Uniform K-stability and asymptotics of energy functionals in Kähler geometry

Abstract

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...

A valuative criterion for uniform K-stability of ℚ-Fano varieties

“…Moreover, if X is a complex Fano manifold, then uniform K-stability of X is equivalent to K-stability of X by [CDS15a,CDS15b,CDS15c,Tia15] and [BBJ15]. We remark that the holomorphic automorphism group of X is finite if X is uniformly K-stable; see [BHJ16, Corollary E] for details. We will define those stability notions in Section 2.1.…”

Tropical and non-Archimedean limits of degenerating families of volume forms

“…Specifically, the estimate for ν t (X t ), suitably averaged over t, is essentially equivalent to [CLT10,Theorem 1.2]. It also appears in [KS01, §3.1] and is exploited in [BHJ16].…”

On the strict convexity of the K-energy

“…A generalized slope formula for the K-energy. We conclude the paper by observing that a by-product of Lemma 7 is the following generalization of the slope formula for the Kenergy in [11] (which concerns the case when Φ t is defined by a bona fide metric on a test configuration) to the present singular setting: Proof. Since M(Φ t ) < ∞ Lemma 7 shows that u t = u 0 + tv ∈ L 1 (∂P ) for all t ≥ 0, where v := u 1 − u 0 .…”

“…In the terminology of [10,11,4] this formula shows that the slope of the Mabuchi functional along a finite energy geodesic is equal to the Non-Archimedean Mabuchi functional of the corresponding (singular) Non-Archimedean metric. It would be very interesting to extend this slope formula to the non-toric setting.…”