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
Consider a polarized complex manifold (X, L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X, L). For many common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper [BHJ15]) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability, as defined in [Der15,BHJ15]. As a partial converse, we show that uniform Kstability implies coercivity of the Mabuchi functional when restricted to Bergman metrics. SÉBASTIEN BOUCKSOM, TOMOYUKI HISAMOTO, AND MATTIAS JONSSONSzékelyhidi [Szé06] proposed to use a version of K-stability in which, for any test configuration (X , L) for (X, L), the Donaldson-Futaki invariant DF(X , L) is bounded below by a positive constant times a suitable norm of (X , L). (See also [Szé15] for a related notion.)Following this lead, we defined in the prequel [BHJ15] to this paper, (X, L) to be uniformly K-stable if there exists δ > 0 such thatfor any normal and ample test configuration (X , L). Here J NA (X , L) is a non-Archimedean analogue of Aubin's J-functional. It is equivalent to the L 1 -norm of (X , L) as well as the minimum norm considered by Dervan [Der15]. The norm is zero iff the normalization of (X , L) is trivial, so uniform K-stability implies K-stability.In [BHJ15] we advocated the point of view that a test configuration defines a non-Archimedean metric on L, that is, a metric on the Berkovich analytification of (X, L) with respect to the trivial norm on the ground field C. Further, we defined non-Archimedean analogues of many classical functionals in Kähler geometry. One example is the functional J NA above. Another is M NA , a non-Archimedean analogue of the Mabuchi K-energy functional M . It agrees with the Donaldson-Futaki invariant, up to an explicit error term, and uniform K-stability is equivalent tofor any ample test configuration (X , L). In [BHJ15] we proved that canonically polarized manifolds and polarized Calabi-Yau manifolds are always uniformly K-stable.A first goal of this paper is to exhibit precise relations between the non-Archimedean functionals and their classical counterparts. From now on we do not a priori assume that the reduced automorphism group of (X, L) is discrete. We prove Theorem A. Let (X , L) be an ample test configuration for a polarized complex manifold (X, L). Consider any smooth strictly positive S 1 -invariant metric Φ on L defined near the central fiber, and let (φ s ) s be the corresponding ray of smooth positive metrics on L. Denoting by M and J the Mabuchi K-energy functional and Aubin J-functional, respectively, we then haveThe corresponding equalities also hold for several other functionals, see Theorem 3.6. More generally, we prove that these asymptotic properties hold in the logarithmic setting, for subklt pairs (X, B) and with weaker positivity assum...
We prove that a toric polarized manifold is uniformly K-stable in the toric sense if and only if the K-energy functional is coercive modulo the torus action. Our key ingredient describes slope of the reduced J-functional along a torus-equivariant test configuration. It further provides a formulation for general polarized manifolds. We explain how in this situation the sub-torus, actually the center, controls the whole automorphism group. ContentsIntroduction 1 1. Slope formula for the reduced J-functional 3 2. Coercivity for the toric case 10 3. Remarks about stability and coercivity in the general setting 14 References 17
We apply the result of [His12] to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by [WN10] so that the sequence of spectral measures for the induced C * -action on the central fiber converges to the canonical Duistermatt-Heckman measure defined by the associated weak geodesic ray. As a consequence, we show that the algebraic p-norm of the test configuration equals to the L p -norm of tangent vectors. Using this result, We may give a natural energy theoretic explanation for the lower bound estimate on the Calabi functional by [Don05] and prove the analogous result for the Kähler-Einstein metric.2000 Mathematics Subject Classification. Primary 32A25, Secondary 32L10, 32W20.
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