On K-stability of Calabi-Yau fibrations

Abstract

We study K-stability of algebraic fiber spaces. In this paper, we prove that Calabi-Yau fibrations over curves are uniformly K-stable in an adiabatic sense if and only if the base curves are log-twisted K-stable, where log-twisted K-stability is a variant of K-stability. Moreover, we prove that there are cscK metrics on rational elliptic surfaces that have only reduced fibers. Contents 14 4.2. Klt-trivial fibrations and log-twisted stability 20 5. Applications to Rational Elliptic Surfaces 27 5.1. Log-twisted K-stability of the base curve 27 5.2. GIT-stability in the sense of Miranda I: Weierstrass fibrations 29 5.3. GIT-stability in the sense of Miranda II: Cubic pencils 32 5.4. Applications to Miranda's conjecture on Chow stability 35 Appendix A. Some results on log-twisted Fano pairs 38 A.1. Notation 38 A.2. Ding stability of log-twisted Fano pairs and δ-invariant 40 A.3. Log-twisted K-semistable Fano pair and destabilizing boundary 44 References 49

Special representatives of complexified Kähler classes

“…Let us choose [ω Z ] sufficiently close to the adiabatic limit. It seems reasonable that some analogue of the results of [29] for rational elliptic surfaces holds in this case, so that if the singularities of f : Z → P 1 are sufficiently generic then it is K-stable with respect to [ω Z ].…”

“…Note also that geometric characterisations of (uniform, log) K-stability for Calabi-Yau fibrations are known, at least in the adiabatic limit when the volume of the fibres, with respect to [ω Z ], is sufficiently small [29]; these would apply to suitable polarisations on (Z, f ).…”

Minimization of Arakelov K-energy for many cases

“…Recently the first author [Hat22b] introduces the notion of "special K-stability" which, nevertheless of its name, include many cases such as K-trivial case, K-ample case, Kstable Fano varieties case, minimal models and some fibrations for instance. The notion is defined by using J-stability (see [Hat21]) and the δ-invariant [FO18,BlJ20] but the first author showed the special K-stability implies the usual K-stability [Hat22a].…”

Minimizing CM degree and specially K-stable varieties

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We prove that the degree of the CM line bundle for a normal family over a curve with fixed general fibers is strictly minimized if the special fiber is either• a smooth projective manifold with a unique cscK metric or• "specially K-stable", which is a new class we introduce in this paper. This phenomenon, as conjectured by Odaka (cf., [Oda20]), is a quantitative strengthening of the separatedness conjecture of moduli spaces of polarized K-stable varieties.The above mentioned special K-stability implies the original K-stability and a lot of cases satisfy it e.g., K-stable log Fano, klt Calabi-Yau (i.e., K X ≡ 0), lc varieties with the ample canonical divisor and uniformly adiabatically K-stable klt-trivial fibrations over curves (cf.,[Hat22]).

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“…The above mentioned special K-stability implies the original K-stability and a lot of cases satisfy it e.g., K-stable log Fano, klt Calabi-Yau (i.e., K X ≡ 0), lc varieties with the ample canonical divisor and uniformly adiabatically K-stable klt-trivial fibrations over curves (cf.,[Hat22]).…”