X/(dim X + 1) (resp. ≥ dim X/(dim X + 1)), then X is K-stable (resp.…”