An example of asymptotically Chow unstable manifolds with constant scalar curvature

By Hajime Ono, Yuji Sano, N. Yotsutani. 2009

58 citations recorded by the source corpus.

Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature K\"ahler metrics in $c_1(L)$ and its automorphism group Aut$(M,L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case when Aut$(V,L)$ is not discrete.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a5d68ba28d97a9e2873e7db33b997f511b71475a

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In Applied Mathematics

1. Robert J. Berman
2. Chi Li
3. Chenyang Xu
4. Song Sun
5. Kento Fujita
6. Mattias Jonsson
7. S. Boucksom
8. Y. Odaka

In Geometry and Topology

1. Robert J. Berman
2. Chi Li
3. Song Sun
4. Chenyang Xu
5. Mattias Jonsson
6. S. Boucksom
7. Kento Fujita
8. Y. Odaka

Citations in the corpus

• On the K-stability of Fano varieties and anticanonical divisors (2016, methodology): “. It is not true that (X,D) is always log canonical for any basis type anticanoncal Q-divisor on any K-semistable Fano variety X. Indeed, we checked by Macaulay2 that the Ono-Sano-Yotsutani’s example [OSY12] does not satisfy the condition. VIA ANTICANONICAL DIVISORS 7 3.2. Relation with the Berman-G”
• On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties (2014, background): “8, there exist smooth Fano varieties admitting Kähler–Einstein metrics which are not asymptotically Chow stable (see [40]).”
• K-stability of Fano varieties: an algebro-geometric approach (2020, background): “ptotic GIT stability also makes one daunted, since the latter is notoriously known to be hard to check. Later people found examples which are K-stable but not asymptotically GIT stable varieties (see [OSY12]), which made the picture even less clear. However, it is remarkable that in [Oda13a], concep”
• The GIT-stability of Polarised Varieties via discrepancy (2008, unknown): “counterexamples with discrete automorphism groups in [Od10]. We should note here that, smooth examples with not discrete automorphisms, which are K-polystable but asymptotically unstable, was made in [OSY09] and [DVZ10]. Claim 2.6 (ii) is an easy corollary of Stoppa’s result [Stp09, Proposition2.13]”
• Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, background): “tomorphism group can 41 not be finite; and it is known that the existence of cscK metric(or even KE metric) does not necessarily imply asymptotic Chow poly-stability, see the recent counter-example in [OSY], [DZ]. It seems to the authors that more delicate work is required to proceed by the quantizat”
• A generalization of the Ross--Thomas slope theory (2009, background): “rete automorphism groups and only quotient singularities in our forthcoming paper [Od]. We should note here that, a smooth toric example which is K-polystable but asymptotically unstable, was made in [OSY09]. We end this section by proving the equivalence of two asymptotic stability notions, followi”
• The Calabi conjecture and K-stability (2010, background): “yimplies asymptotic (poly)stability”. We should note that the first counterexample (with not discrete automorphism 4 YUJIODAKA groups) had been found as a smooth toric Fano 7-fold by Ono-SanoYotsutani [OSY09]. Recently, another example was found by Della Vedova and Zucca [DVZ10] which is a smooth rat”
• Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, background): “n, which is a necessary condition for the existence of balanced metrics Hk for k large (and the condition is not implied by the analytic K-polystability assumption even in the case when X is toric, by the example in [29]).”